:: SCPQSORT semantic presentation

begin


registration
let I, J be shiftable Program of SCMPDS;
let a be Int_position;
let k1 be Integer;
cluster if>0 (a,k1,I,J) ->  shiftable ;
correctness
coherence
if>0 (a,k1,I,J) is shiftable ;
proof
set i = (a,k1) <=0_goto ((card I) + 2);
set G = Goto ((card J) + 1);
 (I ';' (Goto ((card J) + 1))) ';' J = (I ';' (Load (goto ((card J) + 1)))) ';' J by SCMPDS_6:def_1;
then reconsider IJ = (I ';' (Goto ((card J) + 1))) ';' J as shiftable Program of SCMPDS ;
if>0 (a,k1,I,J) = ((((a,k1) <=0_goto ((card I) + 2)) ';' I) ';' (Goto ((card J) + 1))) ';' J by SCMPDS_6:def_5
.= (((a,k1) <=0_goto ((card I) + 2)) ';' (I ';' (Goto ((card J) + 1)))) ';' J by SCMPDS_4:14
.= ((a,k1) <=0_goto ((card I) + 2)) ';' IJ by SCMPDS_4:14
.= (Load ((a,k1) <=0_goto ((card I) + 2))) ';' IJ by SCMPDS_4:def_2 ;
hence  if>0 (a,k1,I,J) is shiftable ; ::_thesis: verum
end;
end;


theorem Th1: :: SCPQSORT:1
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for J being shiftable Program of SCMPDS
for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let J be shiftable Program of SCMPDS; ::_thesis: for a, b being Int_position
for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let a, b be Int_position; ::_thesis: for k1 being Integer st s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P holds
(IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
let k1 be Integer; ::_thesis: ( s . (DataLoc ((s . a),k1)) > 0 & I is_closed_on s,P & I is_halting_on s,P implies (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b )
assume that
A1: s . (DataLoc ((s . a),k1)) > 0 and
A2: I is_closed_on s,P and
A3: I is_halting_on s,P ; ::_thesis: (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b
A4: not b in dom (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by SCMPDS_4:18;
 IExec ((if>0 (a,k1,I,J)),P,s) = (IExec (I,P,s)) +* (Start-At ((((card I) + (card J)) + 2),SCMPDS)) by A1, A2, A3, SCMPDS_6:70;
hence  (IExec ((if>0 (a,k1,I,J)),P,s)) . b = (IExec (I,P,s)) . b by A4, FUNCT_4:11; ::_thesis: verum
end;

set A = NAT ;

set D = SCM-Data-Loc ;

Lm2:  for a being Int_position
for i being Integer
for I being Program of SCMPDS holds while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
proof
let a be Int_position; ::_thesis: for i being Integer
for I being Program of SCMPDS holds while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
let i be Integer; ::_thesis: for I being Program of SCMPDS holds while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
let I be Program of SCMPDS; ::_thesis: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1))))
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
thus  while>0 (a,i,I) = (((a,i) <=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1))) by SCMPDS_8:def_3
.= ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by SCMPDS_4:15 ; ::_thesis: verum
end;

Lm3:  for I being Program of SCMPDS
for a being Int_position
for i being Integer holds Shift (I,1) c= while>0 (a,i,I)
proof
let I be Program of SCMPDS; ::_thesis: for a being Int_position
for i being Integer holds Shift (I,1) c= while>0 (a,i,I)
let a be Int_position; ::_thesis: for i being Integer holds Shift (I,1) c= while>0 (a,i,I)
let i be Integer; ::_thesis: Shift (I,1) c= while>0 (a,i,I)
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
A1: while>0 (a,i,I) = (((a,i) <=0_goto ((card I) + 2)) ';' I) ';' (goto (- ((card I) + 1))) by SCMPDS_8:def_3
.= (((a,i) <=0_goto ((card I) + 2)) ';' I) ';' (Load (goto (- ((card I) + 1)))) by SCMPDS_4:def_3
.= ((Load ((a,i) <=0_goto ((card I) + 2))) ';' I) ';' (Load (goto (- ((card I) + 1)))) by SCMPDS_4:def_2 ;
 card (Load ((a,i) <=0_goto ((card I) + 2))) = 1 by COMPOS_1:54;
hence  Shift (I,1) c= while>0 (a,i,I) by A1, SCMPDS_7:3; ::_thesis: verum
end;

theorem Th2: :: SCPQSORT:2
for P being Instruction-Sequence of SCMPDS
for s, sm being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s, sm being State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let s, sm be State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let a be Int_position; ::_thesis: for i being Integer
for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let i be Integer; ::_thesis: for m being Element of NAT st I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) holds
( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
let m be Element of NAT ; ::_thesis: ( I is_closed_on s,P & I is_halting_on s,P & s . (DataLoc ((s . a),i)) > 0 & m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 & sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) implies ( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm ) )
set b = DataLoc ((s . a),i);
set WH = while>0 (a,i,I);
set iWH = stop (while>0 (a,i,I));
set IsI = stop I;
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
set s2 = Initialize s;
set P2 = P +* (stop I);
set s3 = Initialize s;
set P3 = P +* (stop (while>0 (a,i,I)));
set s4 = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1);
set P4 = P +* (stop (while>0 (a,i,I)));
assume that
A4: I is_closed_on s,P and
A5: I is_halting_on s,P and
A6: s . (DataLoc ((s . a),i)) > 0 and
A7: m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 2 and
A8: sm = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),m) ; ::_thesis: ( DataPart sm = DataPart (IExec (I,P,(Initialize s))) & Initialize sm = sm )
A9: stop I c= P +* (stop I) by FUNCT_4:25;
A10: stop (while>0 (a,i,I)) c= P +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
 while>0 (a,i,I) c= stop (while>0 (a,i,I)) by AFINSQ_1:74;
then A11: while>0 (a,i,I) c= P +* (stop (while>0 (a,i,I))) by A10, XBOOLE_1:1;
 Shift (I,1) c= while>0 (a,i,I) by Lm3;
then A12: Shift (I,1) c= P +* (stop (while>0 (a,i,I))) by A11, XBOOLE_1:1;
A13: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by Lm2;
A14: Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(0 + 1)) = Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),0))) by EXTPRO_1:3
.= Following ((P +* (stop (while>0 (a,i,I)))),(Initialize s))
.= Exec (((a,i) <=0_goto ((card I) + 2)),(Initialize s)) by A13, SCMPDS_6:11 ;
set m2 = LifeSpan ((P +* (stop I)),(Initialize s));
set s5 = Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))));
set P5 = P +* (stop (while>0 (a,i,I)));
set l1 = (card I) + 1;
A15: IC (Initialize s) = 0 by MEMSTR_0:def_11;
TT: not DataLoc ((s . a),i) in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
then A16: (Initialize s) . (DataLoc (((Initialize s) . a),i)) = (Initialize s) . (DataLoc ((s . a),i)) by FUNCT_4:11
.= s . (DataLoc ((s . a),i)) by TT, FUNCT_4:11 ;
set m3 = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1;
set s6 = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1));
set P6 = P +* (stop (while>0 (a,i,I)));
A17: Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)) = Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s)))) by EXTPRO_1:4;
set s7 = Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(((LifeSpan ((P +* (stop I)),(Initialize s))) + 1) + 1));
 (card I) + 1 < (card I) + 2 by XREAL_1:6;
then A18: (card I) + 1 in dom (while>0 (a,i,I)) by SCMPDS_8:18;
A20: I is_closed_on Initialize s,P +* (stop I) by A4, SCMPDS_6:24;
 for x being Int_position holds (Initialize s) . x = (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)) . x by A14, SCMPDS_2:56;
then A22: DataPart (Initialize s) = DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)) by SCMPDS_4:8;
A23: P +* (stop I) halts_on Initialize s by A5, SCMPDS_6:def_3;
 (P +* (stop I)) +* (stop I) halts_on Initialize (Initialize s) by A5, SCMPDS_6:def_3;
then A25: I is_halting_on Initialize s,P +* (stop I) by SCMPDS_6:def_3;
A26: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)) = (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)) . (IC )
.= succ 0 by A6, A15, A14, A16, SCMPDS_2:56
.= 0 + 1 ;
then  DataPart (Comput ((P +* (stop I)),(Initialize s),(LifeSpan ((P +* (stop I)),(Initialize s))))) = DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) by A9, A25, A20, A22, A12, SCMPDS_7:18;
then A27: DataPart (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) = DataPart (Result ((P +* (stop I)),(Initialize s))) by A23, EXTPRO_1:23
.= DataPart (Result ((P +* (stop I)),(Initialize s)))
.= DataPart (IExec (I,P,(Initialize s))) by SCMPDS_4:def_5 ;
A28: IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) = (card I) + 1 by A9, A25, A20, A26, A22, A12, SCMPDS_7:18;
then A29: CurInstr ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) = (P +* (stop (while>0 (a,i,I)))) . ((card I) + 1) by A17, PBOOLE:143
.= (P +* (stop (while>0 (a,i,I)))) . ((card I) + 1)
.= (P +* (stop (while>0 (a,i,I)))) . ((card I) + 1)
.= (while>0 (a,i,I)) . ((card I) + 1) by A18, A11, GRFUNC_1:2
.= goto (- ((card I) + 1)) by SCMPDS_8:19 ;
A30: Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(((LifeSpan ((P +* (stop I)),(Initialize s))) + 1) + 1)) = Following ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) by EXTPRO_1:3
.= Exec ((goto (- ((card I) + 1))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1)))) by A29 ;
now__::_thesis:_for_x_being_Int_position_holds_sm_._x_=_(Comput_((P_+*_(stop_(while>0_(a,i,I)))),(Comput_((P_+*_(stop_(while>0_(a,i,I)))),(Initialize_s),1)),(LifeSpan_((P_+*_(stop_I)),(Initialize_s)))))_._x
let x be Int_position; ::_thesis: sm . x = (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) . x
thus sm . x = (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) . x by A7, A8, A30, SCMPDS_2:54
.= (Comput ((P +* (stop (while>0 (a,i,I)))),(Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),1)),(LifeSpan ((P +* (stop I)),(Initialize s))))) . x by EXTPRO_1:4 ; ::_thesis: verum
end;
hence  DataPart sm = DataPart (IExec (I,P,(Initialize s))) by A27, SCMPDS_4:8; ::_thesis: Initialize sm = sm
IC (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(((LifeSpan ((P +* (stop I)),(Initialize s))) + 1) + 1))) = (Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),(((LifeSpan ((P +* (stop I)),(Initialize s))) + 1) + 1))) . (IC )
.= ICplusConst ((Comput ((P +* (stop (while>0 (a,i,I)))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))),(0 - ((card I) + 1))) by A30, SCMPDS_2:54
.= 0 by A28, A17, SCMPDS_7:1 ;
hence  Initialize sm = sm by A7, A8, MEMSTR_0:46; ::_thesis: verum
end;

theorem Th3: :: SCPQSORT:3
for P being Instruction-Sequence of SCMPDS
for s being State of SCMPDS
for I being Program of SCMPDS st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ) holds
I is_closed_on s,P
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being State of SCMPDS
for I being Program of SCMPDS st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ) holds
I is_closed_on s,P
let s be State of SCMPDS; ::_thesis: for I being Program of SCMPDS st ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ) holds
I is_closed_on s,P
let I be Program of SCMPDS; ::_thesis: ( ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ) implies I is_closed_on s,P )
assume A1: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st DataPart t = DataPart s holds
I is_halting_on t,Q ; ::_thesis: I is_closed_on s,P
set pI = stop I;
set sI = Initialize s;
set PI = P +* (stop I);
defpred S1[ Nat] means  not IC (Comput ((P +* (stop I)),(Initialize s),$1)) in dom (stop I);
A2: for a being Int_position holds s . a = (Initialize s) . a
proof
let a be Int_position; ::_thesis: s . a = (Initialize s) . a
 not a in dom (Start-At (0,SCMPDS)) by SCMPDS_4:18;
hence  s . a = (Initialize s) . a by FUNCT_4:11; ::_thesis: verum
end;
assume  not I is_closed_on s,P ; ::_thesis: contradiction
then  ex k being Element of NAT st S1[k] by SCMPDS_6:def_2;
then A4: ex k being Nat st S1[k] ;
consider n being Nat such that
A5: S1[n] and
A6: for m being Nat st S1[m] holds
n <= m from NAT_1:sch_5(A4);
reconsider n = n as Element of NAT by ORDINAL1:def_12;
set s2 = Comput ((P +* (stop I)),(Initialize s),n);
set P2 = P +* (stop I);
set Ig = ((IC (Comput ((P +* (stop I)),(Initialize s),n))),(succ (IC (Comput ((P +* (stop I)),(Initialize s),n))))) --> ((goto 1),(goto (- 1)));
reconsider P0 = (P +* (stop I)) +* (((IC (Comput ((P +* (stop I)),(Initialize s),n))),(succ (IC (Comput ((P +* (stop I)),(Initialize s),n))))) --> ((goto 1),(goto (- 1)))) as Instruction-Sequence of SCMPDS ;
reconsider P1 = (P +* (stop I)) +* (((IC (Comput ((P +* (stop I)),(Initialize s),n))),(succ (IC (Comput ((P +* (stop I)),(Initialize s),n))))) --> ((goto 1),(goto (- 1)))) as Instruction-Sequence of SCMPDS ;
reconsider P3 = (P +* (stop I)) +* ((IC (Comput ((P +* (stop I)),(Initialize s),n))),(goto 1)) as Instruction-Sequence of SCMPDS ;
reconsider P4 = P3 +* ((succ (IC (Comput ((P +* (stop I)),(Initialize s),n)))),(goto (- 1))) as Instruction-Sequence of SCMPDS ;
U: P0 = P4 by FUNCT_7:139;
A13: for m being Element of NAT st m < n holds
IC (Comput ((P +* (stop I)),(Initialize s),m)) in dom (stop I) by A6;
A17: stop I c= P +* (stop I) by FUNCT_4:25;
 stop I c= P3 by A5, FUNCT_4:25, FUNCT_7:89;
then A18: stop I c= P0 by U, A5, AFINSQ_1:73, FUNCT_7:89;
then XX: Comput (P0,(Initialize s),n) = Comput ((P +* (stop I)),(Initialize s),n) by A17, A13, SCMPDS_4:21;
DataPart (Initialize s) = DataPart (Initialize s)
.= DataPart s by A2, SCMPDS_4:8 ;
then  I is_halting_on Initialize s,P0 by A1;
then A20: P0 +* (stop I) halts_on Initialize (Initialize s) by SCMPDS_6:def_3;
A21: not P0 halts_on Comput ((P +* (stop I)),(Initialize s),n) by SCMPDS_4:20;
 P0 +* (stop I) = P0 by A18, FUNCT_4:98;
hence  contradiction by A21, XX, A20, EXTPRO_1:22; ::_thesis: verum
end;

theorem Th4: :: SCPQSORT:4
for i1, i2, i3, i4 being Instruction of SCMPDS holds card (((i1 ';' i2) ';' i3) ';' i4) = 4
proof
let x1, x2, x3, x4 be Instruction of SCMPDS; ::_thesis: card (((x1 ';' x2) ';' x3) ';' x4) = 4
thus  card (((x1 ';' x2) ';' x3) ';' x4) = (card ((x1 ';' x2) ';' x3)) + 1 by SCMP_GCD:4
.= ((card (x1 ';' x2)) + 1) + 1 by SCMP_GCD:4
.= (2 + 1) + 1 by SCMP_GCD:5
.= 4 ; ::_thesis: verum
end;

theorem Th5: :: SCPQSORT:5
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c + (t . (DataLoc ((s . a),i))) & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c + (t . (DataLoc ((s . a),i))) & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c + (t . (DataLoc ((s . a),i))) & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a, x, y being Int_position
for i, c being Integer st s . x >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c + (t . (DataLoc ((s . a),i))) & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let a, x1, y1 be Int_position; ::_thesis: for i, c being Integer st s . x1 >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x1 >= c + (t . (DataLoc ((s . a),i))) & t . y1 = s . y1 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y1 = t . y1 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let i, c be Integer; ::_thesis: ( s . x1 >= c + (s . (DataLoc ((s . a),i))) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x1 >= c + (t . (DataLoc ((s . a),i))) & t . y1 = s . y1 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y1 = t . y1 ) ) implies ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
set b = DataLoc ((s . a),i);
assume  s . x1 >= c + (s . (DataLoc ((s . a),i))) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x1 >= c + (t . (DataLoc ((s . a),i))) & t . y1 = s . y1 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y1 = t . y1 ) ) or ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
then A2: for x being Int_position st x in {x1} holds
s . x >= c + (s . (DataLoc ((s . a),i))) by TARSKI:def_1;
assume A3: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x1 >= c + (t . (DataLoc ((s . a),i))) & t . y1 = s . y1 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) & (IExec (I,Q,t)) . y1 = t . y1 ) ; ::_thesis: ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_(_for_x_being_Int_position_st_x_in_{x1}_holds_
t_._x_>=_c_+_(t_._(DataLoc_((s_._a),i)))_)_&_(_for_x_being_Int_position_st_x_in_{y1}_holds_
t_._x_=_s_._x_)_&_t_._a_=_s_._a_&_t_._(DataLoc_((s_._a),i))_>_0_holds_
(_(IExec_(I,Q,t))_._a_=_t_._a_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_(IExec_(I,Q,t))_._(DataLoc_((s_._a),i))_<_t_._(DataLoc_((s_._a),i))_&_(_for_x_being_Int_position_st_x_in_{x1}_holds_
(IExec_(I,Q,t))_._x_>=_c_+_((IExec_(I,Q,t))_._(DataLoc_((s_._a),i)))_)_&_(_for_x_being_Int_position_st_x_in_{y1}_holds_
(IExec_(I,Q,t))_._x_=_t_._x_)_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st ( for x being Int_position st x in {x1} holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & ( for x being Int_position st x in {x1} holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( ( for x being Int_position st x in {x1} holds
t . x >= c + (t . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
t . x = s . x ) & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 implies ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & ( for x being Int_position st x in {x1} holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x ) ) )
assume that
A4: for x being Int_position st x in {x1} holds
t . x >= c + (t . (DataLoc ((s . a),i))) and
A5: for x being Int_position st x in {y1} holds
t . x = s . x and
A6: t . a = s . a and
A7: t . (DataLoc ((s . a),i)) > 0 ; ::_thesis: ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & ( for x being Int_position st x in {x1} holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x ) )
 y1 in {y1} by TARSKI:def_1;
then A8: t . y1 = s . y1 by A5;
 x1 in {x1} by TARSKI:def_1;
then A9: t . x1 >= c + (t . (DataLoc ((s . a),i))) by A4;
hence  ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) ) by A3, A6, A7, A8; ::_thesis: ( ( for x being Int_position st x in {x1} holds
(IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) ) & ( for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x ) )
hereby  ::_thesis: for x being Int_position st x in {y1} holds
(IExec (I,Q,t)) . x = t . x
let x be Int_position; ::_thesis: ( x in {x1} implies (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) )
assume A10: x in {x1} ; ::_thesis: (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i)))
 (IExec (I,Q,t)) . x1 >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) by A3, A6, A7, A9, A8;
hence  (IExec (I,Q,t)) . x >= c + ((IExec (I,Q,t)) . (DataLoc ((s . a),i))) by A10, TARSKI:def_1; ::_thesis: verum
end;
let x be Int_position; ::_thesis: ( x in {y1} implies (IExec (I,Q,t)) . x = t . x )
assume A11: x in {y1} ; ::_thesis: (IExec (I,Q,t)) . x = t . x
hence (IExec (I,Q,t)) . x = (IExec (I,Q,t)) . y1 by TARSKI:def_1
.= t . y1 by A3, A6, A7, A9, A8
.= t . x by A11, TARSKI:def_1 ;
::_thesis: verum
end;
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) by A2, SCMPDS_8:27; ::_thesis: verum
end;

theorem Th6: :: SCPQSORT:6
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a, x, y being Int_position
for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a, x, y being Int_position
for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let a, x, y be Int_position; ::_thesis: for i, c being Integer st s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let i, c be Integer; ::_thesis: ( s . x >= c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) implies ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
set b = DataLoc ((s . a),i);
defpred S1[ set ] means  ex t being State of SCMPDS st
( t = $1 & t . x >= c & t . y = s . y );
consider f being Function of (product (the_Values_of SCMPDS)),NAT such that
A2: for s being State of SCMPDS holds
( ( s . (DataLoc ((s . a),i)) <= 0 implies f . s = 0 ) & ( s . (DataLoc ((s . a),i)) > 0 implies f . s = s . (DataLoc ((s . a),i)) ) ) by SCMPDS_8:5;
deffunc H1( State of SCMPDS) ->  Element of NAT = f . $1;
A5: for t being 0 -started State of SCMPDS st S1[t] & H1(t) = 0 holds
t . (DataLoc ((s . a),i)) <= 0 by A2;
assume A6: s . x >= c ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ) or ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
A7: S1[s] by A6;
assume A8: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x >= c & t . y = s . y & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x >= c & (IExec (I,Q,t)) . y = t . y ) ; ::_thesis: ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
A9: now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_S1[t]_&_t_._a_=_s_._a_&_t_._(DataLoc_((s_._a),i))_>_0_holds_
(_(IExec_(I,Q,t))_._a_=_t_._a_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_H1(_Initialize_(IExec_(I,Q,t)))_<_H1(t)_&_S1[_Initialize_(IExec_(I,Q,t))]_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st S1[t] & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( S1[t] & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 implies ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] ) )
assume that
A10: S1[t] and
A11: t . a = s . a and
A12: t . (DataLoc ((s . a),i)) > 0 ; ::_thesis: ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
set It = IExec (I,Q,t);
set t2 = Initialize (IExec (I,Q,t));
thus  ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q ) by A8, A11, A12, A10; ::_thesis: ( H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
thus  H1( Initialize (IExec (I,Q,t))) < H1(t) ::_thesis: S1[ Initialize (IExec (I,Q,t))]
proof
A15: H1(t) = t . (DataLoc ((s . a),i)) by A2, A12
.= t . (DataLoc ((s . a),i)) ;
assume A16: H1( Initialize (IExec (I,Q,t))) >= H1(t) ; ::_thesis: contradiction
then  (Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)) > 0 by A2, A12, A15;
then H1( Initialize (IExec (I,Q,t))) = (Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)) by A2
.= (IExec (I,Q,t)) . (DataLoc ((s . a),i)) by SCMPDS_5:15 ;
hence  contradiction by A8, A11, A12, A10, A16, A15; ::_thesis: verum
end;
thus  S1[ Initialize (IExec (I,Q,t))] ::_thesis: verum
proof
take v = Initialize (IExec (I,Q,t)); ::_thesis: ( v = Initialize (IExec (I,Q,t)) & v . x >= c & v . y = s . y )
thus  v = Initialize (IExec (I,Q,t)) ; ::_thesis: ( v . x >= c & v . y = s . y )
 (IExec (I,Q,t)) . x >= c by A8, A11, A12, A10;
hence  v . x >= c by SCMPDS_5:15; ::_thesis: v . y = s . y
 (IExec (I,Q,t)) . y = t . y by A8, A11, A12, A10;
hence  v . y = s . y by A10, SCMPDS_5:15; ::_thesis: verum
end;
end;
 ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) from SCMPDS_8:sch_3(A5, A7, A9);
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) ; ::_thesis: ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
assume A17: s . (DataLoc ((s . a),i)) > 0 ; ::_thesis: IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
 IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) from SCMPDS_8:sch_4(A17, A5, A7, A9);
hence  IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ; ::_thesis: verum
end;

theorem Th7: :: SCPQSORT:7
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a, x1, x2, x3, x4 being Int_position
for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let a, x1, x2, x3, x4 be Int_position; ::_thesis: for i, c, md being Integer st s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) holds
( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
let i, c, md be Integer; ::_thesis: ( s . x4 = ((s . x3) - c) + (s . x1) & md <= (s . x3) - c & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) implies ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
set b = DataLoc ((s . a),i);
defpred S1[ set ] means  ex t being State of SCMPDS st
( t = $1 & t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 );
assume that
A2: s . x4 = ((s . x3) - c) + (s . x1) and
A3: md <= (s . x3) - c ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS st
( t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 & not ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ) or ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) ) )
consider f being Function of (product (the_Values_of SCMPDS)),NAT such that
A4: for s being State of SCMPDS holds
( ( s . (DataLoc ((s . a),i)) <= 0 implies f . s = 0 ) & ( s . (DataLoc ((s . a),i)) > 0 implies f . s = s . (DataLoc ((s . a),i)) ) ) by SCMPDS_8:5;
deffunc H1( State of SCMPDS) ->  Element of NAT = f . $1;
A7: for t being 0 -started State of SCMPDS st S1[t] & H1(t) = 0 holds
t . (DataLoc ((s . a),i)) <= 0 by A4;
assume A8: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . x4 = ((t . x3) - c) + (t . x1) & md <= (t . x3) - c & t . x2 = s . x2 & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . (DataLoc ((s . a),i)) < t . (DataLoc ((s . a),i)) & (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) & md <= ((IExec (I,Q,t)) . x3) - c & (IExec (I,Q,t)) . x2 = t . x2 ) ; ::_thesis: ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P & ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ) )
A9: now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_S1[t]_&_t_._a_=_s_._a_&_t_._(DataLoc_((s_._a),i))_>_0_holds_
(_(IExec_(I,Q,t))_._a_=_t_._a_&_I_is_closed_on_t,Q_&_I_is_halting_on_t,Q_&_H1(_Initialize_(IExec_(I,Q,t)))_<_H1(t)_&_S1[_Initialize_(IExec_(I,Q,t))]_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st S1[t] & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 holds
( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( S1[t] & t . a = s . a & t . (DataLoc ((s . a),i)) > 0 implies ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] ) )
assume that
A10: S1[t] and
A11: t . a = s . a and
A12: t . (DataLoc ((s . a),i)) > 0 ; ::_thesis: ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q & H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
set It = IExec (I,Q,t);
set t2 = Initialize (IExec (I,Q,t));
consider v being State of SCMPDS such that
A13: v = t and
A14: v . x4 = ((v . x3) - c) + (v . x1) and
A15: md <= (v . x3) - c and
A16: v . x2 = s . x2 by A10;
A17: t . x2 = s . x2 by A13, A16;
A18: t . x4 = ((t . x3) - c) + (t . x1) by A13, A14;
A19: md <= (t . x3) - c by A13, A15;
thus  ( (IExec (I,Q,t)) . a = t . a & I is_closed_on t,Q & I is_halting_on t,Q ) by A8, A11, A12, A18, A16, A13, A15; ::_thesis: ( H1( Initialize (IExec (I,Q,t))) < H1(t) & S1[ Initialize (IExec (I,Q,t))] )
thus  H1( Initialize (IExec (I,Q,t))) < H1(t) ::_thesis: S1[ Initialize (IExec (I,Q,t))]
proof
A20: H1(t) = t . (DataLoc ((s . a),i)) by A4, A12
.= t . (DataLoc ((s . a),i)) ;
assume A21: H1( Initialize (IExec (I,Q,t))) >= H1(t) ; ::_thesis: contradiction
then  (Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)) > 0 by A4, A12, A20;
then H1( Initialize (IExec (I,Q,t))) = (Initialize (IExec (I,Q,t))) . (DataLoc ((s . a),i)) by A4
.= (IExec (I,Q,t)) . (DataLoc ((s . a),i)) by SCMPDS_5:15 ;
hence  contradiction by A8, A11, A12, A18, A19, A13, A16, A21, A20; ::_thesis: verum
end;
thus  S1[ Initialize (IExec (I,Q,t))] ::_thesis: verum
proof
take v = Initialize (IExec (I,Q,t)); ::_thesis: ( v = Initialize (IExec (I,Q,t)) & v . x4 = ((v . x3) - c) + (v . x1) & md <= (v . x3) - c & v . x2 = s . x2 )
thus  v = Initialize (IExec (I,Q,t)) ; ::_thesis: ( v . x4 = ((v . x3) - c) + (v . x1) & md <= (v . x3) - c & v . x2 = s . x2 )
 (IExec (I,Q,t)) . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) by A8, A11, A12, A18, A19, A17;
then  v . x4 = (((IExec (I,Q,t)) . x3) - c) + ((IExec (I,Q,t)) . x1) by SCMPDS_5:15;
then  v . x4 = ((v . x3) - c) + ((IExec (I,Q,t)) . x1) by SCMPDS_5:15;
hence  v . x4 = ((v . x3) - c) + (v . x1) by SCMPDS_5:15; ::_thesis: ( md <= (v . x3) - c & v . x2 = s . x2 )
 md <= ((IExec (I,Q,t)) . x3) - c by A8, A11, A12, A18, A19, A17;
hence  md <= (v . x3) - c by SCMPDS_5:15; ::_thesis: v . x2 = s . x2
 (IExec (I,Q,t)) . x2 = t . x2 by A8, A11, A12, A18, A19, A13, A16;
hence  v . x2 = s . x2 by A13, A16, SCMPDS_5:15; ::_thesis: verum
end;
end;
A22: S1[s] by A2, A3;
 ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) from SCMPDS_8:sch_3(A7, A22, A9);
hence  ( while>0 (a,i,I) is_closed_on s,P & while>0 (a,i,I) is_halting_on s,P ) ; ::_thesis: ( s . (DataLoc ((s . a),i)) > 0 implies IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) )
assume A23: s . (DataLoc ((s . a),i)) > 0 ; ::_thesis: IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s))))
 IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) from SCMPDS_8:sch_4(A23, A7, A22, A9);
hence  IExec ((while>0 (a,i,I)),P,s) = IExec ((while>0 (a,i,I)),P,(Initialize (IExec (I,P,s)))) ; ::_thesis: verum
end;

theorem Th8: :: SCPQSORT:8
for f being FinSequence of INT
for m, k1, k, n being Element of NAT st k1 = k - 1 & f is_non_decreasing_on m,k1 & f is_non_decreasing_on k + 1,n & ( for i being Element of NAT st m <= i & i < k holds
f . i <= f . k ) & ( for i being Element of NAT st k < i & i <= n holds
f . k <= f . i ) holds
f is_non_decreasing_on m,n
proof
let f be FinSequence of INT ; ::_thesis: for m, k1, k, n being Element of NAT st k1 = k - 1 & f is_non_decreasing_on m,k1 & f is_non_decreasing_on k + 1,n & ( for i being Element of NAT st m <= i & i < k holds
f . i <= f . k ) & ( for i being Element of NAT st k < i & i <= n holds
f . k <= f . i ) holds
f is_non_decreasing_on m,n
let m, k1, k, n be Element of NAT ; ::_thesis: ( k1 = k - 1 & f is_non_decreasing_on m,k1 & f is_non_decreasing_on k + 1,n & ( for i being Element of NAT st m <= i & i < k holds
f . i <= f . k ) & ( for i being Element of NAT st k < i & i <= n holds
f . k <= f . i ) implies f is_non_decreasing_on m,n )
assume A1: k1 = k - 1 ; ::_thesis: ( not f is_non_decreasing_on m,k1 or not f is_non_decreasing_on k + 1,n or ex i being Element of NAT st
( m <= i & i < k & not f . i <= f . k ) or ex i being Element of NAT st
( k < i & i <= n & not f . k <= f . i ) or f is_non_decreasing_on m,n )
assume A2: f is_non_decreasing_on m,k1 ; ::_thesis: ( not f is_non_decreasing_on k + 1,n or ex i being Element of NAT st
( m <= i & i < k & not f . i <= f . k ) or ex i being Element of NAT st
( k < i & i <= n & not f . k <= f . i ) or f is_non_decreasing_on m,n )
assume A3: f is_non_decreasing_on k + 1,n ; ::_thesis: ( ex i being Element of NAT st
( m <= i & i < k & not f . i <= f . k ) or ex i being Element of NAT st
( k < i & i <= n & not f . k <= f . i ) or f is_non_decreasing_on m,n )
assume A4: for i being Element of NAT st m <= i & i < k holds
f . i <= f . k ; ::_thesis: ( ex i being Element of NAT st
( k < i & i <= n & not f . k <= f . i ) or f is_non_decreasing_on m,n )
assume A5: for i being Element of NAT st k < i & i <= n holds
f . k <= f . i ; ::_thesis: f is_non_decreasing_on m,n
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_m_<=_i_&_i_<=_j_&_j_<=_n_holds_
f_._i_<=_f_._j
let i, j be Element of NAT ; ::_thesis: ( m <= i & i <= j & j <= n implies f . b1 <= f . b2 )
assume that
A6: m <= i and
A7: i <= j and
A8: j <= n ; ::_thesis: f . b1 <= f . b2
percases ( j < k or j = k or j > k ) by XXREAL_0:1;
suppose j < k ; ::_thesis: f . b1 <= f . b2
then  j + 1 <= k by INT_1:7;
then  j <= k1 by A1, XREAL_1:19;
hence  f . i <= f . j by A2, A6, A7, GRAPH_2:def_12; ::_thesis: verum
end;
supposeA9: j = k ; ::_thesis: f . b1 <= f . b2
percases ( i = j or i <> j ) ;
suppose i = j ; ::_thesis: f . b1 <= f . b2
hence  f . i <= f . j ; ::_thesis: verum
end;
suppose i <> j ; ::_thesis: f . b1 <= f . b2
then  i < j by A7, XXREAL_0:1;
hence  f . i <= f . j by A4, A6, A9; ::_thesis: verum
end;
end;
end;
supposeA10: j > k ; ::_thesis: f . b1 <= f . b2
percases ( i < k or i = k or i > k ) by XXREAL_0:1;
suppose i < k ; ::_thesis: f . b1 <= f . b2
then A11: f . k >= f . i by A4, A6;
 f . j >= f . k by A5, A8, A10;
hence  f . i <= f . j by A11, XXREAL_0:2; ::_thesis: verum
end;
supposeA12: i = k ; ::_thesis: f . b1 <= f . b2
percases ( i = j or i <> j ) ;
suppose i = j ; ::_thesis: f . b1 <= f . b2
hence  f . i <= f . j ; ::_thesis: verum
end;
suppose i <> j ; ::_thesis: f . b1 <= f . b2
then  i < j by A7, XXREAL_0:1;
hence  f . i <= f . j by A5, A8, A12; ::_thesis: verum
end;
end;
end;
suppose i > k ; ::_thesis: f . b1 <= f . b2
then  i >= k + 1 by INT_1:7;
hence  f . i <= f . j by A3, A7, A8, GRAPH_2:def_12; ::_thesis: verum
end;
end;
end;
end;
end;
hence  f is_non_decreasing_on m,n by GRAPH_2:def_12; ::_thesis: verum
end;

Lm4:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let a be Int_position; ::_thesis: for i, c being Integer
for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let i, c be Integer; ::_thesis: for f, g being FinSequence of INT
for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let f, g be FinSequence of INT ; ::_thesis: for m, n, m1 being Element of NAT st s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
let m, n, m1 be Element of NAT ; ::_thesis: ( s . a = c & len f = n & len g = n & f is_FinSequence_on s,m & g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) implies ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
I: Initialize s = s by MEMSTR_0:44;
set b = DataLoc (c,i);
assume that
A2: s . a = c and
A3: len f = n and
A4: len g = n ; ::_thesis: ( not f is_FinSequence_on s,m or not g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m or not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
set WH = while>0 (a,i,I);
set sWH = stop (while>0 (a,i,I));
assume A5: f is_FinSequence_on s,m ; ::_thesis: ( not g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m or not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= $1 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c );
assume A6: g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m ; ::_thesis: ( not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
assume A7: 1 = s . (DataLoc (c,i)) ; ::_thesis: ( not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
assume that
A8: m1 = (m + n) + 1 and
A9: m + 1 = s . (intpos m1) and
A10: m + n = s . (intpos (m1 + 1)) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
A11: m1 = ((m + n) + (2 * 0)) + 1 by A8;
assume A12: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ; ::_thesis: ( while>0 (a,i,I) is_halting_on s,P & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
A13: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= 0 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= 0 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let f1 be FinSequence of INT ; ::_thesis: for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= 0 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let k1, k2, y1, yn be Element of NAT ; ::_thesis: ( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= 0 & f1 is_FinSequence_on t,m & len f1 = n implies ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c ) )
T: Initialize t = t by MEMSTR_0:44;
assume that
A14: t . a = c and
A15: (2 * k1) + 1 = t . (DataLoc (c,i)) and
A16: k2 = ((m + n) + (2 * k1)) + 1 and
A17: ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) and
A18: m + y1 = t . (intpos k2) and
A19: m + yn = t . (intpos (k2 + 1)) and
A20: yn - y1 <= 0 and
A21: f1 is_FinSequence_on t,m and
A22: len f1 = n ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
A23: I is_halting_on t,Q by A12, A14, A15, A16, A17, A18, A19;
take k = (LifeSpan ((Q +* (stop I)),t)) + 2; ::_thesis: ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
set tk = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k);
A24: I is_closed_on t,Q by A12, A14, A15, A16, A17, A18, A19;
then A25: DataPart (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = DataPart (IExec (I,Q,t)) by A14, A15, A23, Th2, T;
consider f2 being FinSequence of INT  such that
A26: len f2 = n and
A27: for i being Element of NAT st 1 <= i & i <= len f2 holds
f2 . i = (IExec (I,Q,t)) . (intpos (m + i)) by SCPISORT:1;
take  f2 ; ::_thesis: ( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
thus  Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) by A14, A15, A24, A23, Th2, T; ::_thesis: ( f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f2_holds_
f2_._i_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,k))_._(intpos_(m_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f2 implies f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos (m + i)) )
assume that
A28: 1 <= i and
A29: i <= len f2 ; ::_thesis: f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos (m + i))
thus f2 . i = (IExec (I,Q,t)) . (intpos (m + i)) by A27, A28, A29
.= (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos (m + i)) by A25, SCMPDS_4:8 ; ::_thesis: verum
end;
hence  f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m by SCPISORT:def_1; ::_thesis: ( len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
thus  len f2 = n by A26; ::_thesis: ( f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
 f2 is_FinSequence_on IExec (I,Q,t),m by A27, SCPISORT:def_1;
hence  f1,f2 are_fiberwise_equipotent by A12, A14, A15, A16, A17, A18, A19, A21, A22, A26; ::_thesis: ( f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
thus  f2 is_non_decreasing_on y1,yn by A20, GRAPH_2:63, XREAL_1:50; ::_thesis: ( ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
thus  for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) by A20, XREAL_1:50; ::_thesis: ( ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
hereby  ::_thesis: ( ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let j be Element of NAT ; ::_thesis: ( y1 >= yn & 1 <= j & j <= n implies f2 . j = t . (intpos (m + j)) )
assume that
A30: y1 >= yn and
A31: 1 <= j and
A32: j <= n ; ::_thesis: f2 . j = t . (intpos (m + j))
thus f2 . j = (IExec (I,Q,t)) . (intpos (m + j)) by A26, A27, A31, A32
.= t . (intpos (m + j)) by A12, A14, A15, A16, A18, A19, A30, A31, A32 ; ::_thesis: verum
end;
hereby  ::_thesis: ( (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < (2 * k1) + 1 implies (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) )
assume that
A33: 1 <= j and
A34: j < (2 * k1) + 1 ; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j))
thus (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = (IExec (I,Q,t)) . (intpos ((m + n) + j)) by A25, SCMPDS_4:8
.= t . (intpos ((m + n) + j)) by A12, A14, A15, A16, A17, A18, A19, A33, A34 ; ::_thesis: verum
end;
 ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) by A12, A14, A15, A16, A18, A19;
hence  (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 by A15, A20, A25, SCMPDS_4:8, XREAL_1:50; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c
 (IExec (I,Q,t)) . a = t . a by A12, A14, A15, A16, A17, A18, A19;
hence  (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c by A14, A25, SCMPDS_4:8; ::_thesis: verum
end;
A35: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A36: S1[k] ; ::_thesis: S1[k + 1]
 S1[k + 1]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= k + 1 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for f1 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= k + 1 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let f1 be FinSequence of INT ; ::_thesis: for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= k + 1 & f1 is_FinSequence_on t,m & len f1 = n holds
ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
let k1, k2, y1, yn be Element of NAT ; ::_thesis: ( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & yn - y1 <= k + 1 & f1 is_FinSequence_on t,m & len f1 = n implies ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c ) )
T: Initialize t = t by MEMSTR_0:44;
assume that
A37: t . a = c and
A38: (2 * k1) + 1 = t . (DataLoc (c,i)) and
A39: k2 = ((m + n) + (2 * k1)) + 1 and
A40: ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) and
A41: m + y1 = t . (intpos k2) and
A42: m + yn = t . (intpos (k2 + 1)) and
A43: yn - y1 <= k + 1 and
A44: f1 is_FinSequence_on t,m and
A45: len f1 = n ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
percases ( yn - y1 <= 0 or yn - y1 > 0 ) ;
suppose yn - y1 <= 0 ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
hence  ex kk being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,kk)) . a = c ) by A13, A37, A38, A39, A40, A41, A42, A44, A45; ::_thesis: verum
end;
supposeA46: yn - y1 > 0 ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,k) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,k),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,k)) . a = c )
set m1 = (LifeSpan ((Q +* (stop I)),t)) + 2;
set t1 = Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2));
set Q1 = Q +* (stop (while>0 (a,i,I)));
A47: I is_halting_on t,Q by A12, A37, A38, A39, A40, A41, A42;
 y1 - 1 >= 0 by A40, A46, XREAL_1:47, XREAL_1:48;
then reconsider y0 = y1 - 1 as Element of NAT by INT_1:3;
set jj = (2 * k1) + 1;
A48: (yn - y1) - 1 <= (k + 1) - 1 by A43, XREAL_1:9;
A49: yn <= y1 + (k + 1) by A43, XREAL_1:20;
consider f2 being FinSequence of INT  such that
A50: len f2 = n and
A51: for i being Element of NAT st 1 <= i & i <= len f2 holds
f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by SCPISORT:1;
set It = IExec (I,Q,t);
A53: ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) by A12, A37, A38, A39, A40, A41, A42;
then consider ym being Element of NAT  such that
A54: y1 <= ym and
A55: ym <= yn and
A56: m + y1 = (IExec (I,Q,t)) . (intpos k2) and
A57: (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) and
A58: (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) and
A59: m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) and
A60: for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) and
A61: for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) by A46, XREAL_1:47;
A62: I is_closed_on t,Q by A12, A37, A38, A39, A40, A41, A42;
then A63: DataPart (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) = DataPart (IExec (I,Q,t)) by A37, A38, A47, Th2, T;
then A64: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (DataLoc (c,i)) = (2 * k1) + 3 by A46, A53, SCMPDS_4:8, XREAL_1:47;
A65: ym >= 1 by A40, A46, A54, XREAL_1:47, XXREAL_0:2;
then reconsider yc = ym - 1 as Element of NAT by INT_1:3, XREAL_1:48;
A66: yc <= yn by A55, XREAL_1:146, XXREAL_0:2;
then A67: yc <= n by A40, A46, XREAL_1:47, XXREAL_0:2;
A68: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (DataLoc (c,i)) = (2 * (k1 + 1)) + 1 by A46, A53, A63, SCMPDS_4:8, XREAL_1:47;
 (IExec (I,Q,t)) . a = t . a by A12, A37, A38, A39, A40, A41, A42;
then A69: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . a = c by A37, A63, SCMPDS_4:8;
set k3 = ((m + n) + (2 * (k1 + 1))) + 1;
set yd = ym + 1;
A70: ym + 1 > ym by XREAL_1:29;
then A71: ym + 1 >= y1 by A54, XXREAL_0:2;
then A72: ym + 1 >= 1 by A40, A46, XREAL_1:47, XXREAL_0:2;
A73: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((((m + n) + (2 * (k1 + 1))) + 1) + 1)) = m + yn by A39, A63, A59, SCMPDS_4:8;
 ym + (1 + k) >= y1 + (1 + k) by A54, XREAL_1:6;
then  yn <= (ym + 1) + k by A49, XXREAL_0:2;
then A74: yn - (ym + 1) <= k by XREAL_1:20;
AA: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (((m + n) + (2 * (k1 + 1))) + 1)) = m + (ym + 1) by A39, A63, A58, SCMPDS_4:8;
X1: (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . a = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . a by SCMPDS_5:15;
X2: (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (DataLoc (c,i)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (DataLoc (c,i)) by SCMPDS_5:15;
X3: (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (((m + n) + (2 * (k1 + 1))) + 1)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (((m + n) + (2 * (k1 + 1))) + 1)) by SCMPDS_5:15;
X4: (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos ((((m + n) + (2 * (k1 + 1))) + 1) + 1)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((((m + n) + (2 * (k1 + 1))) + 1) + 1)) by SCMPDS_5:15;
 f2 is_FinSequence_on Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))),m
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f2 or f2 . i = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + i)) )
assume  ( 1 <= i & i <= len f2 ) ; ::_thesis: f2 . i = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + i))
then  f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A51;
hence  f2 . i = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + i)) by SCMPDS_5:15; ::_thesis: verum
end;
then consider kl being Element of NAT , f3 being FinSequence of INT  such that
W1: Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) = Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl) and
W2: f3 is_FinSequence_on Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl),m and
W3: len f3 = n and
W4: f2,f3 are_fiberwise_equipotent and
W5: f3 is_non_decreasing_on ym + 1,yn and
W6: for j being Element of NAT st ym + 1 < yn & ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) holds
f3 . j = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + j)) and
W7: for j being Element of NAT st ym + 1 >= yn & 1 <= j & j <= n holds
f3 . j = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + j)) and
W8: for j being Element of NAT st 1 <= j & j < (2 * (k1 + 1)) + 1 holds
(Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos ((m + n) + j)) and
W9: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (DataLoc (c,i)) = ((Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (DataLoc (c,i))) - 2 and
W10: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . a = c by A36, A72, A74, A50, A40, A46, A69, X1, A68, X2, AA, X3, A73, X4, XREAL_1:47;
A75: Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) = Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl) by W1;
A76: f3 is_FinSequence_on Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl),m by W2;
A77: len f3 = n by W3;
A78: f2,f3 are_fiberwise_equipotent by W4;
A79: f3 is_non_decreasing_on ym + 1,yn by W5;
A80: for j being Element of NAT st ym + 1 < yn & ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) holds
f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
proof
let j be Element of NAT ; ::_thesis: ( ym + 1 < yn & ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) implies f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) )
assume  ( ym + 1 < yn & ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) ) ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
then  f3 . j = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + j)) by W6;
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A81: for j being Element of NAT st ym + 1 >= yn & 1 <= j & j <= n holds
f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
proof
let j be Element of NAT ; ::_thesis: ( ym + 1 >= yn & 1 <= j & j <= n implies f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) )
assume  ( ym + 1 >= yn & 1 <= j & j <= n ) ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
then  f3 . j = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos (m + j)) by W7;
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A82: for j being Element of NAT st 1 <= j & j < (2 * (k1 + 1)) + 1 holds
(Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j))
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < (2 * (k1 + 1)) + 1 implies (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j)) )
assume  ( 1 <= j & j < (2 * (k1 + 1)) + 1 ) ; ::_thesis: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j))
then  (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))) . (intpos ((m + n) + j)) by W8;
hence  (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j)) by SCMPDS_5:15; ::_thesis: verum
end;
set t2 = Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl);
set Q2 = (Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)));
A83: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (DataLoc (c,i)) = ((Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (DataLoc (c,i))) - 2 by W9, X2;
A84: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . a = c by W10;
A86: (2 * k1) + 3 = (2 * (k1 + 1)) + 1 ;
then  (2 * k1) + 1 < (2 * (k1 + 1)) + 1 by XREAL_1:6;
then A87: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos k2) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + ((2 * k1) + 1))) by A39, A82, NAT_1:11
.= m + y1 by A39, A63, A56, SCMPDS_4:8 ;
A88: ym <= n by A40, A46, A55, XREAL_1:47, XXREAL_0:2;
A89: now__::_thesis:_f3_._ym_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_ym))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . ym = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + ym))
hence  f3 . ym = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + ym)) by A70, A80, A65; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . ym = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + ym))
hence  f3 . ym = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + ym)) by A81, A65, A88; ::_thesis: verum
end;
end;
end;
A90: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_ym_holds_
f3_._i_=_f2_._i
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= ym implies f3 . i = f2 . i )
assume that
A91: 1 <= i and
A92: i <= ym ; ::_thesis: f3 . i = f2 . i
A93: i <= n by A88, A92, XXREAL_0:2;
A94: i < ym + 1 by A70, A92, XXREAL_0:2;
now__::_thesis:_f3_._i_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_i))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i))
hence  f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A80, A91, A94; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i))
hence  f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A81, A91, A93; ::_thesis: verum
end;
end;
end;
hence  f3 . i = f2 . i by A50, A51, A91, A93; ::_thesis: verum
end;
 yc <= yn - 1 by A55, XREAL_1:9;
then  yc - y1 <= (yn - 1) - y1 by XREAL_1:9;
then A95: yc - y1 <= k by A48, XXREAL_0:2;
A96: yc < ym + 1 by A70, XREAL_1:146, XXREAL_0:2;
set jj = (2 * k1) + 2;
 (2 * k1) + 2 >= 2 by NAT_1:11;
then A97: (2 * k1) + 2 >= 1 by XXREAL_0:2;
 (2 * k1) + 2 < (2 * (k1 + 1)) + 1 by A86, XREAL_1:6;
then AA: (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (k2 + 1)) = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + ((2 * k1) + 2))) by A39, A82, A97
.= m + yc by A39, A63, A57, SCMPDS_4:8 ;
YY: ( ( 1 <= y1 & yc <= n ) or y1 >= yc ) by A46, A67, A40, XREAL_1:47;
X1: (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . a = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . a by SCMPDS_5:15;
X2: (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (DataLoc (c,i)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (DataLoc (c,i)) by SCMPDS_5:15;
X3: (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos k2) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos k2) by SCMPDS_5:15;
X4: (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (k2 + 1)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (k2 + 1)) by SCMPDS_5:15;
 f3 is_FinSequence_on Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)),m
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f3 or f3 . i = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + i)) )
assume  ( 1 <= i & i <= len f3 ) ; ::_thesis: f3 . i = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + i))
then  f3 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A76, SCPISORT:def_1;
hence  f3 . i = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + i)) by SCMPDS_5:15; ::_thesis: verum
end;
then consider km being Element of NAT , f4 being FinSequence of INT  such that
W1: Initialize (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) = Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km) and
W2: f4 is_FinSequence_on Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km),m and
W3: len f4 = n and
W4: f3,f4 are_fiberwise_equipotent and
W5: f4 is_non_decreasing_on y1,yc and
W6: for j being Element of NAT st y1 < yc & ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) holds
f4 . j = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + j)) and
W7: for j being Element of NAT st y1 >= yc & 1 <= j & j <= n holds
f4 . j = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + j)) and
W8: for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos ((m + n) + j)) and
W9: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (DataLoc (c,i)) = ((Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (DataLoc (c,i))) - 2 and
W10: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . a = c by A36, A95, A77, YY, A39, X1, A84, X2, A64, A83, X3, A87, X4, AA;
A98: Initialize (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) = Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km) by W1;
A99: f4 is_FinSequence_on Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km),m by W2;
A100: len f4 = n by W3;
A101: f3,f4 are_fiberwise_equipotent by W4;
A102: f4 is_non_decreasing_on y1,yc by W5;
A103: for j being Element of NAT st y1 < yc & ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) holds
f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
proof
let j be Element of NAT ; ::_thesis: ( y1 < yc & ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) implies f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) )
assume  ( y1 < yc & ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) ) ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
then  f4 . j = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + j)) by W6;
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A104: for j being Element of NAT st y1 >= yc & 1 <= j & j <= n holds
f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
proof
let j be Element of NAT ; ::_thesis: ( y1 >= yc & 1 <= j & j <= n implies f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) )
assume  ( y1 >= yc & 1 <= j & j <= n ) ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
then  f4 . j = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos (m + j)) by W7;
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A105: for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j))
proof
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < (2 * k1) + 1 implies (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) )
assume  ( 1 <= j & j < (2 * k1) + 1 ) ; ::_thesis: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j))
then  (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))) . (intpos ((m + n) + j)) by W8;
hence  (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) by SCMPDS_5:15; ::_thesis: verum
end;
A106: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . (DataLoc (c,i)) = ((Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (DataLoc (c,i))) - 2 by W9, X2;
A107: (Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km)) . a = c by W10;
A108: now__::_thesis:_for_i_being_Element_of_NAT_st_yc_<_i_&_i_<=_len_f4_holds_
f4_._i_=_f3_._i
let i be Element of NAT ; ::_thesis: ( yc < i & i <= len f4 implies f4 . i = f3 . i )
assume that
A109: yc < i and
A110: i <= len f4 ; ::_thesis: f4 . i = f3 . i
A111: 1 + 0 <= i by A109, INT_1:7;
now__::_thesis:_f4_._i_=_(Comput_(((Q_+*_(stop_(while>0_(a,i,I))))_+*_(stop_(while>0_(a,i,I)))),(Initialize_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))),kl))_._(intpos_(m_+_i))
percases ( y1 < yc or y1 >= yc ) ;
suppose y1 < yc ; ::_thesis: f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i))
hence  f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A100, A103, A109, A110; ::_thesis: verum
end;
suppose y1 >= yc ; ::_thesis: f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i))
hence  f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A100, A104, A110, A111; ::_thesis: verum
end;
end;
end;
hence  f4 . i = f3 . i by A76, A77, A100, A110, A111, SCPISORT:def_1; ::_thesis: verum
end;
then  f4 . ym = f3 . ym by A100, A88, XREAL_1:146;
then A112: f4 . ym = (IExec (I,Q,t)) . (intpos (m + ym)) by A63, A89, SCMPDS_4:8;
A113: now__::_thesis:_for_i_being_Element_of_NAT_st_yn_<_i_&_i_<=_len_f3_holds_
f3_._i_=_f2_._i
let i be Element of NAT ; ::_thesis: ( yn < i & i <= len f3 implies f3 . i = f2 . i )
assume that
A114: yn < i and
A115: i <= len f3 ; ::_thesis: f3 . i = f2 . i
A116: 1 + 0 <= i by A114, INT_1:7;
now__::_thesis:_f3_._i_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_i))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i))
hence  f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A77, A80, A114, A115; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i))
hence  f3 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A77, A81, A115, A116; ::_thesis: verum
end;
end;
end;
hence  f3 . i = f2 . i by A50, A51, A77, A115, A116; ::_thesis: verum
end;
A117: now__::_thesis:_for_i_being_Element_of_NAT_st_ym_<_i_&_i_<=_yn_holds_
f4_._ym_<=_f4_._i
let i be Element of NAT ; ::_thesis: ( ym < i & i <= yn implies f4 . ym <= f4 . i )
assume that
A118: ym < i and
A119: i <= yn ; ::_thesis: f4 . ym <= f4 . i
consider j being Element of NAT  such that
A120: ym < j and
A121: j <= yn and
A122: f3 . i = f2 . j by A40, A46, A55, A77, A78, A90, A113, A118, A119, RFINSEQ:32, XREAL_1:47;
A123: yc < i by A118, XREAL_1:146, XXREAL_0:2;
A124: 1 <= j by A65, A120, XXREAL_0:2;
A125: j <= len f2 by A40, A46, A50, A121, XREAL_1:47, XXREAL_0:2;
 i <= len f4 by A40, A46, A100, A119, XREAL_1:47, XXREAL_0:2;
then f4 . i = f2 . j by A108, A123, A122
.= (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A51, A124, A125
.= (IExec (I,Q,t)) . (intpos (m + j)) by A63, SCMPDS_4:8 ;
hence  f4 . ym <= f4 . i by A61, A112, A120, A121; ::_thesis: verum
end;
A126: yn > y1 by A46, XREAL_1:47;
A127: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_y0_holds_
f4_._i_=_f3_._i
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= y0 implies f4 . i = f3 . i )
assume that
A128: 1 <= i and
A129: i <= y0 ; ::_thesis: f4 . i = f3 . i
 i - 1 < y1 - 1 by A129, XREAL_1:146, XXREAL_0:2;
then A130: i < y1 by XREAL_1:9;
 y1 <= n by A40, A126, XXREAL_0:2;
then A131: i <= n by A130, XXREAL_0:2;
now__::_thesis:_f4_._i_=_(Comput_(((Q_+*_(stop_(while>0_(a,i,I))))_+*_(stop_(while>0_(a,i,I)))),(Initialize_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))),kl))_._(intpos_(m_+_i))
percases ( y1 < yc or y1 >= yc ) ;
suppose y1 < yc ; ::_thesis: f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i))
hence  f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A103, A128, A130; ::_thesis: verum
end;
suppose y1 >= yc ; ::_thesis: f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i))
hence  f4 . i = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + i)) by A104, A128, A131; ::_thesis: verum
end;
end;
end;
hence  f4 . i = f3 . i by A76, A77, A128, A131, SCPISORT:def_1; ::_thesis: verum
end;
A132: y0 <= yc by A54, XREAL_1:9;
A133: now__::_thesis:_for_i_being_Element_of_NAT_st_y1_<=_i_&_i_<_ym_holds_
f4_._i_<=_f4_._ym
let i be Element of NAT ; ::_thesis: ( y1 <= i & i < ym implies f4 . i <= f4 . ym )
assume that
A134: y1 <= i and
A135: i < ym ; ::_thesis: f4 . i <= f4 . ym
 i + 1 <= ym by A135, INT_1:7;
then A136: i <= yc by XREAL_1:19;
 y0 < i by A134, XREAL_1:146, XXREAL_0:2;
then consider j being Element of NAT  such that
A137: y0 < j and
A138: j <= yc and
A139: f4 . i = f3 . j by A67, A100, A101, A132, A127, A108, A136, RFINSEQ:32;
A140: 1 + 0 <= j by A137, INT_1:7;
A141: j <= n by A67, A138, XXREAL_0:2;
A142: j < ym + 1 by A96, A138, XXREAL_0:2;
now__::_thesis:_f3_._j_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_j))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A80, A140, A142; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A81, A140, A141; ::_thesis: verum
end;
end;
end;
then A143: f4 . i = (IExec (I,Q,t)) . (intpos (m + j)) by A63, A139, SCMPDS_4:8;
A144: j < ym by A138, XREAL_1:146, XXREAL_0:2;
 (y1 - 1) + 1 <= j by A137, INT_1:7;
hence  f4 . i <= f4 . ym by A60, A112, A144, A143; ::_thesis: verum
end;
take mm = ((LifeSpan ((Q +* (stop I)),t)) + 2) + (kl + km); ::_thesis: ex f2 being FinSequence of INT st
( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm) & f2 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm),m & len f2 = n & f1,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f2 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
set tm = Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm);
take  f4 ; ::_thesis: ( Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm) & f4 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm),m & len f4 = n & f1,f4 are_fiberwise_equipotent & f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
 Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)) by A37, A38, A62, A47, Th2, T;
then A146: Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm) = Comput ((Q +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),(kl + km)) by EXTPRO_1:4
.= Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),(kl + km))
.= Comput ((((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl))),km) by A75, EXTPRO_1:4 ;
hence  Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) = Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm) by A98; ::_thesis: ( f4 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm),m & len f4 = n & f1,f4 are_fiberwise_equipotent & f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
thus  f4 is_FinSequence_on Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm),m by A99, A146; ::_thesis: ( len f4 = n & f1,f4 are_fiberwise_equipotent & f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
thus  len f4 = n by A100; ::_thesis: ( f1,f4 are_fiberwise_equipotent & f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f2_holds_
f2_._i_=_(IExec_(I,Q,t))_._(intpos_(m_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f2 implies f2 . i = (IExec (I,Q,t)) . (intpos (m + i)) )
assume that
A147: 1 <= i and
A148: i <= len f2 ; ::_thesis: f2 . i = (IExec (I,Q,t)) . (intpos (m + i))
thus f2 . i = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + i)) by A51, A147, A148
.= (IExec (I,Q,t)) . (intpos (m + i)) by A63, SCMPDS_4:8 ; ::_thesis: verum
end;
then  f2 is_FinSequence_on IExec (I,Q,t),m by SCPISORT:def_1;
then  f1,f2 are_fiberwise_equipotent by A12, A37, A38, A39, A40, A41, A42, A44, A45, A50;
then  f1,f3 are_fiberwise_equipotent by A78, CLASSES1:76;
hence  f1,f4 are_fiberwise_equipotent by A101, CLASSES1:76; ::_thesis: ( f4 is_non_decreasing_on y1,yn & ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
A149: now__::_thesis:_for_j_being_Element_of_NAT_st_ym_+_1_<=_j_&_j_<=_yn_holds_
f4_._j_=_f3_._j
let j be Element of NAT ; ::_thesis: ( ym + 1 <= j & j <= yn implies f4 . j = f3 . j )
assume that
A150: ym + 1 <= j and
A151: j <= yn ; ::_thesis: f4 . j = f3 . j
A152: 1 <= j by A72, A150, XXREAL_0:2;
A153: j <= n by A40, A46, A151, XREAL_1:47, XXREAL_0:2;
A154: yc < j by A96, A150, XXREAL_0:2;
now__::_thesis:_f4_._j_=_(Comput_(((Q_+*_(stop_(while>0_(a,i,I))))_+*_(stop_(while>0_(a,i,I)))),(Initialize_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))),kl))_._(intpos_(m_+_j))
percases ( y1 < yc or y1 >= yc ) ;
suppose y1 < yc ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by A103, A153, A154; ::_thesis: verum
end;
suppose y1 >= yc ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by A104, A153, A152; ::_thesis: verum
end;
end;
end;
hence  f4 . j = f3 . j by A76, A77, A153, A152, SCPISORT:def_1; ::_thesis: verum
end;
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_ym_+_1_<=_i_&_i_<=_j_&_j_<=_yn_holds_
f4_._i_<=_f4_._j
let i, j be Element of NAT ; ::_thesis: ( ym + 1 <= i & i <= j & j <= yn implies f4 . i <= f4 . j )
assume that
A155: ym + 1 <= i and
A156: i <= j and
A157: j <= yn ; ::_thesis: f4 . i <= f4 . j
 ym + 1 <= j by A155, A156, XXREAL_0:2;
then A158: f4 . j = f3 . j by A149, A157;
 i <= yn by A156, A157, XXREAL_0:2;
then  f4 . i = f3 . i by A149, A155;
hence  f4 . i <= f4 . j by A79, A155, A156, A157, A158, GRAPH_2:def_12; ::_thesis: verum
end;
then  f4 is_non_decreasing_on ym + 1,yn by GRAPH_2:def_12;
hence  f4 is_non_decreasing_on y1,yn by A102, A133, A117, Th8; ::_thesis: ( ( for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
thus  for j being Element of NAT st y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
f4 . j = t . (intpos (m + j)) ::_thesis: ( ( for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
proof
let j be Element of NAT ; ::_thesis: ( y1 < yn & ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) implies f4 . j = t . (intpos (m + j)) )
assume that
A159: y1 < yn and
A160: ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) ; ::_thesis: f4 . j = t . (intpos (m + j))
A161: ( 1 <= j & j <= n )
proof
percases ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) by A160;
supposeA162: ( 1 <= j & j < y1 ) ; ::_thesis: ( 1 <= j & j <= n )
then  j < yn by A159, XXREAL_0:2;
hence  ( 1 <= j & j <= n ) by A40, A46, A162, XREAL_1:47, XXREAL_0:2; ::_thesis: verum
end;
supposeA163: ( yn < j & j <= n ) ; ::_thesis: ( 1 <= j & j <= n )
then  y1 < j by A159, XXREAL_0:2;
hence  ( 1 <= j & j <= n ) by A40, A46, A163, XREAL_1:47, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A164: ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) )
proof
percases ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) by A160;
suppose ( 1 <= j & j < y1 ) ; ::_thesis: ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) )
hence  ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) by A71, XXREAL_0:2; ::_thesis: verum
end;
suppose ( yn < j & j <= n ) ; ::_thesis: ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) )
hence  ( ( 1 <= j & j < ym + 1 ) or ( yn < j & j <= n ) ) ; ::_thesis: verum
end;
end;
end;
A165: now__::_thesis:_f3_._j_=_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))_._(intpos_(m_+_j))
percases ( ym + 1 < yn or ym + 1 >= yn ) ;
suppose ym + 1 < yn ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A80, A164; ::_thesis: verum
end;
suppose ym + 1 >= yn ; ::_thesis: f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j))
hence  f3 . j = (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos (m + j)) by A81, A161; ::_thesis: verum
end;
end;
end;
A166: ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) )
proof
percases ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) by A160;
suppose ( 1 <= j & j < y1 ) ; ::_thesis: ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) )
hence  ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) ; ::_thesis: verum
end;
suppose ( yn < j & j <= n ) ; ::_thesis: ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) )
hence  ( ( 1 <= j & j < y1 ) or ( yc < j & j <= n ) ) by A66, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
now__::_thesis:_f4_._j_=_(Comput_(((Q_+*_(stop_(while>0_(a,i,I))))_+*_(stop_(while>0_(a,i,I)))),(Initialize_(Comput_((Q_+*_(stop_(while>0_(a,i,I)))),t,((LifeSpan_((Q_+*_(stop_I)),t))_+_2)))),kl))_._(intpos_(m_+_j))
percases ( y1 < yc or y1 >= yc ) ;
suppose y1 < yc ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by A103, A166; ::_thesis: verum
end;
suppose y1 >= yc ; ::_thesis: f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j))
hence  f4 . j = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos (m + j)) by A104, A161; ::_thesis: verum
end;
end;
end;
hence f4 . j = f3 . j by A76, A77, A161, SCPISORT:def_1
.= (IExec (I,Q,t)) . (intpos (m + j)) by A63, A165, SCMPDS_4:8
.= t . (intpos (m + j)) by A12, A37, A38, A39, A40, A41, A42, A159, A160 ;
::_thesis: verum
end;
thus  for j being Element of NAT st y1 >= yn & 1 <= j & j <= n holds
f4 . j = t . (intpos (m + j)) by A46, XREAL_1:47; ::_thesis: ( ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
hereby  ::_thesis: ( (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 & (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c )
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < (2 * k1) + 1 implies (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) )
assume that
A167: 1 <= j and
A168: j < (2 * k1) + 1 ; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j))
 (2 * k1) + 1 < (2 * (k1 + 1)) + 1 by A86, XREAL_1:6;
then A169: j < (2 * (k1 + 1)) + 1 by A168, XXREAL_0:2;
thus (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (intpos ((m + n) + j)) = (Comput (((Q +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2)))),kl)) . (intpos ((m + n) + j)) by A105, A146, A167, A168
.= (Comput ((Q +* (stop (while>0 (a,i,I)))),t,((LifeSpan ((Q +* (stop I)),t)) + 2))) . (intpos ((m + n) + j)) by A82, A167, A169
.= (IExec (I,Q,t)) . (intpos ((m + n) + j)) by A63, SCMPDS_4:8
.= t . (intpos ((m + n) + j)) by A12, A37, A38, A39, A40, A41, A42, A167, A168 ; ::_thesis: verum
end;
thus  (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . (DataLoc (c,i)) = (t . (DataLoc (c,i))) - 2 by A38, A64, A83, A106, A146; ::_thesis: (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c
thus  (Comput ((Q +* (stop (while>0 (a,i,I)))),t,mm)) . a = c by A107, A146; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A170: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A13, A35);
 ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c )
proof
percases ( n - 1 <= 0 or n - 1 > 0 ) ;
suppose n - 1 <= 0 ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c )
hence  ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c ) by A2, A3, A5, A7, A9, A10, A13, A11; ::_thesis: verum
end;
suppose n - 1 > 0 ; ::_thesis: ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c )
then reconsider nn = n - 1 as Element of NAT by INT_1:3;
 S1[nn] by A170;
hence  ex k being Element of NAT ex f2 being FinSequence of INT st
( Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) & f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m & len f2 = n & f,f2 are_fiberwise_equipotent & f2 is_non_decreasing_on 1,n & ( for j being Element of NAT st 1 < n & ( ( 1 <= j & j < 1 ) or ( n < j & j <= n ) ) holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 >= n & 1 <= j & j <= n holds
f2 . j = s . (intpos (m + j)) ) & ( for j being Element of NAT st 1 <= j & j < (2 * 0) + 1 holds
(Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (intpos ((m + n) + j)) = s . (intpos ((m + n) + j)) ) & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 & (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c ) by A2, A3, A5, A7, A9, A10, A11; ::_thesis: verum
end;
end;
end;
then consider k being Element of NAT , f2 being FinSequence of INT  such that
A171: Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) = Comput ((P +* (stop (while>0 (a,i,I)))),s,k) and
A172: f2 is_FinSequence_on Comput ((P +* (stop (while>0 (a,i,I)))),s,k),m and
A173: len f2 = n and
A174: f,f2 are_fiberwise_equipotent and
A175: f2 is_non_decreasing_on 1,n and
A176: (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . (DataLoc (c,i)) = (s . (DataLoc (c,i))) - 2 and
A177: (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)) . a = c ;
set sk = Comput ((P +* (stop (while>0 (a,i,I)))),s,k);
set s1 = Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k));
set P1 = (P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)));
set s2 = Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1);
set P2 = (P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)));
A179: IC (Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))) = 0 by MEMSTR_0:def_11;
set i1 = (a,i) <=0_goto ((card I) + 2);
set i2 = goto (- ((card I) + 1));
A180: card (while>0 (a,i,I)) = (card I) + 2 by SCMPDS_8:17;
then A181: (card I) + 2 in dom (stop (while>0 (a,i,I))) by COMPOS_1:64;
A182: dom g = Seg n by A4, FINSEQ_1:def_3;
 stop (while>0 (a,i,I)) c= (P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I))) by FUNCT_4:25;
then A183: ((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))) . ((card I) + 2) = (stop (while>0 (a,i,I))) . ((card I) + 2) by A181, GRFUNC_1:2
.= halt SCMPDS by A180, COMPOS_1:64 ;
A184: while>0 (a,i,I) = ((a,i) <=0_goto ((card I) + 2)) ';' (I ';' (goto (- ((card I) + 1)))) by Lm2;
A185: Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),(0 + 1)) = Following (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),0))) by EXTPRO_1:3
.= Following (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))))
.= Exec (((a,i) <=0_goto ((card I) + 2)),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k)))) by A184, SCMPDS_6:11 ;
IC (Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1)) = (Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1)) . (IC )
.= ICplusConst ((Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),((card I) + 2)) by A7, A171, A176, A177, A185, SCMPDS_2:56
.= 0 + ((card I) + 2) by A179, SCMPDS_6:12 ;
then A187: CurInstr (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1))) = halt SCMPDS by A183, PBOOLE:143;
A188: Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1) = Comput ((P +* (stop (while>0 (a,i,I)))),s,(k + 1)) by A171, EXTPRO_1:4;
A190: P +* (stop (while>0 (a,i,I))) halts_on s by A187, A188, EXTPRO_1:29;
hence  while>0 (a,i,I) is_halting_on s,P by I, SCMPDS_6:def_3; ::_thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
A191: Result ((P +* (stop (while>0 (a,i,I)))),s) = Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1) by A187, A188, A190, EXTPRO_1:def_9;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_g_holds_
g_._i_=_f2_._i
let i be Nat; ::_thesis: ( i in dom g implies g . i = f2 . i )
reconsider a = i as Element of NAT by ORDINAL1:def_12;
set xi = intpos (m + a);
assume A194: i in dom g ; ::_thesis: g . i = f2 . i
then A195: 1 <= i by A182, FINSEQ_1:1;
A196: i <= n by A182, A194, FINSEQ_1:1;
 IExec ((while>0 (a,i,I)),P,s) = Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1) by A191, SCMPDS_4:def_5;
hence g . i = (Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1)) . (intpos (m + a)) by A4, A6, A195, A196, SCPISORT:def_1
.= (Comput (((P +* (stop (while>0 (a,i,I)))) +* (stop (while>0 (a,i,I)))),(Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))),1)) . (intpos (m + a))
.= (Initialize (Comput ((P +* (stop (while>0 (a,i,I)))),s,k))) . (intpos (m + a)) by A185, SCMPDS_2:56
.= f2 . i by A171, A172, A173, A195, A196, SCPISORT:def_1 ;
::_thesis: verum
end;
hence  ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) by A4, A173, A174, A175, FINSEQ_2:9; ::_thesis: verum
end;

Lm5:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let a be Int_position; ::_thesis: for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let i, c be Integer; ::_thesis: for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
while>0 (a,i,I) is_halting_on s,P
let m, n, m1 be Element of NAT ; ::_thesis: ( s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) implies while>0 (a,i,I) is_halting_on s,P )
set b = DataLoc (c,i);
assume A2: s . a = c ; ::_thesis: ( not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or while>0 (a,i,I) is_halting_on s,P )
consider f being FinSequence of INT  such that
A3: len f = n and
A4: for i being Element of NAT st 1 <= i & i <= len f holds
f . i = s . (intpos (m + i)) by SCPISORT:1;
A5: f is_FinSequence_on s,m by A4, SCPISORT:def_1;
set ss = IExec ((while>0 (a,i,I)),P,s);
assume A6: 1 = s . (DataLoc (c,i)) ; ::_thesis: ( not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or while>0 (a,i,I) is_halting_on s,P )
consider g being FinSequence of INT  such that
A7: len g = n and
A8: for i being Element of NAT st 1 <= i & i <= len g holds
g . i = (IExec ((while>0 (a,i,I)),P,s)) . (intpos (m + i)) by SCPISORT:1;
A9: g is_FinSequence_on IExec ((while>0 (a,i,I)),P,s),m by A8, SCPISORT:def_1;
assume that
A10: m1 = (m + n) + 1 and
A11: m + 1 = s . (intpos m1) and
A12: m + n = s . (intpos (m1 + 1)) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or while>0 (a,i,I) is_halting_on s,P )
assume  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ; ::_thesis: while>0 (a,i,I) is_halting_on s,P
hence  while>0 (a,i,I) is_halting_on s,P by A2, A6, A10, A11, A12, A3, A5, A7, A9, Lm4; ::_thesis: verum
end;

Lm6:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let s be 0 -started State of SCMPDS; ::_thesis: for I being halt-free shiftable Program of SCMPDS
for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let I be halt-free shiftable Program of SCMPDS; ::_thesis: for a being Int_position
for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let a be Int_position; ::_thesis: for i, c being Integer
for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let i, c be Integer; ::_thesis: for m, n, m1 being Element of NAT st s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) holds
( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
let m, n, m1 be Element of NAT ; ::_thesis: ( s . a = c & 1 = s . (DataLoc (c,i)) & m1 = (m + n) + 1 & m + 1 = s . (intpos m1) & m + n = s . (intpos (m1 + 1)) & ( for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) implies ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P ) )
set b = DataLoc (c,i);
assume A2: s . a = c ; ::_thesis: ( not 1 = s . (DataLoc (c,i)) or not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P ) )
assume A3: 1 = s . (DataLoc (c,i)) ; ::_thesis: ( not m1 = (m + n) + 1 or not m + 1 = s . (intpos m1) or not m + n = s . (intpos (m1 + 1)) or ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P ) )
assume that
A4: m1 = (m + n) + 1 and
A5: m + 1 = s . (intpos m1) and
A6: m + n = s . (intpos (m1 + 1)) ; ::_thesis: ( ex t being 0 -started State of SCMPDS ex Q being Instruction-Sequence of SCMPDS ex f1, f2 being FinSequence of INT ex k1, k2, y1, yn being Element of NAT st
( t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) & not ( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ) or ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P ) )
assume A7: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . a = c & (2 * k1) + 1 = t . (DataLoc (c,i)) & k2 = ((m + n) + (2 * k1)) + 1 & m + y1 = t . (intpos k2) & m + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( I is_closed_on t,Q & I is_halting_on t,Q & (IExec (I,Q,t)) . a = t . a & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (I,Q,t)) . (intpos ((m + n) + j)) = t . (intpos ((m + n) + j)) ) & ( y1 >= yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) ) ) & ( y1 < yn implies ( (IExec (I,Q,t)) . (DataLoc (c,i)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (I,Q,t)) . (intpos (m + j)) = t . (intpos (m + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & m + y1 = (IExec (I,Q,t)) . (intpos k2) & (m + ym) - 1 = (IExec (I,Q,t)) . (intpos (k2 + 1)) & (m + ym) + 1 = (IExec (I,Q,t)) . (intpos (k2 + 2)) & m + yn = (IExec (I,Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (I,Q,t)) . (intpos (m + j)) <= (IExec (I,Q,t)) . (intpos (m + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (I,Q,t)) . (intpos (m + j)) >= (IExec (I,Q,t)) . (intpos (m + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,m & f2 is_FinSequence_on IExec (I,Q,t),m & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) ; ::_thesis: ( while>0 (a,i,I) is_halting_on s,P & while>0 (a,i,I) is_closed_on s,P )
A8: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_DataPart_v_=_DataPart_s_holds_
while>0_(a,i,I)_is_halting_on_v,V
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st DataPart v = DataPart s holds
while>0 (a,i,I) is_halting_on v,V
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( DataPart v = DataPart s implies while>0 (a,i,I) is_halting_on v,V )
assume A9: DataPart v = DataPart s ; ::_thesis: while>0 (a,i,I) is_halting_on v,V
then A10: 1 = v . (DataLoc (c,i)) by A3, SCMPDS_4:8;
A11: m + n = v . (intpos (m1 + 1)) by A6, A9, SCMPDS_4:8;
A12: m + 1 = v . (intpos m1) by A5, A9, SCMPDS_4:8;
 v . a = c by A2, A9, SCMPDS_4:8;
hence  while>0 (a,i,I) is_halting_on v,V by A4, A7, A10, A12, A11, Lm5; ::_thesis: verum
end;
hence  while>0 (a,i,I) is_halting_on s,P ; ::_thesis: while>0 (a,i,I) is_closed_on s,P
thus  while>0 (a,i,I) is_closed_on s,P by A8, Th3; ::_thesis: verum
end;

begin

definition
func Partition  ->  Program of SCMPDS equals :: SCPQSORT:def 1
((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))) ';' (((intpos 2),0) := (GBP,6));
coherence
 ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))) ';' (((intpos 2),0) := (GBP,6)) is Program of SCMPDS ;
end;

:: deftheorem  defines Partition SCPQSORT:def_1_:_
 Partition = ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))) ';' (((intpos 2),0) := (GBP,6));

begin

definition
let n, p0 be Element of NAT ;
func QuickSort (n,p0) ->  Program of SCMPDS equals :: SCPQSORT:def 2
((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))) ';' (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Partition) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))));
coherence
 ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))) ';' (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Partition) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))) is Program of SCMPDS ;
end;

:: deftheorem  defines QuickSort SCPQSORT:def_2_:_
 for n, p0 being Element of NAT holds QuickSort (n,p0) = ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))) ';' (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Partition) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))));

set i1 = (GBP,7) := (GBP,5);

set i2 = AddTo (GBP,5,(- 1));

set i3 = (GBP,6) := ((intpos 4),0);

set i4 = SubFrom (GBP,6,(intpos 2),0);

set i5 = AddTo (GBP,4,(- 1));

set i6 = AddTo (GBP,7,(- 1));

set i7 = Load ((GBP,5) := 0);

set IF1 = if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)));

set WB1 = (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))));

set WH1 = while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))));

set j1 = (GBP,5) := (GBP,7);

set j2 = AddTo (GBP,7,(- 1));

set j3 = (GBP,6) := ((intpos 2),0);

set j4 = SubFrom (GBP,6,(intpos 3),0);

set j5 = AddTo (GBP,3,1);

set j6 = AddTo (GBP,5,(- 1));

set j7 = Load ((GBP,7) := 0);

set IF2 = if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)));

set WB2 = (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))));

set WH2 = while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))));

set k1 = (GBP,5) := (GBP,4);

set k2 = SubFrom (GBP,5,GBP,2);

set k3 = (GBP,3) := (GBP,2);

set k4 = AddTo (GBP,3,1);

set K4 = ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1));

set k5 = (GBP,6) := ((intpos 4),0);

set k6 = ((intpos 4),0) := ((intpos 3),0);

set k7 = ((intpos 3),0) := (GBP,6);

set k8 = AddTo (GBP,5,(- 2));

set k9 = AddTo (GBP,3,1);

set k0 = AddTo (GBP,4,(- 1));

set IF3 = if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))));

set WB3 = ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))));

set WH3 = while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))));

set j8 = (GBP,6) := ((intpos 4),0);

set j9 = ((intpos 4),0) := ((intpos 2),0);

set j0 = ((intpos 2),0) := (GBP,6);

set a1 = intpos 1;

set a2 = intpos 2;

set a3 = intpos 3;

set a4 = intpos 4;

set a5 = intpos 5;

set a6 = intpos 6;

set a7 = intpos 7;

Lm7:  card ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))) = 9
proof
thus  card ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))) = (card (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0)))) + (card (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))) by AFINSQ_1:17
.= 4 + (card (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))) by Th4
.= 4 + (((card ((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1))))) + (card (Load ((GBP,5) := 0)))) + 2) by SCMPDS_6:65
.= 4 + ((2 + (card (Load ((GBP,5) := 0)))) + 2) by SCMP_GCD:5
.= 4 + ((2 + 1) + 2) by COMPOS_1:54
.= 9 ; ::_thesis: verum
end;

Lm8:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 4) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 4) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 4) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
let md, me be Element of NAT ; ::_thesis: ( s . (intpos 2) = md & s . (intpos 4) = me & md >= 8 & me >= 8 & s . GBP = 0 implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) ) )
assume that
A1: s . (intpos 2) = md and
A2: s . (intpos 4) = me and
A3: md >= 8 and
A4: me >= 8 and
A5: s . GBP = 0 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
set t0 = s;
set Q0 = P;
set t1 = IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s);
set Q1 = P;
set t2 = IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s);
set t3 = IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s);
set Q2 = P;
set t4 = Exec (((GBP,7) := (GBP,5)),s);
set Q4 = P;
A6: s . GBP = 0 by A5;
A7: DataLoc ((s . GBP),7) = intpos (0 + 7) by A5, SCMP_GCD:1;
then A8: (Exec (((GBP,7) := (GBP,5)),s)) . GBP = 0 by A6, AMI_3:10, SCMPDS_2:47;
then A9: DataLoc (((Exec (((GBP,7) := (GBP,5)),s)) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A10: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 4) = me by A7, A2, AMI_3:10, SCMPDS_2:47;
A11: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 4) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 4) by SCMPDS_5:42
.= me by A10, A9, AMI_3:10, SCMPDS_2:48 ;
then A12: DataLoc (((IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 4)),0) = intpos (me + 0) by SCMP_GCD:1;
A13: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . GBP = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . GBP by SCMPDS_5:42
.= 0 by A8, A9, AMI_3:10, SCMPDS_2:48 ;
then A14: DataLoc (((IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
A15: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 2) = md by A7, A1, AMI_3:10, SCMPDS_2:47;
A16: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 2) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 2) by SCMPDS_5:42
.= md by A15, A9, AMI_3:10, SCMPDS_2:48 ;
A18: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 5) = s . (intpos 5) by A7, AMI_3:10, SCMPDS_2:47;
set t01 = Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)));
set ii7 = (GBP,5) := 0;
set t5 = Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))));
A19: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 2) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= md by A16, A14, AMI_3:10, SCMPDS_2:47 ;
A20: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . GBP = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A13, A14, AMI_3:10, SCMPDS_2:47 ;
then A21: DataLoc (((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
 DataLoc ((s . GBP),5) = intpos (0 + 5) by A6, SCMP_GCD:1;
then A22: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 7) = s . (intpos 5) by A7, SCMPDS_2:47;
A23: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 7) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 7) by SCMPDS_5:42
.= s . (intpos 5) by A22, A9, AMI_3:10, SCMPDS_2:48 ;
A24: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 7) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 7) by SCMPDS_5:41
.= s . (intpos 5) by A23, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 7) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 7) by SCMPDS_5:41
.= s . (intpos 5) by A24, A21, AMI_3:10, SCMPDS_2:50 ;
then A25: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 7) = s . (intpos 5) by SCMPDS_5:15;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 2) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= md by A19, A21, AMI_3:10, SCMPDS_2:50 ;
then A26: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 2) = s . (intpos 2) by A1, SCMPDS_5:15;
A27: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(Exec_(((GBP,7)_:=_(GBP,5)),s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (Exec (((GBP,7) := (GBP,5)),s)) . (intpos i) = s . (intpos i) )
assume  i >= 8 ; ::_thesis: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos i) = s . (intpos i)
then  i > 7 by XXREAL_0:2;
hence (Exec (((GBP,7) := (GBP,5)),s)) . (intpos i) = s . (intpos i) by A7, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) ;
::_thesis: verum
end;
A28: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1)))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos i) = s . (intpos i) )
assume A29: i >= 8 ; ::_thesis: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos i) = s . (intpos i)
then A30: i > 5 by XXREAL_0:2;
thus (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos i) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,7) := (GBP,5)),s)) . (intpos i) by A9, A30, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by A27, A29 ; ::_thesis: verum
end;
A31: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) = s . (intpos i) )
assume A32: i >= 8 ; ::_thesis: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) = s . (intpos i)
then A33: i > 6 by XXREAL_0:2;
thus (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos i) by A14, A33, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A28, A32 ; ::_thesis: verum
end;
A34: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) = s . (intpos i) )
assume A35: i >= 8 ; ::_thesis: (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) = s . (intpos i)
then A36: i > 6 by XXREAL_0:2;
thus (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) by A21, A36, AMI_3:10, SCMPDS_2:50
.= s . (intpos i) by A31, A35 ; ::_thesis: verum
end;
A37: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 3) = s . (intpos 3) by A7, AMI_3:10, SCMPDS_2:47;
A38: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 3) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 3) by SCMPDS_5:42
.= s . (intpos 3) by A37, A9, AMI_3:10, SCMPDS_2:48 ;
A39: (Exec (((GBP,7) := (GBP,5)),s)) . (intpos 1) = s . (intpos 1) by A7, AMI_3:10, SCMPDS_2:47;
A40: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 1) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A39, A9, AMI_3:10, SCMPDS_2:48 ;
A41: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 1) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A40, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 1) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A41, A21, AMI_3:10, SCMPDS_2:50 ;
then A42: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 1) = s . (intpos 1) by SCMPDS_5:15;
A43: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 3) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= s . (intpos 3) by A38, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 3) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= s . (intpos 3) by A43, A21, AMI_3:10, SCMPDS_2:50 ;
then A44: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 3) = s . (intpos 3) by SCMPDS_5:15;
A45: (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A20, A21, AMI_3:10, SCMPDS_2:50 ;
then A46: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP = 0 by SCMPDS_5:15;
then A47: DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
 intpos 3 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
then A48: (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 3) = s . (intpos 3) by A44, SCMPDS_2:48;
 intpos 2 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
then A49: (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 2) = s . (intpos 2) by A26, SCMPDS_2:48;
 GBP <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
then A50: (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . GBP = 0 by A46, SCMPDS_2:48;
then A51: DataLoc (((Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . GBP),7) = intpos (0 + 7) by SCMP_GCD:1;
A52: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 6) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 6) by SCMPDS_5:41
.= (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos me) by A14, A12, SCMPDS_2:47
.= s . (intpos me) by A4, A28 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 6) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 6) by SCMPDS_5:41
.= ((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 6)) - ((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (DataLoc (((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 2)),0))) by A21, SCMPDS_2:50
.= ((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 6)) - ((IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos (md + 0))) by A19, SCMP_GCD:1
.= (s . (intpos me)) - (s . (intpos md)) by A3, A52, A31 ;
then A53: (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) = (s . (intpos me)) - (s . (intpos md)) by A45, SCMP_GCD:1;
 intpos 1 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
then A54: (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 1) = s . (intpos 1) by A42, SCMPDS_2:48;
A56: now__::_thesis:_(_(IExec_((if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))),P,(Initialize_(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s)))))_._GBP_=_0_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))),P,(Initialize_(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s)))))_._(intpos_1)_=_s_._(intpos_1)_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))),P,(Initialize_(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s)))))_._(intpos_2)_=_s_._(intpos_2)_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))),P,(Initialize_(IExec_((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0))),P,s)))))_._(intpos_3)_=_s_._(intpos_3)_&_(_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),P,s))_._(intpos_i)_=_s_._(intpos_i)_)_)
percases ( (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) <= 0 or (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) > 0 ) ;
supposeA57: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) <= 0 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP = 0 & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
hence (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP = (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP by SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . GBP by SCMPDS_5:40
.= 0 by A46, A47, AMI_3:10, SCMPDS_2:46 ;
::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) by A57, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 1) by SCMPDS_5:40
.= s . (intpos 1) by A42, A47, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) by A57, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 2) by SCMPDS_5:40
.= s . (intpos 2) by A26, A47, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) by A57, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 3) by SCMPDS_5:40
.= s . (intpos 3) by A44, A47, AMI_3:10, SCMPDS_2:46 ; ::_thesis: for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) )
assume A58: i >= 8 ; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i)
then A59: i > 5 by XXREAL_0:2;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos i) by SCMPDS_5:35
.= (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos i) by A57, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos i) by SCMPDS_5:40
.= (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos i) by A47, A59, AMI_3:10, SCMPDS_2:46
.= (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A34, A58 ; ::_thesis: verum
end;
supposeA60: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) > 0 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP = 0 & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP = (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . GBP by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . GBP by SCMPDS_5:42
.= 0 by A50, A51, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) = (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 1) by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A54, A51, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) = (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 2) by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A49, A51, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) = (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 3) by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 3) by SCMPDS_5:42
.= s . (intpos 3) by A48, A51, AMI_3:10, SCMPDS_2:48 ; ::_thesis: for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) )
assume A61: i >= 8 ; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i)
then A62: i > 7 by XXREAL_0:2;
 i > 4 by A61, XXREAL_0:2;
then A63: intpos i <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos i) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos i) by A60, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos i) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos i) by A51, A62, AMI_3:10, SCMPDS_2:48
.= (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos i) by A63, SCMPDS_2:48
.= (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A34, A61 ; ::_thesis: verum
end;
end;
end;
hence  ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) ) by SCMPDS_5:35; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
thus  for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos i) = s . (intpos i) by A56; ::_thesis: ( ( s . (intpos md) < s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 ) ) & ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) ) )
A64: (IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)) . (intpos 5) = (Exec ((AddTo (GBP,5,(- 1))),(Exec (((GBP,7) := (GBP,5)),s)))) . (intpos 5) by SCMPDS_5:42
.= ((Exec (((GBP,7) := (GBP,5)),s)) . (intpos 5)) + (- 1) by A9, SCMPDS_2:48
.= (s . (intpos 5)) - 1 by A18 ;
A65: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 5) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 5)) - 1 by A64, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 5) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 5)) - 1 by A65, A21, AMI_3:10, SCMPDS_2:50 ;
then A66: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 5) = (s . (intpos 5)) - 1 by SCMPDS_5:15;
A67: (IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 4) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec ((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= me by A11, A14, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (intpos 4) = (Exec ((SubFrom (GBP,6,(intpos 2),0)),(IExec (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= me by A67, A21, AMI_3:10, SCMPDS_2:50 ;
then A68: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 4) = s . (intpos 4) by A2, SCMPDS_5:15;
A69: DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) = intpos (0 + 4) by A46, SCMP_GCD:1;
C71: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) = (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) by SCMPDS_5:15
.= (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) by SCMPDS_5:15 ;
hereby  ::_thesis: ( s . (intpos md) >= s . (intpos me) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) ) )
A70: intpos 5 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
assume  s . (intpos md) < s . (intpos me) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (s . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 )
then B71: 0 < (s . (intpos me)) - (s . (intpos md)) by XREAL_1:50;
C71: (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),6)) = (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) by SCMPDS_5:15
.= (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . (DataLoc (((IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)) . GBP),6)) by SCMPDS_5:15 ;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 5) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 5) by A53, B71, C71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 5) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 5) by A51, AMI_3:10, SCMPDS_2:48
.= (s . (intpos 5)) - 1 by A66, A70, SCMPDS_2:48 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1 )
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))))) . (intpos 4) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))))) . (intpos 4) by A53, C71, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 4) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 4) by A51, AMI_3:10, SCMPDS_2:48
.= ((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . (intpos 4)) + (- 1) by A69, SCMPDS_2:48
.= (s . (intpos 4)) - 1 by A68 ; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (s . (intpos 5)) - 1
A72: intpos 7 <> DataLoc (((Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))) . GBP),4) by A46, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 7) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 7) by A53, C71, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,7,(- 1))),(Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))))) . (intpos 7) by SCMPDS_5:42
.= ((Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 7)) + (- 1) by A51, SCMPDS_2:48
.= ((Exec ((AddTo (GBP,4,(- 1))),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 7)) - 1
.= (s . (intpos 5)) - 1 by A25, A72, SCMPDS_2:48 ; ::_thesis: verum
end;
assume  s . (intpos md) >= s . (intpos me) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) )
then B73: (s . (intpos me)) - (s . (intpos md)) <= 0 by XREAL_1:47;
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 5) by SCMPDS_5:35
.= (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 5) by A53, C71, B73, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 5) by SCMPDS_5:40
.= 0 by A47, SCMPDS_2:46 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5) )
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 4) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 4) by SCMPDS_5:35
.= (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 4) by A53, C71, B73, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 4) by SCMPDS_5:40
.= s . (intpos 4) by A47, A68, AMI_3:10, SCMPDS_2:46 ; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = s . (intpos 5)
thus (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),P,s)) . (intpos 7) = (IExec ((if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 7) by SCMPDS_5:35
.= (IExec ((Load ((GBP,5) := 0)),P,(Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s))))) . (intpos 7) by A53, C71, B73, SCMPDS_6:74
.= (Exec (((GBP,5) := 0),(Initialize (Initialize (IExec ((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))),P,s)))))) . (intpos 7) by SCMPDS_5:40
.= s . (intpos 5) by A25, A47, AMI_3:10, SCMPDS_2:46 ; ::_thesis: verum
end;

Lm9:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) )
let m4, md be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 implies ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 5) > 0 and
A3: s . (intpos 4) = m4 + (s . (intpos 5)) and
A4: m4 >= 8 and
A5: s . (intpos 2) = md and
A6: md >= 8 ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) )
 s . (intpos 5) >= 1 + 0 by A2, INT_1:7;
then reconsider m5 = (s . (intpos 5)) - 1 as Element of NAT by INT_1:3, XREAL_1:48;
A7: s . (intpos 5) = m5 + 1 ;
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) = $1 + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) );
A8: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A9: S1[k] ; ::_thesis: S1[k + 1]
thus  S1[k + 1] ::_thesis: verum
proof
set me = m4 + ((k + 1) + 1);
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) = (k + 1) + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & t . (intpos 5) = (k + 1) + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) ) )
set b = DataLoc ((t . GBP),5);
assume that
A10: t . GBP = 0 and
A11: t . (intpos 5) = (k + 1) + 1 and
A12: t . (intpos 4) = m4 + (t . (intpos 5)) and
A13: t . (intpos 2) = md ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
set It = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t);
set IT = Q;
set It0 = Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t));
A23: m4 + ((k + 1) + 1) >= m4 by NAT_1:11;
then A24: m4 + ((k + 1) + 1) >= 8 by A4, XXREAL_0:2;
then A25: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A10, A11, A12, A13, Lm8;
A26: ( t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = t . (intpos 5) ) ) by A6, A10, A11, A12, A13, A24, Lm8;
A27: DataLoc ((t . GBP),5) = intpos (0 + 5) by A10, SCMP_GCD:1;
A28: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_4)_>=_m4_+_(v_._(DataLoc_((t_._GBP),5)))_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),5))_>_0_holds_
(_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_closed_on_v,V_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),5))_<_v_._(DataLoc_((t_._GBP),5))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_4)_>=_m4_+_((IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),5)))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A29: v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) and
A30: v . (intpos 2) = t . (intpos 2) and
A31: v . GBP = t . GBP and
A32: v . (DataLoc ((t . GBP),5)) > 0 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A33: m4 + (v . (DataLoc ((t . GBP),5))) > m4 + 0 by A32, XREAL_1:6;
then reconsider ME = v . (intpos 4) as Element of NAT by A29, INT_1:3;
 v . (intpos 4) >= m4 by A29, A33, XXREAL_0:2;
then A41: ME >= 8 by A4, XXREAL_0:2;
then A42: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A6, A10, A13, A30, A31, Lm8;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP by A6, A10, A13, A30, A31, A41, Lm8; ::_thesis: ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A43: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = v . (intpos 5) ) ) by A6, A10, A13, A30, A31, A41, Lm8;
hereby  ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) < v . (intpos ME) or v . (intpos md) >= v . (intpos ME) ) ;
supposeA44: v . (intpos md) < v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A27, A42, XREAL_1:146; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)))
 (m4 + (v . (DataLoc ((t . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A27, A42, A44;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A29, A42, A44, XREAL_1:9; ::_thesis: verum
end;
supposeA45: v . (intpos md) >= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A6, A10, A13, A27, A30, A31, A32, A41, Lm8; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)))
 m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) < m4 + (v . (DataLoc ((t . GBP),5))) by A27, A32, A43, A45, XREAL_1:6;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A29, A43, A45, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A6, A10, A13, A30, A31, A41, Lm8; ::_thesis: verum
end;
A46: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) by A6, A10, A11, A12, A13, A24, Lm8;
A47: ( t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = (t . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 5)) - 1 ) ) by A6, A10, A11, A12, A13, A24, Lm8;
A48: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP = 0 by A6, A10, A11, A12, A13, A24, Lm8;
then B48: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . GBP = 0 by SCMPDS_5:15;
A49: DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5) = intpos (0 + 5) by A48, SCMP_GCD:1;
A50: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A10, A11, A12, A13, A24, Lm8;
then B50: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 2) = t . (intpos 2) by SCMPDS_5:15;
percases ( t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) or t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) ) ;
supposeA51: t . (intpos md) < t . (intpos (m4 + ((k + 1) + 1))) ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
then A52: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5)) by A12, A47;
B52: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 4) = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5)) by A12, A47, A51, SCMPDS_5:15
.= m4 + ((Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 5)) by SCMPDS_5:15 ;
YY: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 5) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) by SCMPDS_5:15;
Y1: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 1) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 1) by SCMPDS_5:15;
Y3: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 3) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 3) by SCMPDS_5:15;
Y4: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos 4) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) by SCMPDS_5:15;
 (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = k + 1 by A11, A47, A51;
then consider mE being Element of NAT  such that
A53: mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos 7) and
A54: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos 4) = m4 + mE and
A55: mE <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) and
A56: for i being Element of NAT st m4 + mE < i & i <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos i) and
A57: ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . (intpos (m4 + mE)) ) by A9, A12, A13, A47, A26, B48, B50, B52, YY;
A58: IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t) = IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)))) by A11, A12, A27, A28, Th5;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 by A9, A11, A12, A13, A47, A26, A52, YY, B50, B48, Y4; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A9, A11, A12, A13, A25, A47, A26, A52, A58, YY, B50, B48, Y1, Y4; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 by A9, A11, A12, A13, A47, A26, A52, A58, YY, B50, B48, Y4; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A9, A11, A12, A13, A47, A26, A52, A58, YY, B50, B48, Y4; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) by A9, A11, A12, A13, A46, A47, A26, A52, A58, YY, B50, B48, Y3, Y4; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A59: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A61: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
hence (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (intpos i) by A9, A11, A12, A13, A47, A26, A52, A58, YY, B50, B48, Y4
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos i) by SCMPDS_5:15
.= t . (intpos i) by A6, A10, A11, A12, A13, A24, A61, Lm8 ;
::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
take  mE ; ::_thesis: ( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
thus  mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) by A11, A12, A27, A28, A53, Th5; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE by A11, A12, A27, A28, A54, Th5; ::_thesis: ( mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
thus  mE <= t . (intpos 5) by A47, A51, A55, XREAL_1:146, XXREAL_0:2; ::_thesis: ( ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
A62: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = (t . (intpos 4)) - 1 by A6, A10, A11, A12, A13, A24, A51, Lm8;
hereby  ::_thesis: ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) )
let i be Element of NAT ; ::_thesis: ( m4 + mE < i & i <= t . (intpos 4) implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos b1) )
assume that
A63: m4 + mE < i and
A64: i <= t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos b1)
percases ( i = t . (intpos 4) or i <> t . (intpos 4) ) ;
supposeA65: i = t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos b1)
 (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ((k + 1) + 1))) = t . (intpos (m4 + ((k + 1) + 1))) by A4, A23, A59, XXREAL_0:2;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A6, A11, A12, A51, A59, A65; ::_thesis: verum
end;
suppose i <> t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos b1)
then  i < t . (intpos 4) by A64, XXREAL_0:1;
then  i + 1 <= t . (intpos 4) by INT_1:7;
then  i <= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) by A62, XREAL_1:19;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A58, A56, A63; ::_thesis: verum
end;
end;
end;
thus  ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) by A58, A57; ::_thesis: verum
end;
supposeA66: t . (intpos md) >= t . (intpos (m4 + ((k + 1) + 1))) ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A67: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._x_=_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),Q,t))_._x
X1: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) by SCMPDS_5:15;
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . x by A11, A12, A27, A28, Th5
.= (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . x by A26, A49, A66, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 by A48; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A25, A67; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 by A26, A66, A67; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A50, A67; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) by A46, A67; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A68: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A70: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos i) by A67
.= t . (intpos i) by A6, A10, A11, A12, A13, A24, A70, Lm8 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
take ME = (k + 1) + 1; ::_thesis: ( ME = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) = ME by A11, A26, A66, A67; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME by A11, A12, A26, A66, A67; ::_thesis: ( ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  ME <= t . (intpos 5) by A11; ::_thesis: ( ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A11, A12; ::_thesis: ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) )
 (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ((k + 1) + 1))) = t . (intpos (m4 + ((k + 1) + 1))) by A4, A23, A68, XXREAL_0:2;
hence  ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) by A6, A66, A68; ::_thesis: verum
end;
end;
end;
end;
A118: S1[ 0 ]
proof
set me = m4 + 1;
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) = 0 + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & t . (intpos 5) = 0 + 1 & t . (intpos 4) = m4 + (t . (intpos 5)) & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) ) )
set It = IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t);
set IT = Q;
set b = DataLoc ((t . GBP),5);
assume that
A80: t . GBP = 0 and
A81: t . (intpos 5) = 0 + 1 and
A82: t . (intpos 4) = m4 + (t . (intpos 5)) and
A83: t . (intpos 2) = md ; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A84: m4 + 1 >= m4 by NAT_1:11;
then A85: m4 + 1 >= 8 by A4, XXREAL_0:2;
then A86: ( t . (intpos md) < t . (intpos (m4 + 1)) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = (t . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 5)) - 1 ) ) by A6, A80, A81, A82, A83, Lm8;
A87: ( t . (intpos md) >= t . (intpos (m4 + 1)) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 7) = t . (intpos 5) ) ) by A6, A80, A81, A82, A83, A85, Lm8;
A88: DataLoc ((t . GBP),5) = intpos (0 + 5) by A80, SCMP_GCD:1;
A89: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_4)_>=_m4_+_(v_._(DataLoc_((t_._GBP),5)))_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),5))_>_0_holds_
(_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_closed_on_v,V_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),5))_<_v_._(DataLoc_((t_._GBP),5))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_4)_>=_m4_+_((IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),5)))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A90: v . (intpos 4) >= m4 + (v . (DataLoc ((t . GBP),5))) and
A91: v . (intpos 2) = t . (intpos 2) and
A92: v . GBP = t . GBP and
A93: v . (DataLoc ((t . GBP),5)) > 0 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A94: m4 + (v . (DataLoc ((t . GBP),5))) > m4 + 0 by A93, XREAL_1:6;
then reconsider ME = v . (intpos 4) as Element of NAT by A90, INT_1:3;
 v . (intpos 4) >= m4 by A90, A94, XXREAL_0:2;
then A102: ME >= 8 by A4, XXREAL_0:2;
then A103: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A6, A80, A83, A91, A92, Lm8;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP by A6, A80, A83, A91, A92, A102, Lm8; ::_thesis: ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A104: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = v . (intpos 5) ) ) by A6, A80, A83, A91, A92, A102, Lm8;
hereby  ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) < v . (intpos ME) or v . (intpos md) >= v . (intpos ME) ) ;
supposeA105: v . (intpos md) < v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A88, A103, XREAL_1:146; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)))
 (m4 + (v . (DataLoc ((t . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A88, A103, A105;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A90, A103, A105, XREAL_1:9; ::_thesis: verum
end;
supposeA106: v . (intpos md) >= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A6, A80, A83, A88, A91, A92, A93, A102, Lm8; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5)))
 m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) < m4 + (v . (DataLoc ((t . GBP),5))) by A88, A93, A104, A106, XREAL_1:6;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((t . GBP),5))) by A90, A104, A106, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A6, A80, A83, A91, A92, A102, Lm8; ::_thesis: verum
end;
A107: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP = 0 by A6, A80, A81, A82, A83, A85, Lm8;
then A108: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) = 0 by A81, A86, A87, SCMP_GCD:1;
A109: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._x_=_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),Q,t))_._x
X1: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . GBP),5)) by SCMPDS_5:15;
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,(Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))))) . x by A81, A82, A88, A89, Th5
.= (Initialize (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t))) . x by A108, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . GBP = 0 by A107; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
 (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A80, A81, A82, A83, A85, Lm8;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A109; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 5) = 0 by A81, A86, A87, A109; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
 (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A80, A81, A82, A83, A85, Lm8;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A109; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
 (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) by A6, A80, A81, A82, A83, A85, Lm8;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 3) = t . (intpos 3) by A109; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) ) )
A110: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A111: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),Q,t)) . (intpos i) by A109
.= t . (intpos i) by A6, A80, A81, A82, A83, A85, A111, Lm8 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
A113: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + 1)) = t . (intpos (m4 + 1)) by A4, A84, A110, XXREAL_0:2;
percases ( t . (intpos md) < t . (intpos (m4 + 1)) or t . (intpos md) >= t . (intpos (m4 + 1)) ) ;
supposeA114: t . (intpos md) < t . (intpos (m4 + 1)) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
take ME = 0 ; ::_thesis: ( ME = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) = ME by A81, A86, A109, A114; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME by A81, A82, A86, A109, A114; ::_thesis: ( ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  ME <= t . (intpos 5) by A81; ::_thesis: ( ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
hereby  ::_thesis: ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) )
let i be Element of NAT ; ::_thesis: ( m4 + ME < i & i <= t . (intpos 4) implies (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) )
assume that
A115: m4 + ME < i and
A116: i <= t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i)
 m4 + 1 <= i by A115, INT_1:7;
then  i = m4 + 1 by A81, A82, A116, XXREAL_0:1;
hence  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A6, A110, A113, A114; ::_thesis: verum
end;
thus  ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) ; ::_thesis: verum
end;
supposeA117: t . (intpos md) >= t . (intpos (m4 + 1)) ; ::_thesis: ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + mE & mE <= t . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + mE)) ) )
take ME = 1; ::_thesis: ( ME = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 7) = ME by A81, A87, A109, A117; ::_thesis: ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME & ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos 4) = m4 + ME by A81, A82, A87, A109, A117; ::_thesis: ( ME <= t . (intpos 5) & ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  ME <= t . (intpos 5) by A81; ::_thesis: ( ( for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) ) & ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) )
thus  for i being Element of NAT st m4 + ME < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos i) by A81, A82; ::_thesis: ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) )
thus  ( ME = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),Q,t)) . (intpos (m4 + ME)) ) by A6, A110, A113, A117; ::_thesis: verum
end;
end;
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A118, A8);
hence  ( (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex mE being Element of NAT st
( mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) & (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m4 + mE & mE <= s . (intpos 5) & ( for i being Element of NAT st m4 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) ) & ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m4 + mE)) ) ) ) by A1, A3, A5, A7; ::_thesis: verum
end;

Lm10:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for m4, md being Element of NAT st s . GBP = 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P )
let m4, md be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 4) = m4 + (s . (intpos 5)) & m4 >= 8 & s . (intpos 2) = md & md >= 8 implies ( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 4) = m4 + (s . (intpos 5)) and
A3: m4 >= 8 and
A4: s . (intpos 2) = md and
A5: md >= 8 ; ::_thesis: ( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P )
set b = DataLoc ((s . GBP),5);
A6: DataLoc ((s . GBP),5) = intpos (0 + 5) by A1, SCMP_GCD:1;
now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_4)_>=_m4_+_(v_._(DataLoc_((s_._GBP),5)))_&_v_._(intpos_2)_=_s_._(intpos_2)_&_v_._GBP_=_s_._GBP_&_v_._(DataLoc_((s_._GBP),5))_>_0_holds_
(_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_closed_on_v,V_&_(((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((s_._GBP),5))_<_v_._(DataLoc_((s_._GBP),5))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_4)_>=_m4_+_((IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(DataLoc_((s_._GBP),5)))_&_(IExec_(((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 4) >= m4 + (v . (DataLoc ((s . GBP),5))) & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),5)) > 0 holds
( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 4) >= m4 + (v . (DataLoc ((s . GBP),5))) & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),5)) > 0 implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A7: v . (intpos 4) >= m4 + (v . (DataLoc ((s . GBP),5))) and
A8: v . (intpos 2) = s . (intpos 2) and
A9: v . GBP = s . GBP and
A10: v . (DataLoc ((s . GBP),5)) > 0 ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A11: m4 + (v . (DataLoc ((s . GBP),5))) > m4 + 0 by A10, XREAL_1:6;
then reconsider ME = v . (intpos 4) as Element of NAT by A7, INT_1:3;
 v . (intpos 4) >= m4 by A7, A11, XXREAL_0:2;
then A19: ME >= 8 by A3, XXREAL_0:2;
then A20: ( v . (intpos md) < v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = (v . (intpos 5)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = (v . (intpos 4)) - 1 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = (v . (intpos 5)) - 1 ) ) by A1, A4, A5, A8, A9, Lm8;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . GBP = v . GBP by A1, A4, A5, A8, A9, A19, Lm8; ::_thesis: ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_closed_on v,V & (((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A21: ( v . (intpos md) >= v . (intpos ME) implies ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 5) = 0 & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) = v . (intpos 4) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 7) = v . (intpos 5) ) ) by A1, A4, A5, A8, A9, A19, Lm8;
hereby  ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) < v . (intpos ME) or v . (intpos md) >= v . (intpos ME) ) ;
supposeA22: v . (intpos md) < v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) by A6, A20, XREAL_1:146; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)))
 (m4 + (v . (DataLoc ((s . GBP),5)))) - 1 = m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) by A6, A20, A22;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) by A7, A20, A22, XREAL_1:9; ::_thesis: verum
end;
supposeA23: v . (intpos md) >= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) )
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) by A1, A4, A5, A6, A8, A9, A10, A19, Lm8; ::_thesis: (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5)))
 m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) < m4 + (v . (DataLoc ((s . GBP),5))) by A6, A10, A21, A23, XREAL_1:6;
hence  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 4) >= m4 + ((IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (DataLoc ((s . GBP),5))) by A7, A21, A23, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A1, A4, A5, A8, A9, A19, Lm8; ::_thesis: verum
end;
hence  ( while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P & while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P ) by A2, A6, Th5; ::_thesis: verum
end;

Lm11:  card (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) = 11
proof
thus  card (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) = 9 + 2 by Lm7, SCMPDS_8:17
.= 11 ; ::_thesis: verum
end;

Lm12:  card ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))) = 9
proof
thus  card ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))) = (card (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0)))) + (card (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))) by AFINSQ_1:17
.= 4 + (card (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))) by Th4
.= 4 + (((card ((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1))))) + (card (Load ((GBP,7) := 0)))) + 2) by SCMPDS_6:65
.= 4 + ((2 + (card (Load ((GBP,7) := 0)))) + 2) by SCMP_GCD:5
.= 4 + ((2 + 1) + 2) by COMPOS_1:54
.= 9 ; ::_thesis: verum
end;

Lm13:  card (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) = 11
proof
thus  card (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) = 9 + 2 by Lm12, SCMPDS_8:17
.= 11 ; ::_thesis: verum
end;

Lm14:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 3) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 3) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md, me being Element of NAT st s . (intpos 2) = md & s . (intpos 3) = me & md >= 8 & me >= 8 & s . GBP = 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
let md, me be Element of NAT ; ::_thesis: ( s . (intpos 2) = md & s . (intpos 3) = me & md >= 8 & me >= 8 & s . GBP = 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) ) )
assume that
A1: s . (intpos 2) = md and
A2: s . (intpos 3) = me and
A3: md >= 8 and
A4: me >= 8 and
A5: s . GBP = 0 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
set t0 = s;
set t1 = IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s);
set Q1 = P;
set t2 = IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s);
set t3 = IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s);
set t4 = Exec (((GBP,5) := (GBP,7)),s);
A7: s . GBP = 0 by A5;
A8: DataLoc ((s . GBP),5) = intpos (0 + 5) by A5, SCMP_GCD:1;
then A9: (Exec (((GBP,5) := (GBP,7)),s)) . GBP = 0 by A5, AMI_3:10, SCMPDS_2:47;
then A10: DataLoc (((Exec (((GBP,5) := (GBP,7)),s)) . GBP),7) = intpos (0 + 7) by SCMP_GCD:1;
A11: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 2) = md by A8, A1, AMI_3:10, SCMPDS_2:47;
A12: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 2) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 2) by SCMPDS_5:42
.= md by A11, A10, AMI_3:10, SCMPDS_2:48 ;
then A13: DataLoc (((IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 2)),0) = intpos (md + 0) by SCMP_GCD:1;
A14: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . GBP = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . GBP by SCMPDS_5:42
.= 0 by A9, A10, AMI_3:10, SCMPDS_2:48 ;
then A15: DataLoc (((IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
A16: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 3) = me by A8, A2, AMI_3:10, SCMPDS_2:47;
A17: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 3) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 3) by SCMPDS_5:42
.= me by A16, A10, AMI_3:10, SCMPDS_2:48 ;
A19: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 7) = s . (intpos 7) by A8, AMI_3:10, SCMPDS_2:47;
set t01 = Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s));
set jj7 = (GBP,7) := 0;
set t5 = Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))));
A20: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 3) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= me by A17, A15, AMI_3:10, SCMPDS_2:47 ;
A21: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . GBP = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A14, A15, AMI_3:10, SCMPDS_2:47 ;
then A22: DataLoc (((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
A23: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 4) = s . (intpos 4) by A8, AMI_3:10, SCMPDS_2:47;
A24: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 4) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 4) by SCMPDS_5:42
.= s . (intpos 4) by A23, A10, AMI_3:10, SCMPDS_2:48 ;
A25: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 1) = s . (intpos 1) by A8, AMI_3:10, SCMPDS_2:47;
A26: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 1) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A25, A10, AMI_3:10, SCMPDS_2:48 ;
A27: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 1) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A26, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 1) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A27, A22, AMI_3:10, SCMPDS_2:50 ;
then A28: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 1) = s . (intpos 1) by SCMPDS_5:15;
A29: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 4) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= s . (intpos 4) by A24, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 4) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 4) by SCMPDS_5:41
.= s . (intpos 4) by A29, A22, AMI_3:10, SCMPDS_2:50 ;
then A30: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 4) = s . (intpos 4) by SCMPDS_5:15;
A31: (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . GBP = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A21, A22, AMI_3:10, SCMPDS_2:50 ;
then A32: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP = 0 by SCMPDS_5:15;
then A33: DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),7) = intpos (0 + 7) by SCMP_GCD:1;
 DataLoc ((s . GBP),7) = intpos (0 + 7) by A7, SCMP_GCD:1;
then A34: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos 5) = s . (intpos 7) by A8, SCMPDS_2:47;
A35: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 5) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 5) by SCMPDS_5:42
.= s . (intpos 7) by A34, A10, AMI_3:10, SCMPDS_2:48 ;
A36: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 5) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 5) by SCMPDS_5:41
.= s . (intpos 7) by A35, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 5) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 5) by SCMPDS_5:41
.= s . (intpos 7) by A36, A22, AMI_3:10, SCMPDS_2:50 ;
then A37: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 5) = s . (intpos 7) by SCMPDS_5:15;
A38: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 2) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= md by A12, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 2) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 2) by SCMPDS_5:41
.= md by A38, A22, AMI_3:10, SCMPDS_2:50 ;
then A39: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 2) = s . (intpos 2) by A1, SCMPDS_5:15;
 intpos 4 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
then A40: (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) by A30, SCMPDS_2:48;
 intpos 2 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
then A41: (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) by A39, SCMPDS_2:48;
 GBP <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
then A42: (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = 0 by A32, SCMPDS_2:48;
then A43: DataLoc (((Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A44: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(Exec_(((GBP,5)_:=_(GBP,7)),s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (Exec (((GBP,5) := (GBP,7)),s)) . (intpos i) = s . (intpos i) )
assume  i >= 8 ; ::_thesis: (Exec (((GBP,5) := (GBP,7)),s)) . (intpos i) = s . (intpos i)
then  i > 5 by XXREAL_0:2;
hence (Exec (((GBP,5) := (GBP,7)),s)) . (intpos i) = s . (intpos i) by A8, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) ;
::_thesis: verum
end;
A45: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1)))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos i) = s . (intpos i) )
assume A46: i >= 8 ; ::_thesis: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos i) = s . (intpos i)
then A47: i > 7 by XXREAL_0:2;
thus (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos i) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,5) := (GBP,7)),s)) . (intpos i) by A10, A47, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by A44, A46 ; ::_thesis: verum
end;
A48: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos i) = s . (intpos i) )
assume A49: i >= 8 ; ::_thesis: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos i) = s . (intpos i)
then A50: i > 6 by XXREAL_0:2;
thus (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos i) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos i) by A15, A50, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A45, A49 ; ::_thesis: verum
end;
A51: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) = s . (intpos i) )
assume A52: i >= 8 ; ::_thesis: (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) = s . (intpos i)
then A53: i > 6 by XXREAL_0:2;
thus (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos i) by A22, A53, AMI_3:10, SCMPDS_2:50
.= s . (intpos i) by A48, A52 ; ::_thesis: verum
end;
A54: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 6) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 6) by SCMPDS_5:41
.= (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos md) by A15, A13, SCMPDS_2:47
.= s . (intpos md) by A3, A45 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 6) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 6) by SCMPDS_5:41
.= ((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 6)) - ((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (DataLoc (((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 3)),0))) by A22, SCMPDS_2:50
.= ((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 6)) - ((IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos (me + 0))) by A20, SCMP_GCD:1
.= (s . (intpos md)) - (s . (intpos me)) by A4, A54, A48 ;
then A55: (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (DataLoc (((IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . GBP),6)) = (s . (intpos md)) - (s . (intpos me)) by A31, SCMP_GCD:1;
 intpos 1 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
then A56: (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) by A28, SCMPDS_2:48;
A57: now__::_thesis:_(_(IExec_((if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0)))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s)))))_._GBP_=_0_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0)))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s)))))_._(intpos_1)_=_s_._(intpos_1)_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0)))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s)))))_._(intpos_2)_=_s_._(intpos_2)_&_(IExec_((if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0)))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0))),P,s)))))_._(intpos_4)_=_s_._(intpos_4)_&_(_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),P,s))_._(intpos_i)_=_s_._(intpos_i)_)_)
percases ( (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) <= 0 or (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) > 0 ) ;
supposeA58: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) <= 0 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = 0 & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
hence (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP by SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP by SCMPDS_5:40
.= 0 by A32, A33, AMI_3:10, SCMPDS_2:46 ;
::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) by A58, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) by SCMPDS_5:40
.= s . (intpos 1) by A28, A33, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) by A58, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) by SCMPDS_5:40
.= s . (intpos 2) by A39, A33, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) by A58, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) by SCMPDS_5:40
.= s . (intpos 4) by A30, A33, AMI_3:10, SCMPDS_2:46 ; ::_thesis: for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) )
assume A59: i >= 8 ; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i)
then A60: i > 7 by XXREAL_0:2;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by SCMPDS_5:35
.= (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by A58, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by SCMPDS_5:40
.= (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos i) by A33, A60, AMI_3:10, SCMPDS_2:46
.= (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A51, A59 ; ::_thesis: verum
end;
supposeA61: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) > 0 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = 0 & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
hence (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP = (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . GBP by SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . GBP by SCMPDS_5:42
.= 0 by A42, A43, AMI_3:10, SCMPDS_2:48 ;
::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) = (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 1) by A61, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A56, A43, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) = (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 2) by A61, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A41, A43, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) = (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 4) by A61, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 4) by SCMPDS_5:42
.= s . (intpos 4) by A40, A43, AMI_3:10, SCMPDS_2:48 ; ::_thesis: for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) )
assume A62: i >= 8 ; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i)
then A63: i > 5 by XXREAL_0:2;
 i > 3 by A62, XXREAL_0:2;
then A64: intpos i <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by A61, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos i) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos i) by A43, A63, AMI_3:10, SCMPDS_2:48
.= (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos i) by A64, SCMPDS_2:48
.= (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A51, A62 ; ::_thesis: verum
end;
end;
end;
hence  ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . GBP = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 4) = s . (intpos 4) ) by SCMPDS_5:35; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) ) & ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
thus  for i being Element of NAT st i >= 8 holds
(IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos i) = s . (intpos i) by A57; ::_thesis: ( ( s . (intpos md) > s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 ) ) & ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) ) )
A65: (IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)) . (intpos 7) = (Exec ((AddTo (GBP,7,(- 1))),(Exec (((GBP,5) := (GBP,7)),s)))) . (intpos 7) by SCMPDS_5:42
.= ((Exec (((GBP,5) := (GBP,7)),s)) . (intpos 7)) + (- 1) by A10, SCMPDS_2:48
.= (s . (intpos 7)) - 1 by A19 ;
A66: (IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)) . (intpos 7) = (Exec (((GBP,6) := ((intpos 2),0)),(IExec ((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))),P,s)))) . (intpos 7) by SCMPDS_5:41
.= (s . (intpos 7)) - 1 by A65, A15, AMI_3:10, SCMPDS_2:47 ;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 7) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 7) by SCMPDS_5:41
.= (s . (intpos 7)) - 1 by A66, A22, AMI_3:10, SCMPDS_2:50 ;
then A67: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 7) = (s . (intpos 7)) - 1 by SCMPDS_5:15;
(IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (intpos 3) = (Exec ((SubFrom (GBP,6,(intpos 3),0)),(IExec (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))),P,s)))) . (intpos 3) by SCMPDS_5:41
.= me by A20, A22, AMI_3:10, SCMPDS_2:50 ;
then A68: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (intpos 3) = s . (intpos 3) by A2, SCMPDS_5:15;
A69: DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) = intpos (0 + 3) by A32, SCMP_GCD:1;
XX: (Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) = (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),6)) by SCMPDS_5:15
.= (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . (DataLoc (((IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s)) . GBP),6)) by SCMPDS_5:15 ;
hereby  ::_thesis: ( s . (intpos md) <= s . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) ) )
A70: intpos 7 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
assume  s . (intpos md) > s . (intpos me) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (s . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 )
then B71: (s . (intpos md)) - (s . (intpos me)) > 0 by XREAL_1:50;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by A55, XX, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 7) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by A43, AMI_3:10, SCMPDS_2:48
.= (s . (intpos 7)) - 1 by A67, A70, SCMPDS_2:48 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1 )
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by A55, XX, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 3) by SCMPDS_5:42
.= (Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by A43, AMI_3:10, SCMPDS_2:48
.= (s . (intpos 3)) + 1 by A69, A68, SCMPDS_2:48 ; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (s . (intpos 7)) - 1
A72: intpos 5 <> DataLoc (((Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))) . GBP),3) by A32, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by SCMPDS_5:35
.= (IExec (((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by A55, XX, B71, SCMPDS_6:73
.= (Exec ((AddTo (GBP,5,(- 1))),(Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))))) . (intpos 5) by SCMPDS_5:42
.= ((Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5)) + (- 1) by A43, SCMPDS_2:48
.= ((Exec ((AddTo (GBP,3,1)),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5)) - 1
.= (s . (intpos 7)) - 1 by A37, A72, SCMPDS_2:48 ; ::_thesis: verum
end;
hereby  ::_thesis: verum
assume  s . (intpos md) <= s . (intpos me) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) )
then B73: 0 >= (s . (intpos md)) - (s . (intpos me)) by XREAL_1:47;
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 7) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by SCMPDS_5:35
.= (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by A55, XX, B73, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 7) by SCMPDS_5:40
.= 0 by A33, SCMPDS_2:46 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = s . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7) )
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 3) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by SCMPDS_5:35
.= (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by A55, XX, B73, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 3) by SCMPDS_5:40
.= s . (intpos 3) by A33, A68, AMI_3:10, SCMPDS_2:46 ; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = s . (intpos 7)
thus (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by SCMPDS_5:35
.= (IExec ((Load ((GBP,7) := 0)),P,(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by A55, XX, B73, SCMPDS_6:74
.= (Exec (((GBP,7) := 0),(Initialize (IExec ((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))),P,s))))) . (intpos 5) by SCMPDS_5:40
.= s . (intpos 7) by A37, A33, AMI_3:10, SCMPDS_2:46 ; ::_thesis: verum
end;
end;

Lm15:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for m3, md being Element of NAT st s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for m3, md being Element of NAT st s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for m3, md being Element of NAT st s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) )
let m3, md be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 7) > 0 & (s . (intpos 3)) + (s . (intpos 7)) = m3 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 implies ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 7) > 0 and
A3: (s . (intpos 3)) + (s . (intpos 7)) = m3 and
A4: s . (intpos 3) >= 8 and
A5: s . (intpos 2) = md and
A6: md >= 8 ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) )
 s . (intpos 7) >= 1 + 0 by A2, INT_1:7;
then reconsider m7 = (s . (intpos 7)) - 1 as Element of NAT by INT_1:3, XREAL_1:48;
A7: s . (intpos 7) = m7 + 1 ;
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 7) = $1 + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) );
A8: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A9: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_t_._GBP_=_0_&_t_._(intpos_7)_=_(k_+_1)_+_1_&_(t_._(intpos_3))_+_(t_._(intpos_7))_=_m3_&_t_._(intpos_3)_>=_8_&_t_._(intpos_2)_=_md_holds_
(_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._GBP_=_0_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_1)_=_t_._(intpos_1)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_7)_=_0_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_2)_=_t_._(intpos_2)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_4)_=_t_._(intpos_4)_&_(_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)_)_&_ex_m5,_mE3_being_Element_of_NAT_st_
(_m5_=_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_5)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_3)_=_mE3_&_mE3_+_m5_=_m3_&_m5_<=_t_._(intpos_7)_&_(_for_i_being_Element_of_NAT_st_t_._(intpos_3)_<=_i_&_i_<_mE3_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_md)_>_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_)_&_(_m5_=_0_or_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_md)_<=_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_mE3)_)_)_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 7) = (k + 1) + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . (intpos 1) = b2 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . (intpos 2) = b2 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,b2)) . (intpos 4) = b2 . (intpos 4) & ( for i being Element of NAT st b4 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b3,i)) . (intpos b4) = i . (intpos b4) ) & ex m5, mE3 being Element of NAT st
( b4 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b5 & b5 + b4 = m3 & b4 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b6 & b6 < b5 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b6) ) & ( b4 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & t . (intpos 7) = (k + 1) + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b3 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,i)) . (intpos b3) = i . (intpos b3) ) & ex m5, mE3 being Element of NAT st
( b3 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b4 & b4 + b3 = m3 & b3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b5 & b5 < b4 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) & ( b3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b4) ) ) ) )
set b = DataLoc ((t . GBP),7);
assume that
A10: t . GBP = 0 and
A11: t . (intpos 7) = (k + 1) + 1 and
A12: (t . (intpos 3)) + (t . (intpos 7)) = m3 and
A13: t . (intpos 3) >= 8 and
A14: t . (intpos 2) = md ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b3 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,i)) . (intpos b3) = i . (intpos b3) ) & ex m5, mE3 being Element of NAT st
( b3 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b4 & b4 + b3 = m3 & b3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b5 & b5 < b4 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) & ( b3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b4) ) ) )
reconsider me = t . (intpos 3) as Element of NAT by A13, INT_1:3;
set It = IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t);
set IT = Q;
A24: ( t . (intpos md) > t . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = (t . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 7)) - 1 ) ) by A6, A10, A13, A14, Lm14;
A25: DataLoc ((t . GBP),7) = intpos (0 + 7) by A10, SCMP_GCD:1;
A26: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_3)_>=_8_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),7))_>_0_holds_
(_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_closed_on_v,V_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),7))_<_v_._(DataLoc_((t_._GBP),7))_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_3)_>=_8_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A27: v . (intpos 3) >= 8 and
A28: v . (intpos 2) = t . (intpos 2) and
A29: v . GBP = t . GBP and
A30: v . (DataLoc ((t . GBP),7)) > 0 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
reconsider ME = v . (intpos 3) as Element of NAT by A27, INT_1:3;
A38: ME = v . (intpos 3) ;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP by A6, A10, A14, A27, A28, A29, Lm14; ::_thesis: ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A39: ( v . (intpos md) > v . (intpos ME) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 7) = (v . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 5) = (v . (intpos 7)) - 1 ) ) by A6, A10, A14, A27, A28, A29, Lm14;
hereby  ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) > v . (intpos ME) or v . (intpos md) <= v . (intpos ME) ) ;
supposeA40: v . (intpos md) > v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) by A25, A39, XREAL_1:146; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
A41: (v . (intpos 3)) + 1 > v . (intpos 3) by XREAL_1:29;
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 by A6, A10, A14, A27, A28, A29, A40, Lm14;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A27, A41, XXREAL_0:2; ::_thesis: verum
end;
supposeA42: v . (intpos md) <= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) by A6, A10, A14, A25, A27, A28, A29, A30, Lm14; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A6, A10, A14, A27, A28, A29, A42, Lm14; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A6, A10, A14, A27, A28, A29, A38, Lm14; ::_thesis: verum
end;
A43: ( t . (intpos md) <= t . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 5) = t . (intpos 7) ) ) by A6, A10, A13, A14, Lm14;
A44: me = t . (intpos 3) ;
then A45: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP = 0 by A6, A10, A13, A14, Lm14;
then A46: DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7) = intpos (0 + 7) by SCMP_GCD:1;
A47: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A10, A13, A14, A44, Lm14;
A48: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) by A6, A10, A13, A14, A44, Lm14;
A49: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A10, A13, A14, A44, Lm14;
B47: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 1) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 1) by SCMPDS_5:15;
B48: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 4) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 4) by SCMPDS_5:15;
B49: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 2) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 2) by SCMPDS_5:15;
percases ( t . (intpos md) > t . (intpos me) or t . (intpos md) <= t . (intpos me) ) ;
supposeA50: t . (intpos md) > t . (intpos me) ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b3 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,i)) . (intpos b3) = i . (intpos b3) ) & ex m5, mE3 being Element of NAT st
( b3 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b4 & b4 + b3 = m3 & b3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b5 & b5 < b4 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) & ( b3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b4) ) ) )
then  t . (intpos 3) < (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) by A24, XREAL_1:29;
then A51: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) >= 8 by A13, XXREAL_0:2;
A52: ((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3)) + ((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7)) = m3 by A12, A24, A50;
X1: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 3) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) by SCMPDS_5:15;
X3: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos 7) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) by SCMPDS_5:15;
consider m5, mE3 being Element of NAT  such that
A53: m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos 5) and
A54: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos 3) = mE3 and
A55: mE3 + m5 = m3 and
A56: m5 <= (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) and
A57: for i being Element of NAT st (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos i) and
A58: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . (intpos mE3) ) by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, X1, X2, X3;
A59: IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t) = IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)))) by A11, A13, A25, A26, Th6;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, X1, X2, X3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A9, A11, A12, A14, A45, A47, B47, A49, B49, A24, A43, A51, A52, A59, X1, X2, X3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, A59, X1, X2, X3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, A59, X1, X2, X3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) by A9, A11, A12, A14, A45, A49, B49, A48, B48, A24, A43, A51, A52, A59, X1, X2, X3; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
A60: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A62: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
 (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (intpos i) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos i) by SCMPDS_5:15;
hence (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos i) by A9, A11, A12, A14, A45, A49, B49, A24, A43, A51, A52, A59, X1, X2, X3, A62
.= t . (intpos i) by A6, A10, A13, A14, A44, A62, Lm14 ;
::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take m5 = m5; ::_thesis: ex mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take mE3 = mE3; ::_thesis: ( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) by A11, A13, A25, A26, A53, Th6; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 by A11, A13, A25, A26, A54, Th6; ::_thesis: ( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  mE3 + m5 = m3 by A55; ::_thesis: ( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 <= t . (intpos 7) by A24, A50, A56, XREAL_1:146, XXREAL_0:2; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
A63: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = (t . (intpos 3)) + 1 by A6, A10, A13, A14, A50, Lm14;
hereby  ::_thesis: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) )
let i be Element of NAT ; ::_thesis: ( t . (intpos 3) <= i & i < mE3 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos b1) )
assume that
A64: t . (intpos 3) <= i and
A65: i < mE3 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos b1)
percases ( i = t . (intpos 3) or i <> t . (intpos 3) ) ;
supposeA66: i = t . (intpos 3) ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos b1)
 (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos me) = t . (intpos me) by A13, A60;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) by A6, A50, A60, A66; ::_thesis: verum
end;
suppose i <> t . (intpos 3) ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos b1)
then  t . (intpos 3) < i by A64, XXREAL_0:1;
then  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) <= i by A63, INT_1:7;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) by A59, A57, A65; ::_thesis: verum
end;
end;
end;
thus  ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) by A59, A58; ::_thesis: verum
end;
supposeA67: t . (intpos md) <= t . (intpos me) ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 2) = b1 . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,b1)) . (intpos 4) = b1 . (intpos 4) & ( for i being Element of NAT st b3 >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),b2,i)) . (intpos b3) = i . (intpos b3) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 5) = b3 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos 3) = b4 & b4 + b3 = m3 & b3 <= m5 . (intpos 7) & ( for i being Element of NAT st m5 . (intpos 3) <= b5 & b5 < b4 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b5) ) & ( b3 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),mE3,m5)) . (intpos b4) ) ) )
A68: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._x_=_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),Q,t))_._x
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . x = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . x
X1: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) by SCMPDS_5:15;
thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . x by A11, A13, A25, A26, Th6
.= (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . x by A43, A46, A67, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 by A45; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A47, A68; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 by A43, A67, A68; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A49, A68; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) by A48, A68; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
A69: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A71: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos i) by A68
.= t . (intpos i) by A6, A10, A13, A14, A44, A71, Lm14 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take m5 = (k + 1) + 1; ::_thesis: ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take mE3 = me; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 by A11, A43, A67, A68; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 by A43, A67, A68; ::_thesis: ( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  mE3 + m5 = m3 by A11, A12; ::_thesis: ( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 <= t . (intpos 7) by A11; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ; ::_thesis: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) )
 (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos me) = t . (intpos me) by A13, A69;
hence  ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) by A6, A67, A69; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS_st_t_._GBP_=_0_&_t_._(intpos_7)_=_0_+_1_&_(t_._(intpos_3))_+_(t_._(intpos_7))_=_m3_&_t_._(intpos_3)_>=_8_&_t_._(intpos_2)_=_md_holds_
(_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._GBP_=_0_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_1)_=_t_._(intpos_1)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_7)_=_0_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_2)_=_t_._(intpos_2)_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_4)_=_t_._(intpos_4)_&_(_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)_)_&_ex_m5_being_Element_of_NAT_ex_mE3_being_Element_of_NAT_st_
(_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_5)_=_m5_&_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_3)_=_mE3_&_mE3_+_m5_=_m3_&_m5_<=_t_._(intpos_7)_&_(_for_i_being_Element_of_NAT_st_t_._(intpos_3)_<=_i_&_i_<_mE3_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_md)_>_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_)_&_(_m5_=_0_or_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_md)_<=_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_mE3)_)_)_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 7) = 0 + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md holds
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & t . (intpos 7) = 0 + 1 & (t . (intpos 3)) + (t . (intpos 7)) = m3 & t . (intpos 3) >= 8 & t . (intpos 2) = md implies ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) ) )
set b = DataLoc ((t . GBP),7);
assume that
A72: t . GBP = 0 and
A73: t . (intpos 7) = 0 + 1 and
A74: (t . (intpos 3)) + (t . (intpos 7)) = m3 and
A75: t . (intpos 3) >= 8 and
A76: t . (intpos 2) = md ; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
reconsider me = t . (intpos 3) as Element of NAT by A75, INT_1:3;
set It = IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t);
set IT = Q;
A86: ( t . (intpos md) > t . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) = (t . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = (t . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 5) = (t . (intpos 7)) - 1 ) ) by A6, A72, A75, A76, Lm14;
A87: DataLoc ((t . GBP),7) = intpos (0 + 7) by A72, SCMP_GCD:1;
A88: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_3)_>=_8_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),7))_>_0_holds_
(_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_closed_on_v,V_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(DataLoc_((t_._GBP),7))_<_v_._(DataLoc_((t_._GBP),7))_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_3)_>=_8_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 3) >= 8 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),7)) > 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A89: v . (intpos 3) >= 8 and
A90: v . (intpos 2) = t . (intpos 2) and
A91: v . GBP = t . GBP and
A92: v . (DataLoc ((t . GBP),7)) > 0 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
reconsider ME = v . (intpos 3) as Element of NAT by A89, INT_1:3;
A93: ME = v . (intpos 3) ;
A96: v . (intpos ME) = v . (intpos ME) ;
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP by A6, A72, A76, A89, A90, A91, Lm14, A96; ::_thesis: ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A101: ( v . (intpos md) > v . (intpos ME) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 7) = (v . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 5) = (v . (intpos 7)) - 1 ) ) by A6, A72, A76, A89, A90, A91, Lm14;
hereby  ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) > v . (intpos ME) or v . (intpos md) <= v . (intpos ME) ) ;
supposeA102: v . (intpos md) > v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) by A87, A101, XREAL_1:146; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
A103: (v . (intpos 3)) + 1 > v . (intpos 3) by XREAL_1:29;
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 by A6, A72, A76, A89, A90, A91, A102, Lm14;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A89, A103, XXREAL_0:2; ::_thesis: verum
end;
supposeA104: v . (intpos md) <= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((t . GBP),7)) < v . (DataLoc ((t . GBP),7)) by A6, A72, A76, A87, A89, A90, A91, A92, Lm14; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A6, A72, A76, A89, A90, A91, A104, Lm14; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A6, A72, A76, A89, A90, A91, A93, Lm14; ::_thesis: verum
end;
A105: ( t . (intpos md) <= t . (intpos me) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 7) = 0 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 3) = t . (intpos 3) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 5) = t . (intpos 7) ) ) by A6, A72, A75, A76, Lm14;
A106: me = t . (intpos 3) ;
then A107: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP = 0 by A6, A72, A75, A76, Lm14;
then A108: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) = 0 by A73, A86, A105, SCMP_GCD:1;
A109: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._x_=_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),Q,t))_._x
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . x = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . x
X1: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . GBP = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (DataLoc (((IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . GBP),7)) by SCMPDS_5:15;
thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . x = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,(Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))))) . x by A73, A75, A87, A88, Th6
.= (Initialize (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t))) . x by A108, X1, X2, SCMPDS_8:23
.= (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . GBP = 0 by A107; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 1) = t . (intpos 1) by A6, A72, A75, A76, A106, Lm14;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A109; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 7) = 0 by A73, A86, A105, A109; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 2) = t . (intpos 2) by A6, A72, A75, A76, A106, Lm14;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 2) = t . (intpos 2) by A109; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos 4) = t . (intpos 4) by A6, A72, A75, A76, A106, Lm14;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 4) = t . (intpos 4) by A109; ::_thesis: ( ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ) & ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ) )
A110: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume A111: i >= 8 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),Q,t)) . (intpos i) by A109
.= t . (intpos i) by A6, A72, A75, A76, A106, A111, Lm14 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) = t . (intpos i) ; ::_thesis: ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
A113: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos me) = t . (intpos me) by A75, A110;
hereby  ::_thesis: verum
percases ( t . (intpos md) > t . (intpos me) or t . (intpos md) <= t . (intpos me) ) ;
supposeA114: t . (intpos md) > t . (intpos me) ; ::_thesis: ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take m5 = 0 ; ::_thesis: ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take mE3 = m3; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 by A73, A86, A109, A114; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 by A73, A74, A86, A109, A114; ::_thesis: ( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  mE3 + m5 = m3 ; ::_thesis: ( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 <= t . (intpos 7) by A73; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
hereby  ::_thesis: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) )
let i be Element of NAT ; ::_thesis: ( t . (intpos 3) <= i & i < mE3 implies (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) )
assume that
A115: t . (intpos 3) <= i and
A116: i < mE3 ; ::_thesis: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i)
 i <= me by A73, A74, A116, NAT_1:13;
then  i = t . (intpos 3) by A115, XXREAL_0:1;
hence  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) by A6, A110, A113, A114; ::_thesis: verum
end;
thus  ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) ; ::_thesis: verum
end;
supposeA117: t . (intpos md) <= t . (intpos me) ; ::_thesis: ex m5 being Element of NAT ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take m5 = 1; ::_thesis: ex mE3 being Element of NAT st
( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
take mE3 = me; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 5) = m5 by A73, A105, A109, A117; ::_thesis: ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos 3) = mE3 by A105, A109, A117; ::_thesis: ( mE3 + m5 = m3 & m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  mE3 + m5 = m3 by A73, A74; ::_thesis: ( m5 <= t . (intpos 7) & ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  m5 <= t . (intpos 7) by A73; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) )
thus  for i being Element of NAT st t . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos i) ; ::_thesis: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) )
thus  ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),Q,t)) . (intpos mE3) ) by A6, A110, A113, A117; ::_thesis: verum
end;
end;
end;
end;
then A118: S1[ 0 ] ;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A118, A8);
hence  ( (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 7) = 0 & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 4) = s . (intpos 4) & ( for i being Element of NAT st i >= 8 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) = s . (intpos i) ) & ex m5, mE3 being Element of NAT st
( m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 5) & (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos 3) = mE3 & mE3 + m5 = m3 & m5 <= s . (intpos 7) & ( for i being Element of NAT st s . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos i) ) & ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,s)) . (intpos mE3) ) ) ) by A1, A3, A4, A5, A7; ::_thesis: verum
end;

Lm16:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md being Element of NAT st s . GBP = 0 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md being Element of NAT st s . GBP = 0 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md being Element of NAT st s . GBP = 0 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 holds
( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P )
let md be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 3) >= 8 & s . (intpos 2) = md & md >= 8 implies ( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P ) )
set b = DataLoc ((s . GBP),7);
assume that
A1: s . GBP = 0 and
A2: s . (intpos 3) >= 8 and
A3: s . (intpos 2) = md and
A4: md >= 8 ; ::_thesis: ( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P )
A5: DataLoc ((s . GBP),7) = intpos (0 + 7) by A1, SCMP_GCD:1;
now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_3)_>=_8_&_v_._(intpos_2)_=_s_._(intpos_2)_&_v_._GBP_=_s_._GBP_&_v_._(DataLoc_((s_._GBP),7))_>_0_holds_
(_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._GBP_=_v_._GBP_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_closed_on_v,V_&_(((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))_is_halting_on_v,V_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(DataLoc_((s_._GBP),7))_<_v_._(DataLoc_((s_._GBP),7))_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_3)_>=_8_&_(IExec_(((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 3) >= 8 & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),7)) > 0 holds
( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 3) >= 8 & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),7)) > 0 implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A6: v . (intpos 3) >= 8 and
A7: v . (intpos 2) = s . (intpos 2) and
A8: v . GBP = s . GBP and
A9: v . (DataLoc ((s . GBP),7)) > 0 ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
reconsider ME = v . (intpos 3) as Element of NAT by A6, INT_1:3;
A17: ME = v . (intpos 3) ;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . GBP = v . GBP by A1, A3, A4, A6, A7, A8, Lm14; ::_thesis: ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_closed_on v,V & (((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))) is_halting_on v,V ) by SCMPDS_6:20, SCMPDS_6:21; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) )
A18: ( v . (intpos md) > v . (intpos ME) implies ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 7) = (v . (intpos 7)) - 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 5) = (v . (intpos 7)) - 1 ) ) by A1, A3, A4, A6, A7, A8, Lm14;
hereby  ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2)
percases ( v . (intpos md) > v . (intpos ME) or v . (intpos md) <= v . (intpos ME) ) ;
supposeA19: v . (intpos md) > v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) by A5, A18, XREAL_1:146; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
A20: (v . (intpos 3)) + 1 > v . (intpos 3) by XREAL_1:29;
 (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) = (v . (intpos 3)) + 1 by A1, A3, A4, A6, A7, A8, A19, Lm14;
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A6, A20, XXREAL_0:2; ::_thesis: verum
end;
supposeA21: v . (intpos md) <= v . (intpos ME) ; ::_thesis: ( (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) & (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 )
hence  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (DataLoc ((s . GBP),7)) < v . (DataLoc ((s . GBP),7)) by A1, A3, A4, A5, A6, A7, A8, A9, Lm14; ::_thesis: (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 3) >= 8 by A1, A3, A4, A6, A7, A8, A21, Lm14; ::_thesis: verum
end;
end;
end;
thus  (IExec (((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))),V,v)) . (intpos 2) = v . (intpos 2) by A1, A3, A4, A6, A7, A8, A17, Lm14; ::_thesis: verum
end;
hence  ( while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on s,P & while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on s,P ) by A2, Th6; ::_thesis: verum
end;

Lm17:  card (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))) = 29
proof
thus  card (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))) = (card ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))))) + (card (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))) by AFINSQ_1:17
.= (11 + 11) + (card (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))) by Lm11, Lm13, AFINSQ_1:17
.= 22 + ((card (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))) + 1) by SCMPDS_6:75
.= 22 + (((card ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1)))) + 1) + 1) by SCMP_GCD:4
.= 22 + ((((card (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2))))) + 1) + 1) + 1) by SCMP_GCD:4
.= 22 + (((((card ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6)))) + 1) + 1) + 1) + 1) by SCMP_GCD:4
.= 22 + ((((((card (((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0)))) + 1) + 1) + 1) + 1) + 1) by SCMP_GCD:4
.= 22 + (((((2 + 1) + 1) + 1) + 1) + 1) by SCMP_GCD:5
.= 29 ; ::_thesis: verum
end;

Lm18:  card (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))) = 31
proof
thus  card (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))) = 29 + 2 by Lm17, SCMPDS_8:17
.= 31 ; ::_thesis: verum
end;

begin

theorem Th12: :: SCPQSORT:9
card Partition = 38
proof
thus  card Partition = (card ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0)))) + 1 by SCMP_GCD:4
.= ((card (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0)))) + 1) + 1 by SCMP_GCD:4
.= (((card ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))))) + 1) + 1) + 1 by SCMP_GCD:4
.= ((((card (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1)))) + (card (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))))) + 1) + 1) + 1 by AFINSQ_1:17
.= (((4 + 31) + 1) + 1) + 1 by Lm18, Th4
.= 38 ; ::_thesis: verum
end;

Lm19:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for m3, m4 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 3) = m3 & s . (intpos 4) = m4 & m3 > 6 & m4 > 6 holds
( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for m3, m4 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 3) = m3 & s . (intpos 4) = m4 & m3 > 6 & m4 > 6 holds
( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for m3, m4 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 3) = m3 & s . (intpos 4) = m4 & m3 > 6 & m4 > 6 holds
( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
let m3, m4 be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 3) = m3 & s . (intpos 4) = m4 & m3 > 6 & m4 > 6 implies ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 5) > 0 and
A3: s . (intpos 3) = m3 and
A4: s . (intpos 4) = m4 and
A5: m3 > 6 and
A6: m4 > 6 ; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
A7: m4 > 3 by A6, XXREAL_0:2;
set x = intpos m3;
set y = intpos m4;
set t0 = Initialize s;
set t1 = IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s));
set t2 = IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s));
set t3 = IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s));
set t4 = IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s));
set t5 = IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s));
set t6 = Exec (((GBP,6) := ((intpos 4),0)),(Initialize s));
A8: (Initialize s) . GBP = 0 by A1, SCMPDS_5:15;
then A9: DataLoc (((Initialize s) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
then A10: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . GBP = 0 by A8, AMI_3:10, SCMPDS_2:47;
A11: (Initialize s) . (intpos 4) = m4 by A4, SCMPDS_5:15;
then A12: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 4) = m4 by A9, AMI_3:10, SCMPDS_2:47;
then A13: DataLoc (((Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 4)),0) = intpos (m4 + 0) by SCMP_GCD:1;
A14: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . GBP = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . GBP by SCMPDS_5:42
.= 0 by A6, A10, A13, AMI_3:10, SCMPDS_2:47 ;
A15: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 6) = (Initialize s) . (DataLoc (((Initialize s) . (intpos 4)),0)) by A9, SCMPDS_2:47
.= (Initialize s) . (intpos (m4 + 0)) by A11, SCMP_GCD:1
.= s . (intpos m4) by SCMPDS_5:15 ;
(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 6) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 6) by SCMPDS_5:42
.= s . (intpos m4) by A6, A15, A13, AMI_3:10, SCMPDS_2:47 ;
then A16: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . GBP),6)) = s . (intpos m4) by A14, SCMP_GCD:1;
 (Initialize s) . (intpos 3) = m3 by A3, SCMPDS_5:15;
then A17: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 3) = m3 by A9, AMI_3:10, SCMPDS_2:47;
A18: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 3) by SCMPDS_5:42
.= m3 by A7, A17, A13, AMI_3:10, SCMPDS_2:47 ;
then A19: DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) = intpos (m3 + 0) by SCMP_GCD:1;
A20: (Initialize s) . (intpos m3) = s . (intpos m3) by SCMPDS_5:15;
A21: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos m4) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos m4) by SCMPDS_5:42
.= (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (DataLoc (((Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 3)),0)) by A13, SCMPDS_2:47
.= (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos (m3 + 0)) by A17, SCMP_GCD:1
.= s . (intpos m3) by A5, A20, A9, AMI_3:10, SCMPDS_2:47 ;
A22: now__::_thesis:_(IExec_(((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6))),P,(Initialize_s)))_._(intpos_m4)_=_s_._(intpos_m3)
percases ( intpos m4 <> DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) or intpos m4 = DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) ) ;
supposeA23: intpos m4 <> DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) ; ::_thesis: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m4) = s . (intpos m3)
thus (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m4) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A21, A23, SCMPDS_2:47 ; ::_thesis: verum
end;
supposeA24: intpos m4 = DataLoc (((IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 3)),0) ; ::_thesis: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m4) = s . (intpos m3)
thus (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m4) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A19, A16, A24, SCMPDS_2:47 ; ::_thesis: verum
end;
end;
end;
 (Initialize s) . (intpos 2) = s . (intpos 2) by SCMPDS_5:15;
then A25: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 2) = s . (intpos 2) by A9, AMI_3:10, SCMPDS_2:47;
A26: m4 > 2 by A6, XXREAL_0:2;
A27: m3 > 5 by A5, XXREAL_0:2;
A28: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . GBP = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A5, A14, A19, AMI_3:10, SCMPDS_2:47 ;
then A29: DataLoc (((IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A30: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A28, A29, AMI_3:10, SCMPDS_2:48 ;
then A31: GBP <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by AMI_3:10, SCMP_GCD:1;
A32: m3 > 3 by A5, XXREAL_0:2;
then A33: intpos m3 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A34: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 3) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= m3 by A32, A18, A19, AMI_3:10, SCMPDS_2:47 ;
A35: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 3) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= m3 by A34, A29, AMI_3:10, SCMPDS_2:48 ;
A36: DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) = intpos (0 + 3) by A30, SCMP_GCD:1;
A37: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= m3 + 1 by A35, A36, SCMPDS_2:48 ;
A38: m3 > 2 by A5, XXREAL_0:2;
A39: m4 > 5 by A6, XXREAL_0:2;
A40: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 2) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A25, A13, A26, AMI_3:10, SCMPDS_2:47 ;
A41: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos m3) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos m3) by SCMPDS_5:41
.= s . (intpos m4) by A19, A16, SCMPDS_2:47 ;
A42: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos m3) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos m3) by SCMPDS_5:41
.= s . (intpos m4) by A27, A41, A29, AMI_3:10, SCMPDS_2:48 ;
A43: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos m3) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos m3) by SCMPDS_5:41
.= s . (intpos m4) by A42, A33, SCMPDS_2:48 ;
A44: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos m4) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A39, A22, A29, AMI_3:10, SCMPDS_2:48 ;
A45: intpos m4 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A7, A30, AMI_3:10, SCMP_GCD:1;
A46: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos m4) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A44, A45, SCMPDS_2:48 ;
A47: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 2) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A40, A19, A38, AMI_3:10, SCMPDS_2:47 ;
A48: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 2) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A47, A29, AMI_3:10, SCMPDS_2:48 ;
A49: intpos 2 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A50: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A48, A49, SCMPDS_2:48 ;
A51: m4 > 1 by A6, XXREAL_0:2;
A52: DataLoc ((s . GBP),5) = intpos (0 + 5) by A1, SCMP_GCD:1;
 (Initialize s) . (intpos 5) = s . (intpos 5) by SCMPDS_5:15;
then A53: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 5) = s . (intpos 5) by A9, AMI_3:10, SCMPDS_2:47;
A54: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 5) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 5) by SCMPDS_5:42
.= s . (intpos 5) by A39, A53, A13, AMI_3:10, SCMPDS_2:47 ;
A55: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 5) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= s . (intpos 5) by A27, A54, A19, AMI_3:10, SCMPDS_2:47 ;
A56: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 5) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= ((IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 5)) + (- 2) by A29, SCMPDS_2:48
.= (s . (intpos 5)) - 2 by A55 ;
A57: intpos 5 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A58: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 5)) - 2 by A56, A57, SCMPDS_2:48 ;
 (Initialize s) . (intpos 1) = s . (intpos 1) by SCMPDS_5:15;
then A59: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos 1) = s . (intpos 1) by A9, AMI_3:10, SCMPDS_2:47;
A60: m3 > 1 by A5, XXREAL_0:2;
A61: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A30, A31, SCMPDS_2:48 ;
then A62: GBP <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by AMI_3:10, SCMP_GCD:1;
A63: m4 > 4 by A6, XXREAL_0:2;
then A64: intpos m4 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
A65: m3 > 4 by A5, XXREAL_0:2;
then A66: intpos m3 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
A67: intpos 4 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A68: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 4) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 4) by SCMPDS_5:42
.= m4 by A63, A12, A13, AMI_3:10, SCMPDS_2:47 ;
A69: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 4) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= m4 by A65, A68, A19, AMI_3:10, SCMPDS_2:47 ;
A70: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 4) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= m4 by A69, A29, AMI_3:10, SCMPDS_2:48 ;
A71: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= m4 by A70, A67, SCMPDS_2:48 ;
A72: intpos 3 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
A73: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos 1) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A59, A13, A51, AMI_3:10, SCMPDS_2:47 ;
A74: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos 1) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A73, A19, A60, AMI_3:10, SCMPDS_2:47 ;
A75: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos 1) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A74, A29, AMI_3:10, SCMPDS_2:48 ;
A76: intpos 1 <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
A77: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 1) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A75, A76, SCMPDS_2:48 ;
A78: intpos 1 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . GBP = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A61, A62, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . GBP = 0 by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 1) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A77, A78, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
A79: intpos 2 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 2) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A50, A79, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos m3) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos m3) by SCMPDS_5:41
.= s . (intpos m4) by A43, A66, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m3) = s . (intpos m4) by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos m4) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos m4) by SCMPDS_5:41
.= s . (intpos m3) by A46, A64, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos m4) = s . (intpos m3) by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
A80: DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) = intpos (0 + 4) by A61, SCMP_GCD:1;
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 3) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= m3 + 1 by A37, A72, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 3) = (s . (intpos 3)) + 1 by A2, A3, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 & (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 4) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= ((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4)) + (- 1) by A80, SCMPDS_2:48
.= (s . (intpos 4)) - 1 by A4, A71 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 4) = (s . (intpos 4)) - 1 by A2, A52, SCMPDS_6:83; ::_thesis: ( (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 & ( for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
A81: intpos 5 <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
(IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos 5) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 5)) - 2 by A58, A81, SCMPDS_2:48 ;
hence  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos 5) = (s . (intpos 5)) - 2 by A2, A52, SCMPDS_6:83; ::_thesis: for i being Element of NAT st i >= 8 & i <> m3 & i <> m4 holds
(IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i)
A82: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(Exec_(((GBP,6)_:=_((intpos_4),0)),(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A83: i >= 8 and
 i <> m3 and
 i <> m4 ; ::_thesis: (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos i) = s . (intpos i)
 i > 6 by A83, XXREAL_0:2;
hence (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos i) = (Initialize s) . (intpos i) by A9, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by SCMPDS_5:15 ;
::_thesis: verum
end;
A84: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A85: i >= 8 and
A86: i <> m3 and
A87: i <> m4 ; ::_thesis: (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos i) = s . (intpos i)
thus (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos i) = (Exec ((((intpos 4),0) := ((intpos 3),0)),(Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,6) := ((intpos 4),0)),(Initialize s))) . (intpos i) by A13, A87, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A82, A85, A86, A87 ; ::_thesis: verum
end;
A88: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_(((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A89: i >= 8 and
A90: i <> m3 and
A91: i <> m4 ; ::_thesis: (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos i) = s . (intpos i)
thus (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos i) = (Exec ((((intpos 3),0) := (GBP,6)),(IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))),P,(Initialize s))) . (intpos i) by A19, A90, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A84, A89, A90, A91 ; ::_thesis: verum
end;
A92: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2)))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A93: i >= 8 and
A94: i <> m3 and
A95: i <> m4 ; ::_thesis: (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos i) = s . (intpos i)
A96: i > 5 by A93, XXREAL_0:2;
thus (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos i) = (Exec ((AddTo (GBP,5,(- 2))),(IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))),P,(Initialize s))) . (intpos i) by A29, A96, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by A88, A93, A94, A95 ; ::_thesis: verum
end;
A97: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A98: i >= 8 and
A99: i <> m3 and
A100: i <> m4 ; ::_thesis: (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i)
 i > 3 by A98, XXREAL_0:2;
then A101: intpos i <> DataLoc (((IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . GBP),3) by A30, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = (Exec ((AddTo (GBP,3,1)),(IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))),P,(Initialize s))) . (intpos i) by A101, SCMPDS_2:48
.= s . (intpos i) by A92, A98, A99, A100 ; ::_thesis: verum
end;
A102: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m3_&_i_<>_m4_holds_
(IExec_((((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1)))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A103: i >= 8 and
A104: i <> m3 and
A105: i <> m4 ; ::_thesis: (IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos i) = s . (intpos i)
 i > 4 by A103, XXREAL_0:2;
then A106: intpos i <> DataLoc (((IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP),4) by A61, AMI_3:10, SCMP_GCD:1;
thus (IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos i) = (Exec ((AddTo (GBP,4,(- 1))),(IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) by A106, SCMPDS_2:48
.= s . (intpos i) by A97, A103, A104, A105 ; ::_thesis: verum
end;
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m3 & i <> m4 implies (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume that
A107: i >= 8 and
A108: i <> m3 and
A109: i <> m4 ; ::_thesis: (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = s . (intpos i)
thus (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize s))) . (intpos i) = (IExec ((((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))),P,(Initialize s))) . (intpos i) by A2, A52, SCMPDS_6:83
.= s . (intpos i) by A102, A107, A108, A109 ; ::_thesis: verum
end;
end;

Lm20:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, m3 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= 8 & md <= m3 holds
( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, m3 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= 8 & md <= m3 holds
( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md, m3 being Element of NAT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= 8 & md <= m3 holds
( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
let md, m3 be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= 8 & md <= m3 implies ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) ) )
I: Initialize s = s by MEMSTR_0:44;
assume that
A1: s . GBP = 0 and
A2: s . (intpos 5) > 0 and
A3: s . (intpos 4) = m3 + (s . (intpos 5)) and
A4: m3 = (s . (intpos 3)) - 1 and
A5: s . (intpos 2) = md and
A6: md >= 8 and
A7: md <= m3 ; ::_thesis: ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
set s1 = IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s);
A8: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . GBP by SCMPDS_5:15;
A9: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 1) by SCMPDS_5:15;
A10: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 2) by SCMPDS_5:15;
A11: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 3) by SCMPDS_5:15;
A12: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 4) by SCMPDS_5:15;
A13: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 7) by SCMPDS_5:15;
A14: m3 >= 8 by A6, A7, XXREAL_0:2;
then consider mE being Element of NAT  such that
A15: mE = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7) and
A16: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) = m3 + mE and
A17: mE <= s . (intpos 5) and
A18: for i being Element of NAT st m3 + mE < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) and
A19: ( mE = 0 or (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos (m3 + mE)) ) by A1, A2, A3, A5, A6, Lm9;
A20: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP = 0 by A1, A2, A3, A5, A6, A14, Lm9;
A21: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) = s . (intpos 1) by A1, A2, A3, A5, A6, A14, Lm9;
A22: while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_halting_on s,P by A1, A3, A5, A6, A14, Lm10;
set s2 = IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))));
set sm = (m3 + 1) + mE;
set s12 = IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s);
set s3 = IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s);
set P12 = P;
set P3 = P;
set m4 = m3 + mE;
A23: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . GBP = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . GBP by SCMPDS_5:15;
A24: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 1) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 1) by SCMPDS_5:15;
A25: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 2) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 2) by SCMPDS_5:15;
A26: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 3) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 3) by SCMPDS_5:15;
A27: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 4) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 4) by SCMPDS_5:15;
A28: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos 5) by SCMPDS_5:15;
A29: ((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3)) + ((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7)) = (m3 + 1) + mE by A1, A2, A3, A4, A5, A6, A14, A15, Lm9;
A30: m3 + mE >= 8 by A14, NAT_1:12;
A31: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) = s . (intpos 2) by A1, A2, A3, A5, A6, A14, Lm9;
A32: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = s . (intpos 3) by A1, A2, A3, A5, A6, A14, Lm9;
 (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) = m3 + 1 by A1, A2, A3, A4, A5, A6, A14, Lm9;
then  m3 < (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) by XREAL_1:29;
then A33: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) >= 8 by A14, XXREAL_0:2;
A34: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) = 0 by A1, A2, A3, A5, A6, A14, Lm9;
A35: m3 + mE <= m3 + (s . (intpos 5)) by A17, XREAL_1:6;
A36: while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0)))))) is_closed_on s,P by A1, A3, A5, A6, A14, Lm10;
percases ( mE <= 0 or mE > 0 ) ;
supposeA38: mE <= 0 ; ::_thesis: ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
X1: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . GBP = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7)) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7)) by SCMPDS_5:15;
X3: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 5) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 5) by SCMPDS_5:15;
X5: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 1) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) by SCMPDS_5:15;
X6: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 2) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) by SCMPDS_5:15;
X7: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 3) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) by SCMPDS_5:15;
X8: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 4) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) by SCMPDS_5:15;
A39: DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7) = intpos (0 + 7) by A20, SCMP_GCD:1;
then A40: while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s),P by A15, A38, SCMPDS_8:20;
A41: while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s),P by A15, A38, A39, SCMPDS_8:20;
then A42: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 5) by A36, A22, A40, SCMPDS_7:30
.= (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 5) by A15, A38, A39, X1, X2, SCMPDS_8:23
.= 0 by A34, X3 ;
A43: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . GBP = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . GBP by A36, A22, A41, A40, SCMPDS_7:30
.= (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . GBP by A15, A38, A39, X1, X2, SCMPDS_8:23
.= 0 by A20, X1 ;
then A44: DataLoc (((IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A45: (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) is_halting_on s,P by A36, A22, A41, A40, I, SCMPDS_7:24;
A46: (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) is_closed_on s,P by A36, A22, A41, A40, I, SCMPDS_7:24;
hence  ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P ) by A45, SCPISORT:8; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . GBP by A46, A45, SCPISORT:6
.= 0 by A43, A42, A44, SCMPDS_6:84 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 1) by A46, A45, SCPISORT:6
.= (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 1) by A42, A44, SCMPDS_6:84
.= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 1) by A36, A22, A41, A40, SCMPDS_7:30
.= s . (intpos 1) by A21, A15, A38, A39, X1, X2, X5, SCMPDS_8:23 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 2) by A46, A45, SCPISORT:6
.= (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 2) by A42, A44, SCMPDS_6:84
.= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 2) by A36, A22, A41, A40, SCMPDS_7:30
.= md by A5, A31, A15, A38, A39, X1, X2, X6, SCMPDS_8:23 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A47: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 3) by A46, A45, SCPISORT:6
.= (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 3) by A42, A44, SCMPDS_6:84
.= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 3) by A36, A22, A41, A40, SCMPDS_7:30
.= s . (intpos 3) by A32, A15, A38, A39, X1, X2, X7, SCMPDS_8:23 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A48: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 4) by A46, A45, SCPISORT:6
.= (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 4) by A42, A44, SCMPDS_6:84
.= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 4) by A36, A22, A41, A40, SCMPDS_7:30
.= m3 + mE by A15, A16, A38, A39, X1, X2, X8, SCMPDS_8:23 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) by A3, A17, XREAL_1:6; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 by A48, NAT_1:12; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A49: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 5) by A46, A45, SCPISORT:6
.= 0 by A42, A44, SCMPDS_6:84 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) by A2; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 by A49; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) by A4, A38, A47, A48, A49; ::_thesis: ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
take n1 = m3 + mE; ::_thesis: ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
take n2 = (m3 + mE) + 1; ::_thesis: ( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
A50: mE = 0 by A38;
hence  n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 by A4, A47; ::_thesis: ( n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
thus  n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 by A48; ::_thesis: ( ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
A51: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),P,s))_._x_=_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),P,(Initialize_(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),P,s)))))_._x
let x be Int_position; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . x = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . x
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . x = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . x by A46, A45, SCPISORT:6
.= (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . x by A42, A44, SCMPDS_6:84
.= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . x by A36, A22, A41, A40, SCMPDS_7:30 ; ::_thesis: verum
end;
A52: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) )
assume A53: i >= 8 ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i)
X1: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . GBP = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7)) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7)) by SCMPDS_5:15;
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos i) by A51
.= (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos i) by A15, A38, A39, X1, X2, SCMPDS_8:23
.= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) by SCMPDS_5:15
.= s . (intpos i) by A1, A2, A3, A5, A6, A14, A53, Lm9 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ; ::_thesis: ( ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
A54: m3 + mE < n2 by XREAL_1:29;
hence  ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) by A30, A52, XXREAL_0:2; ::_thesis: ( ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
thus  for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A4, A50, XREAL_1:146, XXREAL_0:2; ::_thesis: for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i)
hereby  ::_thesis: verum
X1: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . GBP = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7)) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (DataLoc (((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP),7)) by SCMPDS_5:15;
A55: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos md) by A51
.= (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos md) by A15, A38, A39, X1, X2, SCMPDS_8:23
.= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) by SCMPDS_5:15 ;
let i be Element of NAT ; ::_thesis: ( n2 <= i & i <= s . (intpos 4) implies (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) )
assume that
A56: n2 <= i and
A57: i <= s . (intpos 4) ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i)
A58: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos i) by A51
.= (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos i) by A15, A38, A39, X1, X2, SCMPDS_8:23
.= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) by SCMPDS_5:15 ;
 m3 + mE < i by A54, A56, XXREAL_0:2;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A18, A57, A55, A58; ::_thesis: verum
end;
end;
supposeA59: mE > 0 ; ::_thesis: ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
then consider m5, mE3 being Element of NAT  such that
A60: m5 = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 5) and
A61: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 3) = mE3 and
A62: mE3 + m5 = (m3 + 1) + mE and
A63: m5 <= (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 7) and
A64: for i being Element of NAT st (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 3) <= i & i < mE3 holds
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos i) and
A65: ( m5 = 0 or (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos md) <= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos mE3) ) by A4, A5, A6, A15, A20, A31, A32, A33, Lm15, A8, A10, A13, A11;
A66: mE3 + m5 <= mE3 + ((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7)) by A63, A13, XREAL_1:6;
then A67: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) <= mE3 by A4, A32, A15, A62, XREAL_1:6;
A68: ((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3)) + ((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7)) <= mE3 + ((IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 7)) by A4, A32, A15, A62, A63, A13, XREAL_1:6;
then A69: s . (intpos 3) <= mE3 by A32, XREAL_1:6;
then  mE3 >= 8 by A32, A33, XXREAL_0:2;
then A70: mE3 > 6 by XXREAL_0:2;
X1: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . GBP = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 2) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) by SCMPDS_5:15;
X3: (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos 3) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) by SCMPDS_5:15;
B71: while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)),P by A5, A6, A20, A31, A33, Lm16, X1, X2, X3;
A71: while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s),P
proof
 P +* (stop (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) halts_on Initialize (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) by B71, SCMPDS_6:def_3;
hence  while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_halting_on IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s),P by SCMPDS_6:def_3; ::_thesis: verum
end;
B72: while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)),P by A5, A6, A20, A31, A33, Lm16, X1, X2, X3;
A72: while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s),P
proof
 for k being Element of NAT holds IC (Comput ((P +* (stop (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))))),(Initialize (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)))),k)) in dom (stop (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) by B72, SCMPDS_6:def_2;
hence  while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))) is_closed_on IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s),P by SCMPDS_6:def_2; ::_thesis: verum
end;
then A73: (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) is_closed_on s,P by A36, A22, A71, I, SCMPDS_7:24;
 (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 1) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 1) by A4, A5, A6, A20, A31, A32, A33, A29, A59, Lm15, A8, A9, A10, A11, A13;
then A74: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 1) = s . (intpos 1) by A21, A36, A22, A72, A71, SCMPDS_7:30;
 (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 4) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 4) by A4, A5, A6, A20, A31, A32, A33, A29, A59, Lm15, A8, A10, A11, A12, A13;
then A75: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 4) = m3 + mE by A16, A36, A22, A72, A71, SCMPDS_7:30;
 (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos 2) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 2) by A4, A5, A6, A20, A31, A32, A33, A29, A59, Lm15, A8, A10, A11, A13;
then A76: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 2) = md by A5, A31, A36, A22, A72, A71, SCMPDS_7:30;
 (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . GBP = 0 by A4, A5, A6, A20, A31, A32, A33, A29, A59, Lm15, A8, A10, A11, A13;
then A77: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . GBP = 0 by A36, A22, A72, A71, SCMPDS_7:30;
A78: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) = m5 by A36, A22, A60, A72, A71, SCMPDS_7:30;
A79: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 3) = mE3 by A36, A22, A61, A72, A71, SCMPDS_7:30;
A80: (while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))) is_halting_on s,P by A36, A22, A72, A71, I, SCMPDS_7:24;
hence  ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on s,P & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on s,P ) by A73, SCPISORT:8; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1, n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( n1 >= s . (intpos 3) & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A81: m3 + mE > 6 by A30, XXREAL_0:2;
A82: m3 + mE > m3 + 0 by A59, XREAL_1:6;
then A83: m3 + mE >= m3 + 1 by INT_1:7;
A84: IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)))) = IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) ;
hereby  ::_thesis: verum
percases ( (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) > 0 or (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) <= 0 ) ;
supposeA85: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) > 0 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
then  (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . GBP = 0 by A77, A79, A70, A75, A81, Lm19, A23, A26, A27, A28, A84;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 by A73, A80, SCPISORT:6; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
 (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 1) = (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 1) by A77, A79, A70, A75, A81, A85, Lm19, A23, A24, A26, A27, A28, A84;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) by A73, A80, A74, SCPISORT:6; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
 (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 2) = (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 2) by A77, A79, A70, A75, A81, A85, Lm19, A23, A25, A26, A27, A28, A84;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md by A73, A80, A76, SCPISORT:6; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
 (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 3) = ((IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 3)) + 1 by A77, A79, A70, A75, A81, A85, Lm19, A23, A26, A27, A28, A84;
then A86: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) = mE3 + 1 by A73, A80, A79, SCPISORT:6;
then  mE3 < (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) by XREAL_1:29;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) by A32, A67, XXREAL_0:2; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A87: (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 4) = ((IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 4)) - 1 by A77, A79, A70, A75, A81, A85, Lm19, A23, A26, A27, A28, A84;
then A88: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (m3 + mE) - 1 by A73, A80, A75, SCPISORT:6;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) by A3, A35, XREAL_1:146, XXREAL_0:2; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A89: (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 5) = ((IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5)) - 2 by A77, A79, A70, A75, A81, A85, Lm19, A23, A26, A27, A28, A84;
then A90: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) = ((IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5)) - 2 by A73, A80, SCPISORT:6;
 (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) >= 0 + 1 by A85, INT_1:7;
then A91: ((IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5)) - 2 >= 1 - 2 by XREAL_1:9;
then A92: ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) + 1 >= (- 1) + 1 by A90, XREAL_1:6;
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 by A83, A88, XREAL_1:19; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
 m5 <= s . (intpos 5) by A15, A17, A63, A13, XXREAL_0:2;
then  m5 - 2 < (s . (intpos 5)) - 0 by XREAL_1:15;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) by A36, A22, A60, A72, A71, A90, SCMPDS_7:30; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 by A73, A80, A89, A91, SCPISORT:6; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) by A62, A73, A80, A75, A78, A87, A86, A90, SCPISORT:6; ::_thesis: ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
take n1 = mE3; ::_thesis: ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
take n2 = m3 + mE; ::_thesis: ( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
 n2 = n1 + (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) + 1) by A62, A78, A90;
then A93: n2 >= n1 + 0 by A92, XREAL_1:6;
thus  n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 by A86; ::_thesis: ( n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
thus n2 = ((m3 + mE) + (- 1)) + 1
.= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 by A73, A80, A75, A87, SCPISORT:6 ; ::_thesis: ( ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
A94: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0)))))))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos i) = s . (intpos i) )
A95: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos i) by SCMPDS_5:15;
assume A96: i >= 8 ; ::_thesis: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos i) = s . (intpos i)
thus (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos i) by A36, A22, A72, A71, SCMPDS_7:30
.= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) by A4, A5, A6, A20, A31, A32, A33, A29, A59, A96, Lm15, A8, A10, A11, A13, A95
.= s . (intpos i) by A1, A2, A3, A5, A6, A14, A96, Lm9 ; ::_thesis: verum
end;
A97: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_n1_&_i_<>_n2_holds_
(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> n1 & i <> n2 implies (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) )
assume that
A98: i >= 8 and
A99: i <> n1 and
A100: i <> n2 ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i)
A101: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos i) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos i) by SCMPDS_5:15;
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos i) by A73, A80, SCPISORT:6
.= (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos i) by A77, A79, A70, A75, A81, A85, A98, A99, A100, Lm19, A23, A26, A27, A28, A84, A101
.= s . (intpos i) by A94, A98 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ; ::_thesis: ( ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
A102: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos (m3 + mE)) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos (m3 + mE)) by SCMPDS_5:15;
 (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos mE3) = (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos (m3 + mE)) by A77, A79, A70, A75, A81, A85, Lm19, A23, A26, A27, A28, A84, A102;
then A103: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos n2) by A73, A80, SCPISORT:6
.= s . (intpos n2) by A14, A94, NAT_1:12 ;
 m3 < m3 + 1 by XREAL_1:29;
then  m3 < n1 by A4, A32, A67, XXREAL_0:2;
then A104: md < n1 by A7, XXREAL_0:2;
then  md < n2 by A93, XXREAL_0:2;
then A105: n2 >= 8 by A6, XXREAL_0:2;
A106: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) = s . (intpos md) by A6, A97, A104, A93;
then A107: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) by A1, A2, A3, A5, A6, A14, Lm9;
A108: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos mE3) = (Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))) . (intpos mE3) by SCMPDS_5:15;
 (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos (m3 + mE)) = (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos mE3) by A77, A79, A70, A75, A81, A85, Lm19, A23, A26, A27, A28, A84, A108;
then A109: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos n1) by A73, A80, SCPISORT:6
.= s . (intpos n1) by A32, A33, A69, A94, XXREAL_0:2 ;
hence  ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) by A3, A32, A17, A68, A103, XREAL_1:6; ::_thesis: ( ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
A110: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos md) by SCMPDS_5:15;
A111: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos md) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) by A4, A5, A6, A20, A31, A32, A33, A29, A59, Lm15, A8, A10, A11, A13, A110
.= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) by A1, A2, A3, A5, A6, A14, A106, Lm9 ;
hereby  ::_thesis: for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i)
let i be Element of NAT ; ::_thesis: ( s . (intpos 3) <= i & i <= n1 implies (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos b1) )
assume that
A112: s . (intpos 3) <= i and
A113: i <= n1 ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos b1)
A114: i >= 8 by A32, A33, A112, XXREAL_0:2;
A115: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos 3) <= i by A1, A2, A3, A5, A6, A14, A112, Lm9;
percases ( i < n1 or i >= n1 ) ;
supposeA116: i < n1 ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos b1)
A117: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos i) by SCMPDS_5:15;
(IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos i) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) by A4, A5, A6, A20, A31, A32, A33, A29, A59, A114, A8, A10, A11, A13, A117, Lm15
.= s . (intpos i) by A1, A2, A3, A5, A6, A14, A114, Lm9
.= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A32, A33, A97, A93, A112, A116, XXREAL_0:2 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A64, A111, A115, A116, A11; ::_thesis: verum
end;
suppose i >= n1 ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos b1)
then  i = n1 by A113, XXREAL_0:1;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A1, A2, A3, A5, A6, A14, A19, A59, A103, A105, A107, Lm9; ::_thesis: verum
end;
end;
end;
A118: n1 >= 8 by A6, A104, XXREAL_0:2;
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( n2 <= i & i <= s . (intpos 4) implies (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos b1) )
assume that
A119: n2 <= i and
A120: i <= s . (intpos 4) ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos b1)
A121: i >= 8 by A105, A119, XXREAL_0:2;
A122: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos n1) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos n1) by SCMPDS_5:15;
percases ( n2 < i or n2 >= i ) ;
supposeA123: n2 < i ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos b1)
(IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = s . (intpos i) by A1, A2, A3, A5, A6, A14, A121, Lm9
.= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A97, A93, A105, A123, XXREAL_0:2 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A18, A107, A120, A123; ::_thesis: verum
end;
suppose n2 >= i ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos b1)
then  i = n2 by A119, XXREAL_0:1;
then (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos n1) by A1, A2, A3, A5, A6, A14, A109, A118, Lm9
.= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos n1) by A4, A5, A6, A20, A31, A32, A33, A29, A59, A118, Lm15, A8, A10, A11, A13, A122 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A36, A22, A60, A65, A72, A71, A85, A111, SCMPDS_7:30; ::_thesis: verum
end;
end;
end;
end;
supposeA124: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) <= 0 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = 0 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A125: DataLoc (((IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . GBP),5) = intpos (0 + 5) by A77, SCMP_GCD:1;
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . GBP = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . GBP by A73, A80, SCPISORT:6
.= 0 by A77, A124, A125, SCMPDS_6:84 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 1) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 1) by A73, A80, SCPISORT:6
.= s . (intpos 1) by A74, A124, A125, SCMPDS_6:84 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = md & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 2) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 2) by A73, A80, SCPISORT:6
.= md by A76, A124, A125, SCMPDS_6:84 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A126: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 3) by A73, A80, SCPISORT:6
.= mE3 by A79, A124, A125, SCMPDS_6:84 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3) >= s . (intpos 3) by A4, A15, A62, A66, XREAL_1:6; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A127: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 4) by A73, A80, SCPISORT:6
.= m3 + mE by A75, A124, A125, SCMPDS_6:84 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) <= s . (intpos 4) by A3, A17, XREAL_1:6; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A128: (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . (intpos 5) = 0 by A36, A22, A60, A72, A71, A124, SCMPDS_7:30;
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) >= m3 by A82, A127; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A129: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . (intpos 5) by A73, A80, SCPISORT:6
.= 0 by A125, A128, SCMPDS_6:84 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) < s . (intpos 5) by A2; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5) >= - 1 by A129; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) & ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) ) )
A130: m3 + mE = ((m3 + mE) + 1) - 1
.= (mE3 + 0) - 1 by A36, A22, A60, A62, A72, A71, A128, SCMPDS_7:30 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 5)) by A126, A127, A129; ::_thesis: ex n1 being Element of NAT ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
take n1 = m3 + mE; ::_thesis: ex n2 being Element of NAT st
( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
take n2 = (m3 + mE) + 1; ::_thesis: ( n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 & n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
thus  n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 3)) - 1 by A126, A130; ::_thesis: ( n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 & ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
thus  n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos 4)) + 1 by A127; ::_thesis: ( ( for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ) & ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
A131: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),P,s))_._x_=_(IExec_((while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))),P,(Initialize_(IExec_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0))))))),P,s)))))_._x
let x be Int_position; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . x = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . x
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . x = (IExec ((if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))),P,(Initialize (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s))))) . x by A73, A80, SCPISORT:6
.= (IExec (((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))),P,s)) . x by A124, A125, SCMPDS_6:84
.= (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . x by A36, A22, A72, A71, SCMPDS_7:30 ; ::_thesis: verum
end;
A132: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) )
assume A133: i >= 8 ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i)
A134: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos i) by SCMPDS_5:15;
thus (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos i) by A131
.= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) by A4, A5, A6, A20, A31, A32, A33, A29, A59, A133, Lm15, A8, A10, A11, A13, A134
.= s . (intpos i) by A1, A2, A3, A5, A6, A14, A133, Lm9 ; ::_thesis: verum
end;
hence  for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = s . (intpos i) ; ::_thesis: ( ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) & ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
A135: m3 + mE < n2 by XREAL_1:29;
hence  ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n2) ) or ( s . (intpos 3) <= n1 & n2 <= s . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n1) = s . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos n2) = s . (intpos n1) ) ) by A30, A132, XXREAL_0:2; ::_thesis: ( ( for i being Element of NAT st s . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) ) )
hereby  ::_thesis: for i being Element of NAT st n2 <= i & i <= s . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i)
let i be Element of NAT ; ::_thesis: ( s . (intpos 3) <= i & i <= n1 implies (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) )
assume that
A136: s . (intpos 3) <= i and
A137: i <= n1 ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i)
 i < mE3 by A130, A137, XREAL_1:146, XXREAL_0:2;
then A138: (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos md) > (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos i) by A32, A64, A136, A11;
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos md) by A131;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A131, A138; ::_thesis: verum
end;
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( n2 <= i & i <= s . (intpos 4) implies (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) )
assume that
A139: n2 <= i and
A140: i <= s . (intpos 4) ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i)
A141: m3 + mE < i by A135, A139, XXREAL_0:2;
then A142: i >= 8 by A30, XXREAL_0:2;
A143: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos md) by SCMPDS_5:15;
A144: (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) = (Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))) . (intpos i) by SCMPDS_5:15;
A145: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos md) by A131
.= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos md) by A4, A5, A6, A20, A31, A32, A33, A29, A59, Lm15, A8, A10, A11, A13, A143 ;
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0))))))),P,(Initialize (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s))))) . (intpos i) by A131
.= (IExec ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))),P,s)) . (intpos i) by A4, A5, A6, A20, A31, A32, A33, A29, A59, A142, Lm15, A8, A10, A11, A13, A144 ;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),P,s)) . (intpos i) by A18, A140, A141, A145; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;

Lm21:  for i being Integer st i >= - 1 & i <= 0 & not i = - 1 holds
i = 0
proof
let i be Integer; ::_thesis: ( i >= - 1 & i <= 0 & not i = - 1 implies i = 0 )
assume that
A1: i >= - 1 and
A2: i <= 0 ; ::_thesis: ( i = - 1 or i = 0 )
percases ( i <= - 1 or i > - 1 ) ;
suppose i <= - 1 ; ::_thesis: ( i = - 1 or i = 0 )
hence  ( i = - 1 or i = 0 ) by A1, XXREAL_0:1; ::_thesis: verum
end;
suppose i > - 1 ; ::_thesis: ( i = - 1 or i = 0 )
then  i >= (- 1) + 1 by INT_1:7;
hence  ( i = - 1 or i = 0 ) by A2; ::_thesis: verum
end;
end;
end;

Lm22:  for i1, i2 being Integer
for n1, n2, i being Element of NAT st i1 >= - 1 & i1 <= 0 & n2 = i2 + 1 & i2 = n1 + i1 & i < n2 holds
i <= n1
proof
let i1, i2 be Integer; ::_thesis: for n1, n2, i being Element of NAT st i1 >= - 1 & i1 <= 0 & n2 = i2 + 1 & i2 = n1 + i1 & i < n2 holds
i <= n1
let n1, n2, i be Element of NAT ; ::_thesis: ( i1 >= - 1 & i1 <= 0 & n2 = i2 + 1 & i2 = n1 + i1 & i < n2 implies i <= n1 )
assume that
A1: i1 >= - 1 and
A2: i1 <= 0 and
A3: n2 = i2 + 1 and
A4: i2 = n1 + i1 and
A5: i < n2 ; ::_thesis: i <= n1
percases ( i1 = 0 or i1 = - 1 ) by A1, A2, Lm21;
suppose i1 = 0 ; ::_thesis: i <= n1
hence  i <= n1 by A3, A4, A5, NAT_1:13; ::_thesis: verum
end;
suppose i1 = - 1 ; ::_thesis: i <= n1
hence  i <= n1 by A3, A4, A5; ::_thesis: verum
end;
end;
end;

Lm23:  for i1, i2 being Integer
for n1, n2 being Element of NAT st i1 >= - 1 & n2 = i2 + 1 & i2 = n1 + i1 holds
n1 <= n2
proof
let i1, i2 be Integer; ::_thesis: for n1, n2 being Element of NAT st i1 >= - 1 & n2 = i2 + 1 & i2 = n1 + i1 holds
n1 <= n2
let n1, n2 be Element of NAT ; ::_thesis: ( i1 >= - 1 & n2 = i2 + 1 & i2 = n1 + i1 implies n1 <= n2 )
assume that
A1: i1 >= - 1 and
A2: n2 = i2 + 1 and
A3: i2 = n1 + i1 ; ::_thesis: n1 <= n2
 n1 + i1 >= n1 + (- 1) by A1, XREAL_1:6;
then  i2 + 1 >= (n1 + (- 1)) + 1 by A3, XREAL_1:6;
hence  n1 <= n2 by A2; ::_thesis: verum
end;

Lm24:  for s, s1 being State of SCMPDS
for n0, n1, n2, n being Element of NAT
for f, f1 being FinSequence of INT st f is_FinSequence_on s,n0 & f1 is_FinSequence_on s1,n0 & len f = n & len f1 = n & ( for i being Element of NAT st i >= n0 + 1 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ) & ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) holds
f,f1 are_fiberwise_equipotent
proof
let s, s1 be State of SCMPDS; ::_thesis: for n0, n1, n2, n being Element of NAT
for f, f1 being FinSequence of INT st f is_FinSequence_on s,n0 & f1 is_FinSequence_on s1,n0 & len f = n & len f1 = n & ( for i being Element of NAT st i >= n0 + 1 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ) & ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) holds
f,f1 are_fiberwise_equipotent
let n0, n1, n2, n be Element of NAT ; ::_thesis: for f, f1 being FinSequence of INT st f is_FinSequence_on s,n0 & f1 is_FinSequence_on s1,n0 & len f = n & len f1 = n & ( for i being Element of NAT st i >= n0 + 1 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ) & ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) holds
f,f1 are_fiberwise_equipotent
let f, f1 be FinSequence of INT ; ::_thesis: ( f is_FinSequence_on s,n0 & f1 is_FinSequence_on s1,n0 & len f = n & len f1 = n & ( for i being Element of NAT st i >= n0 + 1 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ) & ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) implies f,f1 are_fiberwise_equipotent )
assume that
A1: f is_FinSequence_on s,n0 and
A2: f1 is_FinSequence_on s1,n0 ; ::_thesis: ( not len f = n or not len f1 = n or ex i being Element of NAT st
( i >= n0 + 1 & i <> n1 & i <> n2 & not s1 . (intpos i) = s . (intpos i) ) or ( not ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) & not ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) or f,f1 are_fiberwise_equipotent )
assume that
A3: len f = n and
A4: len f1 = n ; ::_thesis: ( ex i being Element of NAT st
( i >= n0 + 1 & i <> n1 & i <> n2 & not s1 . (intpos i) = s . (intpos i) ) or ( not ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) & not ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) or f,f1 are_fiberwise_equipotent )
assume A5: for i being Element of NAT st i >= n0 + 1 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ; ::_thesis: ( ( not ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) & not ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) or f,f1 are_fiberwise_equipotent )
assume A6: ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) ; ::_thesis: f,f1 are_fiberwise_equipotent
percases ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) by A6;
supposeA7: ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) ; ::_thesis: f,f1 are_fiberwise_equipotent
A8: dom f1 = Seg n by A4, FINSEQ_1:def_3;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_f1_holds_
f1_._i_=_f_._i
let i be Nat; ::_thesis: ( i in dom f1 implies f1 . b1 = f . b1 )
reconsider a = i as Element of NAT by ORDINAL1:def_12;
assume A9: i in dom f1 ; ::_thesis: f1 . b1 = f . b1
then A10: 1 <= i by A8, FINSEQ_1:1;
then A11: n0 + 1 <= n0 + i by XREAL_1:6;
A12: i <= n by A8, A9, FINSEQ_1:1;
percases ( ( n0 + i <> n1 & n0 + i <> n2 ) or not n0 + i <> n1 or not n0 + i <> n2 ) ;
supposeA13: ( n0 + i <> n1 & n0 + i <> n2 ) ; ::_thesis: f1 . b1 = f . b1
thus f1 . i = s1 . (intpos (n0 + a)) by A2, A4, A10, A12, SCPISORT:def_1
.= s . (intpos (n0 + a)) by A5, A11, A13
.= f . i by A1, A3, A10, A12, SCPISORT:def_1 ; ::_thesis: verum
end;
supposeA14: ( not n0 + i <> n1 or not n0 + i <> n2 ) ; ::_thesis: f1 . b1 = f . b1
hereby  ::_thesis: verum
percases ( n0 + i = n1 or n0 + i = n2 ) by A14;
suppose n0 + i = n1 ; ::_thesis: f1 . i = f . i
hence f1 . i = s . (intpos (n0 + a)) by A2, A4, A7, A10, A12, SCPISORT:def_1
.= f . i by A1, A3, A10, A12, SCPISORT:def_1 ;
::_thesis: verum
end;
suppose n0 + i = n2 ; ::_thesis: f1 . i = f . i
hence f1 . i = s . (intpos (n0 + a)) by A2, A4, A7, A10, A12, SCPISORT:def_1
.= f . i by A1, A3, A10, A12, SCPISORT:def_1 ;
::_thesis: verum
end;
end;
end;
end;
end;
end;
hence  f,f1 are_fiberwise_equipotent by A3, A4, FINSEQ_2:9; ::_thesis: verum
end;
supposeA15: ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ; ::_thesis: f,f1 are_fiberwise_equipotent
then A16: n1 - n0 >= 1 by XREAL_1:19;
then reconsider m1 = n1 - n0 as Element of NAT by INT_1:3;
A17: m1 <= len f by A3, A15, XREAL_1:20;
A18: n2 - n0 >= 1 by A15, XREAL_1:19;
then reconsider m2 = n2 - n0 as Element of NAT by INT_1:3;
A19: m2 <= len f1 by A4, A15, XREAL_1:20;
A20: n2 = m2 + n0 ;
A21: n1 = m1 + n0 ;
then A22: f . m1 = s1 . (intpos n2) by A1, A15, A16, A17, SCPISORT:def_1
.= f1 . m2 by A2, A18, A19, A20, SCPISORT:def_1 ;
A23: now__::_thesis:_for_k_being_Element_of_NAT_st_k_<>_m1_&_k_<>_m2_&_1_<=_k_&_k_<=_len_f_holds_
f_._k_=_f1_._k
let k be Element of NAT ; ::_thesis: ( k <> m1 & k <> m2 & 1 <= k & k <= len f implies f . k = f1 . k )
assume that
A24: k <> m1 and
A25: k <> m2 and
A26: 1 <= k and
A27: k <= len f ; ::_thesis: f . k = f1 . k
A28: k + n0 <> m1 + n0 by A24;
A29: n0 + 1 <= n0 + k by A26, XREAL_1:6;
A30: k + n0 <> m2 + n0 by A25;
thus f . k = s . (intpos (k + n0)) by A1, A26, A27, SCPISORT:def_1
.= s1 . (intpos (k + n0)) by A5, A28, A30, A29
.= f1 . k by A2, A3, A4, A26, A27, SCPISORT:def_1 ; ::_thesis: verum
end;
A31: m2 <= len f by A3, A15, XREAL_1:20;
then f . m2 = s1 . (intpos n1) by A1, A15, A18, A20, SCPISORT:def_1
.= f1 . m1 by A2, A3, A4, A16, A17, A21, SCPISORT:def_1 ;
hence  f,f1 are_fiberwise_equipotent by A3, A4, A16, A18, A17, A31, A22, A23, SCPISORT:3; ::_thesis: verum
end;
end;
end;

Lm25:  for s, s1 being State of SCMPDS
for n0, n1, n2 being Element of NAT
for c1, c2 being Integer st ( for i being Element of NAT st i >= n0 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ) & n1 <= n2 & ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( c1 <= n1 & n2 <= c2 & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) holds
for i being Element of NAT st i >= n0 & ( i < c1 or i > c2 ) holds
s1 . (intpos i) = s . (intpos i)
proof
let s, s1 be State of SCMPDS; ::_thesis: for n0, n1, n2 being Element of NAT
for c1, c2 being Integer st ( for i being Element of NAT st i >= n0 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ) & n1 <= n2 & ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( c1 <= n1 & n2 <= c2 & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) holds
for i being Element of NAT st i >= n0 & ( i < c1 or i > c2 ) holds
s1 . (intpos i) = s . (intpos i)
let n0, n1, n2 be Element of NAT ; ::_thesis: for c1, c2 being Integer st ( for i being Element of NAT st i >= n0 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ) & n1 <= n2 & ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( c1 <= n1 & n2 <= c2 & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) holds
for i being Element of NAT st i >= n0 & ( i < c1 or i > c2 ) holds
s1 . (intpos i) = s . (intpos i)
let c1, c2 be Integer; ::_thesis: ( ( for i being Element of NAT st i >= n0 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ) & n1 <= n2 & ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( c1 <= n1 & n2 <= c2 & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) implies for i being Element of NAT st i >= n0 & ( i < c1 or i > c2 ) holds
s1 . (intpos i) = s . (intpos i) )
assume A1: for i being Element of NAT st i >= n0 & i <> n1 & i <> n2 holds
s1 . (intpos i) = s . (intpos i) ; ::_thesis: ( not n1 <= n2 or ( not ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) & not ( c1 <= n1 & n2 <= c2 & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) or for i being Element of NAT st i >= n0 & ( i < c1 or i > c2 ) holds
s1 . (intpos i) = s . (intpos i) )
assume A2: n1 <= n2 ; ::_thesis: ( ( not ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) & not ( c1 <= n1 & n2 <= c2 & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) or for i being Element of NAT st i >= n0 & ( i < c1 or i > c2 ) holds
s1 . (intpos i) = s . (intpos i) )
assume A3: ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( c1 <= n1 & n2 <= c2 & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) ; ::_thesis: for i being Element of NAT st i >= n0 & ( i < c1 or i > c2 ) holds
s1 . (intpos i) = s . (intpos i)
percases ( ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) or ( c1 <= n1 & n2 <= c2 & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ) by A3;
supposeA4: ( s1 . (intpos n1) = s . (intpos n1) & s1 . (intpos n2) = s . (intpos n2) ) ; ::_thesis: for i being Element of NAT st i >= n0 & ( i < c1 or i > c2 ) holds
s1 . (intpos i) = s . (intpos i)
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i >= n0 & ( i < c1 or i > c2 ) implies s1 . (intpos b1) = s . (intpos b1) )
assume that
A5: i >= n0 and
 ( i < c1 or i > c2 ) ; ::_thesis: s1 . (intpos b1) = s . (intpos b1)
percases ( ( i <> n1 & i <> n2 ) or not i <> n1 or not i <> n2 ) ;
suppose ( i <> n1 & i <> n2 ) ; ::_thesis: s1 . (intpos b1) = s . (intpos b1)
hence  s1 . (intpos i) = s . (intpos i) by A1, A5; ::_thesis: verum
end;
supposeA6: ( not i <> n1 or not i <> n2 ) ; ::_thesis: s1 . (intpos b1) = s . (intpos b1)
hereby  ::_thesis: verum
percases ( i = n1 or i = n2 ) by A6;
suppose i = n1 ; ::_thesis: s1 . (intpos i) = s . (intpos i)
hence  s1 . (intpos i) = s . (intpos i) by A4; ::_thesis: verum
end;
suppose i = n2 ; ::_thesis: s1 . (intpos i) = s . (intpos i)
hence  s1 . (intpos i) = s . (intpos i) by A4; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
supposeA7: ( c1 <= n1 & n2 <= c2 & s1 . (intpos n1) = s . (intpos n2) & s1 . (intpos n2) = s . (intpos n1) ) ; ::_thesis: for i being Element of NAT st i >= n0 & ( i < c1 or i > c2 ) holds
s1 . (intpos i) = s . (intpos i)
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i >= n0 & ( i < c1 or i > c2 ) implies s1 . (intpos b1) = s . (intpos b1) )
assume that
A8: i >= n0 and
A9: ( i < c1 or i > c2 ) ; ::_thesis: s1 . (intpos b1) = s . (intpos b1)
percases ( i < c1 or i > c2 ) by A9;
suppose i < c1 ; ::_thesis: s1 . (intpos b1) = s . (intpos b1)
then  i < n1 by A7, XXREAL_0:2;
hence  s1 . (intpos i) = s . (intpos i) by A1, A2, A8; ::_thesis: verum
end;
suppose i > c2 ; ::_thesis: s1 . (intpos b1) = s . (intpos b1)
then  i > n2 by A7, XXREAL_0:2;
hence  s1 . (intpos i) = s . (intpos i) by A1, A2, A8; ::_thesis: verum
end;
end;
end;
end;
end;
end;

Lm26:  for n being Element of NAT
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, m3, n0 being Element of NAT
for f, f1 being FinSequence of INT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 + 1 <= s . (intpos 3) & s . (intpos 4) <= n0 + n & f is_FinSequence_on s,n0 & f1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s),n0 & n0 >= 7 & len f = n & len f1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) )
proof
let n be Element of NAT ; ::_thesis: for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, m3, n0 being Element of NAT
for f, f1 being FinSequence of INT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 + 1 <= s . (intpos 3) & s . (intpos 4) <= n0 + n & f is_FinSequence_on s,n0 & f1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s),n0 & n0 >= 7 & len f = n & len f1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) )
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, m3, n0 being Element of NAT
for f, f1 being FinSequence of INT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 + 1 <= s . (intpos 3) & s . (intpos 4) <= n0 + n & f is_FinSequence_on s,n0 & f1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s),n0 & n0 >= 7 & len f = n & len f1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md, m3, n0 being Element of NAT
for f, f1 being FinSequence of INT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 + 1 <= s . (intpos 3) & s . (intpos 4) <= n0 + n & f is_FinSequence_on s,n0 & f1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s),n0 & n0 >= 7 & len f = n & len f1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) )
let md, m3, n0 be Element of NAT ; ::_thesis: for f, f1 being FinSequence of INT st s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 + 1 <= s . (intpos 3) & s . (intpos 4) <= n0 + n & f is_FinSequence_on s,n0 & f1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s),n0 & n0 >= 7 & len f = n & len f1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) )
let f, f1 be FinSequence of INT ; ::_thesis: ( s . GBP = 0 & s . (intpos 5) > 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 + 1 <= s . (intpos 3) & s . (intpos 4) <= n0 + n & f is_FinSequence_on s,n0 & f1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s),n0 & n0 >= 7 & len f = n & len f1 = n implies ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 5) > 0 and
A3: s . (intpos 4) = m3 + (s . (intpos 5)) and
A4: m3 = (s . (intpos 3)) - 1 and
A5: s . (intpos 2) = md and
A6: md >= n0 + 1 and
A7: md <= m3 and
A8: n0 + 1 <= s . (intpos 3) and
A9: s . (intpos 4) <= n0 + n ; ::_thesis: ( not f is_FinSequence_on s,n0 or not f1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s),n0 or not n0 >= 7 or not len f = n or not len f1 = n or ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) ) )
defpred S1[ Element of NAT ] means  for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for n3 being Element of NAT
for g, g1 being FinSequence of INT st t . GBP = 0 & t . (intpos 5) > 0 & t . (intpos 5) <= $1 + 1 & t . (intpos 4) = n3 + (t . (intpos 5)) & n3 = (t . (intpos 3)) - 1 & t . (intpos 2) = md & md <= n3 & n0 + 1 <= t . (intpos 3) & t . (intpos 4) <= n0 + n & g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) );
assume A10: f is_FinSequence_on s,n0 ; ::_thesis: ( not f1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s),n0 or not n0 >= 7 or not len f = n or not len f1 = n or ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) ) )
assume A11: f1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s),n0 ; ::_thesis: ( not n0 >= 7 or not len f = n or not len f1 = n or ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) ) )
assume that
A12: n0 >= 7 and
A13: len f = n and
A14: len f1 = n ; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) )
 n0 + 1 >= 7 + 1 by A12, XREAL_1:6;
then A15: md >= 8 by A6, XXREAL_0:2;
A16: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A17: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_n3_being_Element_of_NAT_
for_g,_g1_being_FinSequence_of_INT_
for_Q_being_Instruction-Sequence_of_SCMPDS_st_t_._GBP_=_0_&_t_._(intpos_5)_>_0_&_t_._(intpos_5)_<=_(k_+_1)_+_1_&_t_._(intpos_4)_=_n3_+_(t_._(intpos_5))_&_n3_=_(t_._(intpos_3))_-_1_&_t_._(intpos_2)_=_md_&_md_<=_n3_&_n0_+_1_<=_t_._(intpos_3)_&_t_._(intpos_4)_<=_n0_+_n_&_g_is_FinSequence_on_t,n0_&_g1_is_FinSequence_on_IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t),n0_&_len_g_=_n_&_len_g1_=_n_holds_
(_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._GBP_=_0_&_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_1)_=_t_._(intpos_1)_&_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_2)_=_md_&_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_4)_>=_md_&_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_4)_<=_t_._(intpos_4)_&_g,g1_are_fiberwise_equipotent_&_(_for_i_being_Element_of_NAT_st_t_._(intpos_3)_<=_i_&_i_<=_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_4)_holds_
(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_md)_>=_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_i)_)_&_(_for_i_being_Element_of_NAT_st_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_4)_<_i_&_i_<=_t_._(intpos_4)_holds_
(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_md)_<=_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_i)_)_&_(_for_i_being_Element_of_NAT_st_i_>=_n0_+_1_&_(_i_<_t_._(intpos_3)_or_i_>_t_._(intpos_4)_)_holds_
(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)_)_)
let t be 0 -started State of SCMPDS; ::_thesis: for n3 being Element of NAT
for g, g1 being FinSequence of INT
for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) > 0 & t . (intpos 5) <= (k + 1) + 1 & t . (intpos 4) = n3 + (t . (intpos 5)) & n3 = (t . (intpos 3)) - 1 & t . (intpos 2) = md & md <= n3 & n0 + 1 <= t . (intpos 3) & t . (intpos 4) <= n0 + n & g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,b5)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,b5)) . (intpos 1) = b5 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,b5)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,b5)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,b5)) . (intpos 4) <= b5 . (intpos 4) & b7,b8 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b10 & b10 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,i)) . (intpos b10) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,i)) . (intpos 4) < b10 & b10 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,i)) . (intpos b10) ) & ( for i being Element of NAT st b10 >= n0 + 1 & ( b10 < i . (intpos 3) or b10 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b9,i)) . (intpos b10) = i . (intpos b10) ) )
let n3 be Element of NAT ; ::_thesis: for g, g1 being FinSequence of INT
for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) > 0 & t . (intpos 5) <= (k + 1) + 1 & t . (intpos 4) = n3 + (t . (intpos 5)) & n3 = (t . (intpos 3)) - 1 & t . (intpos 2) = md & md <= n3 & n0 + 1 <= t . (intpos 3) & t . (intpos 4) <= n0 + n & g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,b4)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,b4)) . (intpos 1) = b4 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,b4)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,b4)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,b4)) . (intpos 4) <= b4 . (intpos 4) & b6,b7 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b9 & b9 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,i)) . (intpos b9) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,i)) . (intpos 4) < b9 & b9 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,i)) . (intpos b9) ) & ( for i being Element of NAT st b9 >= n0 + 1 & ( b9 < i . (intpos 3) or b9 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b8,i)) . (intpos b9) = i . (intpos b9) ) )
let g, g1 be FinSequence of INT ; ::_thesis: for Q being Instruction-Sequence of SCMPDS st t . GBP = 0 & t . (intpos 5) > 0 & t . (intpos 5) <= (k + 1) + 1 & t . (intpos 4) = n3 + (t . (intpos 5)) & n3 = (t . (intpos 3)) - 1 & t . (intpos 2) = md & md <= n3 & n0 + 1 <= t . (intpos 3) & t . (intpos 4) <= n0 + n & g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,b2)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,b2)) . (intpos 1) = b2 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,b2)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,b2)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,b2)) . (intpos 4) <= b2 . (intpos 4) & b4,b5 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b7 & b7 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,i)) . (intpos b7) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,i)) . (intpos 4) < b7 & b7 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,i)) . (intpos b7) ) & ( for i being Element of NAT st b7 >= n0 + 1 & ( b7 < i . (intpos 3) or b7 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b6,i)) . (intpos b7) = i . (intpos b7) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: ( t . GBP = 0 & t . (intpos 5) > 0 & t . (intpos 5) <= (k + 1) + 1 & t . (intpos 4) = n3 + (t . (intpos 5)) & n3 = (t . (intpos 3)) - 1 & t . (intpos 2) = md & md <= n3 & n0 + 1 <= t . (intpos 3) & t . (intpos 4) <= n0 + n & g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n implies ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) <= b1 . (intpos 4) & b3,b4 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b6 & b6 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) < b6 & b6 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st b6 >= n0 + 1 & ( b6 < i . (intpos 3) or b6 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) = i . (intpos b6) ) ) )
assume that
A18: t . GBP = 0 and
A19: t . (intpos 5) > 0 and
A20: t . (intpos 5) <= (k + 1) + 1 and
A21: t . (intpos 4) = n3 + (t . (intpos 5)) and
A22: n3 = (t . (intpos 3)) - 1 and
A23: t . (intpos 2) = md and
A24: md <= n3 and
A25: n0 + 1 <= t . (intpos 3) and
A26: t . (intpos 4) <= n0 + n ; ::_thesis: ( g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n implies ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) <= b1 . (intpos 4) & b3,b4 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b6 & b6 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) < b6 & b6 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st b6 >= n0 + 1 & ( b6 < i . (intpos 3) or b6 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) = i . (intpos b6) ) ) )
set t1 = IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t);
set Q1 = Q;
A27: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . GBP by SCMPDS_5:15;
A28: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 1) = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos 1) by SCMPDS_5:15;
A29: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 2) = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos 2) by SCMPDS_5:15;
A30: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos 3) by SCMPDS_5:15;
A31: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos 4) by SCMPDS_5:15;
A32: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos 5) by SCMPDS_5:15;
A33: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5)) by A15, A18, A19, A21, A22, A23, A24, Lm20;
A34: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_4)_=_((v_._(intpos_3))_-_1)_+_(v_._(intpos_5))_&_md_<=_(v_._(intpos_3))_-_1_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),5))_>_0_holds_
(_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._GBP_=_v_._GBP_&_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))_is_closed_on_v,V_&_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))_is_halting_on_v,V_&_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(DataLoc_((t_._GBP),5))_<_v_._(DataLoc_((t_._GBP),5))_&_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_4)_=_(((IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_3))_-_1)_+_((IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_5))_&_md_<=_((IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_3))_-_1_&_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
A35: DataLoc ((t . GBP),5) = intpos (0 + 5) by A18, SCMP_GCD:1;
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 4) = ((v . (intpos 3)) - 1) + (v . (intpos 5)) & md <= (v . (intpos 3)) - 1 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 holds
( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 4) = ((v . (intpos 3)) - 1) + (v . (intpos 5)) & md <= (v . (intpos 3)) - 1 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 implies ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A36: v . (intpos 4) = ((v . (intpos 3)) - 1) + (v . (intpos 5)) and
A37: md <= (v . (intpos 3)) - 1 and
A38: v . (intpos 2) = t . (intpos 2) and
A39: v . GBP = t . GBP and
A40: v . (DataLoc ((t . GBP),5)) > 0 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
reconsider mm = (v . (intpos 3)) - 1 as Element of NAT by A37, INT_1:3;
set Iv = IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v);
A47: v . (intpos 4) = mm + (v . (intpos 5)) by A36;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP by A15, A18, A23, A37, A38, A39, A40, A35, Lm20; ::_thesis: ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V ) by A15, A18, A23, A37, A38, A39, A40, A35, A47, Lm20; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A15, A18, A23, A37, A38, A39, A40, A35, A47, Lm20; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) by A15, A18, A23, A37, A38, A39, A40, A35, A47, Lm20; ::_thesis: ( md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3) >= v . (intpos 3) by A15, A18, A23, A37, A38, A39, A40, A35, A47, Lm20;
then  ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 >= (v . (intpos 3)) - 1 by XREAL_1:9;
hence  md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 by A37, XXREAL_0:2; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2)
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) by A15, A18, A23, A37, A38, A39, A40, A35, A47, Lm20; ::_thesis: verum
end;
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) < t . (intpos 5) by A15, A18, A19, A21, A22, A23, A24, Lm20;
then  ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5)) + 1 <= t . (intpos 5) by INT_1:7;
then  ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5)) + 1 <= (k + 1) + 1 by A20, XXREAL_0:2;
then A48: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) <= k + 1 by XREAL_1:6;
set t2 = IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t);
assume A49: g is_FinSequence_on t,n0 ; ::_thesis: ( g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n implies ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) <= b1 . (intpos 4) & b3,b4 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b6 & b6 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) < b6 & b6 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st b6 >= n0 + 1 & ( b6 < i . (intpos 3) or b6 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) = i . (intpos b6) ) ) )
assume A50: g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 ; ::_thesis: ( len g = n & len g1 = n implies ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) <= b1 . (intpos 4) & b3,b4 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b6 & b6 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) < b6 & b6 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st b6 >= n0 + 1 & ( b6 < i . (intpos 3) or b6 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) = i . (intpos b6) ) ) )
A51: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP = 0 by A15, A18, A19, A21, A22, A23, A24, Lm20;
then A52: DataLoc (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
assume that
A53: len g = n and
A54: len g1 = n ; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) <= b1 . (intpos 4) & b3,b4 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b6 & b6 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) < b6 & b6 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st b6 >= n0 + 1 & ( b6 < i . (intpos 3) or b6 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) = i . (intpos b6) ) )
A55: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 2) = md by A15, A18, A19, A21, A22, A23, A24, Lm20;
A56: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A15, A18, A19, A21, A22, A23, A24, Lm20;
A57: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) >= t . (intpos 3) by A15, A18, A19, A21, A22, A23, A24, Lm20;
A58: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) >= n3 by A15, A18, A19, A21, A22, A23, A24, Lm20;
A59: DataLoc ((t . GBP),5) = intpos (0 + 5) by A18, SCMP_GCD:1;
A60: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) <= t . (intpos 4) by A15, A18, A19, A21, A22, A23, A24, Lm20;
consider n1, n2 being Element of NAT  such that
A61: n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3)) - 1 and
A62: n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4)) + 1 and
A63: for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) = t . (intpos i) and
A64: ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) and
A65: for i being Element of NAT st t . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) and
A66: for i being Element of NAT st n2 <= i & i <= t . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A15, A18, A19, A21, A22, A23, A24, Lm20;
A67: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) >= - 1 by A15, A18, A19, A21, A22, A23, A24, Lm20;
then A68: n1 <= n2 by A33, A61, A62, Lm23;
percases ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) > 0 or (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) <= 0 ) ;
supposeA69: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) > 0 ; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) <= b1 . (intpos 4) & b3,b4 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b6 & b6 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) < b6 & b6 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st b6 >= n0 + 1 & ( b6 < i . (intpos 3) or b6 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) = i . (intpos b6) ) )
set t3 = IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))));
consider f3 being FinSequence of INT  such that
A71: len f3 = n and
A72: for i being Element of NAT st 1 <= i & i <= len f3 holds
f3 . i = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos (n0 + i)) by SCPISORT:1;
A73: f3 is_FinSequence_on IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t),n0 by A72, SCPISORT:def_1;
A74: f3 is_FinSequence_on Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)),n0
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f3 or f3 . i = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos (n0 + i)) )
assume  ( 1 <= i & i <= len f3 ) ; ::_thesis: f3 . i = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos (n0 + i))
then  f3 . i = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos (n0 + i)) by A72;
hence  f3 . i = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos (n0 + i)) by SCMPDS_5:15; ::_thesis: verum
end;
A75: ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
proof
percases ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A64;
suppose ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) ; ::_thesis: ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
hence  ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) ; ::_thesis: verum
end;
supposeA76: ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ; ::_thesis: ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
then A77: n2 <= n0 + n by A26, XXREAL_0:2;
 n1 >= n0 + 1 by A25, A76, XXREAL_0:2;
hence  ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A68, A76, A77, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
A78: n0 + 1 <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) by A25, A57, XXREAL_0:2;
 ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3)) - 1 >= (t . (intpos 3)) - 1 by A57, XREAL_1:9;
then A79: md <= n1 by A22, A24, A61, XXREAL_0:2;
now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_n0_+_1_&_i_<>_n1_&_i_<>_n2_holds_
(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= n0 + 1 & i <> n1 & i <> n2 implies (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume that
A80: i >= n0 + 1 and
A81: i <> n1 and
A82: i <> n2 ; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) = t . (intpos i)
 n0 + 1 >= 7 + 1 by A12, XREAL_1:6;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) = t . (intpos i) by A63, A80, A81, A82, XXREAL_0:2; ::_thesis: verum
end;
then A83: g,f3 are_fiberwise_equipotent by A49, A53, A71, A73, A75, Lm24;
A84: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) <= n0 + n by A26, A60, XXREAL_0:2;
A85: IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t) = IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)))) by A19, A21, A22, A24, A59, A34, Th7;
consider f4 being FinSequence of INT  such that
A86: len f4 = n and
A87: for i being Element of NAT st 1 <= i & i <= len f4 holds
f4 . i = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))))) . (intpos (n0 + i)) by SCPISORT:1;
A88: f4 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)))),n0
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f4 or f4 . i = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))))) . (intpos (n0 + i)) )
assume  ( 1 <= i & i <= len f4 ) ; ::_thesis: f4 . i = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))))) . (intpos (n0 + i))
hence  f4 . i = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))))) . (intpos (n0 + i)) by A87; ::_thesis: verum
end;
thus  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 by A17, A33, A48, A51, A55, A61, A69, A71, A74, A85, A86, A88, A79, A78, A84, A27, A29, A30, A31, A32; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
A89: ( ( md < (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) or md > (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) ) implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos md) ) by A6, A17, A33, A48, A51, A55, A61, A69, A71, A74, A78, A79, A84, A85, A86, A88, A27, A29, A30, A31, A32;
A90: md < (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) by A61, A79, XREAL_1:146, XXREAL_0:2;
A91: dom f4 = Seg n by A86, FINSEQ_1:def_3;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_f4_holds_
f4_._i_=_g1_._i
let i be Nat; ::_thesis: ( i in dom f4 implies f4 . i = g1 . i )
reconsider a = i as Element of NAT by ORDINAL1:def_12;
assume A92: i in dom f4 ; ::_thesis: f4 . i = g1 . i
then A93: i <= n by A91, FINSEQ_1:1;
A94: 1 <= i by A91, A92, FINSEQ_1:1;
hence f4 . i = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos (n0 + a)) by A86, A87, A85, A93
.= g1 . i by A50, A54, A94, A93, SCPISORT:def_1 ;
::_thesis: verum
end;
then A95: f4 = g1 by A54, A86, FINSEQ_2:9;
thus  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A17, A51, A56, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A85, A27, A28, A29, A30, A31, A32; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
thus  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A85, A27, A29, A30, A31, A32; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
thus  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A85, A27, A29, A30, A31, A32; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
 (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))))) . (intpos 4) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A27, A29, A30, A31, A32;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) by A60, A85, XXREAL_0:2; ::_thesis: ( g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
 f3,f4 are_fiberwise_equipotent by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A27, A29, A30, A31, A32;
hence  g,g1 are_fiberwise_equipotent by A83, A95, CLASSES1:76; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
A96: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) = n1 + 1 by A61;
hereby  ::_thesis: ( ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
let i be Element of NAT ; ::_thesis: ( t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) )
assume that
A97: t . (intpos 3) <= i and
A98: i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
A99: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos i) by SCMPDS_5:15;
A100: n0 + 1 <= i by A25, A97, XXREAL_0:2;
hereby  ::_thesis: verum
percases ( i < (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) or i >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) ) ;
supposeA101: i < (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
then A102: i <= n1 by A96, INT_1:7;
A103: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A85, A100, A101, A99, A27, A29, A30, A31, A32;
 (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) by A90, A89, SCMPDS_5:15;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) by A65, A97, A102, A103; ::_thesis: verum
end;
supposeA104: i >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
thus  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A85, A98, A104, A27, A29, A30, A31, A32; ::_thesis: verum
end;
end;
end;
end;
hereby  ::_thesis: for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i)
let i be Element of NAT ; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) )
assume that
A105: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i and
A106: i <= t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
A107: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos i) by SCMPDS_5:15;
hereby  ::_thesis: verum
percases ( i > (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) or i <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) ) ;
supposeA108: i > (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
then A109: i >= n2 by A62, INT_1:7;
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) > n1 + 0 by A33, A61, A69, XREAL_1:6;
then  n2 > n1 + 1 by A62, XREAL_1:6;
then  n2 >= n0 + 1 by A61, A78, XXREAL_0:2;
then  i >= n0 + 1 by A109, XXREAL_0:2;
then A110: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A85, A108, A27, A29, A30, A31, A32, A107;
 (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) by A90, A89, SCMPDS_5:15;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) by A66, A106, A109, A110; ::_thesis: verum
end;
supposeA111: i <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
thus  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A85, A105, A111, A27, A29, A30, A31, A32; ::_thesis: verum
end;
end;
end;
end;
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos b1) = t . (intpos b1) )
A112: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) = (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (intpos i) by SCMPDS_5:15;
assume that
A113: i >= n0 + 1 and
A114: ( i < t . (intpos 3) or i > t . (intpos 4) ) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos b1) = t . (intpos b1)
 n0 + 1 >= 7 + 1 by A12, XREAL_1:6;
then A115: i >= 8 by A113, XXREAL_0:2;
then A116: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) = t . (intpos i) by A63, A64, A68, A114, Lm25;
percases ( i < t . (intpos 3) or i > t . (intpos 4) ) by A114;
suppose i < t . (intpos 3) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos b1) = t . (intpos b1)
then  i < (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3) by A57, XXREAL_0:2;
hence (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A85, A113, A27, A29, A30, A31, A32, A112
.= t . (intpos i) by A63, A64, A68, A114, Lm25, A115 ;
::_thesis: verum
end;
suppose i > t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos b1) = t . (intpos b1)
then  i > (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) by A60, XXREAL_0:2;
hence (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A17, A51, A55, A33, A61, A48, A69, A71, A74, A86, A88, A79, A78, A84, A85, A113, A27, A29, A30, A31, A32, A112
.= t . (intpos i) by A116 ;
::_thesis: verum
end;
end;
end;
end;
supposeA117: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) <= 0 ; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 1) = b1 . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,b1)) . (intpos 4) <= b1 . (intpos 4) & b3,b4 are_fiberwise_equipotent & ( for i being Element of NAT st i . (intpos 3) <= b6 & b6 <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos 4) < b6 & b6 <= i . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) ) & ( for i being Element of NAT st b6 >= n0 + 1 & ( b6 < i . (intpos 3) or b6 > i . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),b5,i)) . (intpos b6) = i . (intpos b6) ) )
A118: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._x_=_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),Q,t))_._x
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . x = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . x
X1: (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . GBP = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (DataLoc (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP),5)) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (DataLoc (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP),5)) by SCMPDS_5:15;
thus (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . x = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))))) . x by A19, A21, A22, A24, A59, A34, Th7
.= (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . x by A52, A117, X1, X2, SCMPDS_8:23
.= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 by A51; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
thus  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A56, A118; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
thus  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md by A55, A118; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
 (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= n3 by A58, A118;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md by A24, XXREAL_0:2; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
thus  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) by A60, A118; ::_thesis: ( g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
A119: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_n0_+_1_&_i_<>_n1_&_i_<>_n2_holds_
(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
A120: n0 + 1 >= 7 + 1 by A12, XREAL_1:6;
let i be Element of NAT ; ::_thesis: ( i >= n0 + 1 & i <> n1 & i <> n2 implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume that
A121: i >= n0 + 1 and
A122: i <> n1 and
A123: i <> n2 ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A118
.= t . (intpos i) by A63, A121, A122, A123, A120, XXREAL_0:2 ; ::_thesis: verum
end;
A124: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
proof
percases ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A64;
suppose ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) ; ::_thesis: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
hence  ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A118; ::_thesis: verum
end;
suppose ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ; ::_thesis: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
hence  ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A118; ::_thesis: verum
end;
end;
end;
 ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
proof
percases ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A124;
suppose ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) ; ::_thesis: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
hence  ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) ; ::_thesis: verum
end;
supposeA125: ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ; ::_thesis: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
then A126: n2 <= n0 + n by A26, XXREAL_0:2;
 n1 >= n0 + 1 by A25, A125, XXREAL_0:2;
hence  ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A68, A125, A126, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  g,g1 are_fiberwise_equipotent by A49, A50, A53, A54, A119, Lm24; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
A127: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) = n2 - 1 by A62, A118;
hereby  ::_thesis: ( ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
let i be Element of NAT ; ::_thesis: ( t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) )
assume that
A128: t . (intpos 3) <= i and
A129: i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
 i < n2 by A127, A129, XREAL_1:146, XXREAL_0:2;
then  i <= n1 by A67, A33, A61, A62, A117, Lm22;
then  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A65, A128;
then  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A118;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) by A118; ::_thesis: verum
end;
hereby  ::_thesis: for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i)
let i be Element of NAT ; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) )
assume that
A130: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i and
A131: i <= t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
 ((IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4)) + 1 = n2 by A62, A118;
then  n2 <= i by A130, INT_1:7;
then  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A66, A131;
then  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A118;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) by A118; ::_thesis: verum
end;
thus  for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) by A68, A119, A124, Lm25; ::_thesis: verum
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
 s . (intpos 5) >= 1 + 0 by A2, INT_1:7;
then reconsider m5 = (s . (intpos 5)) - 1 as Element of NAT by INT_1:3, XREAL_1:48;
A132: s . (intpos 5) = m5 + 1 ;
A133: S1[ 0 ]
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for n3 being Element of NAT
for g, g1 being FinSequence of INT st t . GBP = 0 & t . (intpos 5) > 0 & t . (intpos 5) <= 0 + 1 & t . (intpos 4) = n3 + (t . (intpos 5)) & n3 = (t . (intpos 3)) - 1 & t . (intpos 2) = md & md <= n3 & n0 + 1 <= t . (intpos 3) & t . (intpos 4) <= n0 + n & g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for n3 being Element of NAT
for g, g1 being FinSequence of INT st t . GBP = 0 & t . (intpos 5) > 0 & t . (intpos 5) <= 0 + 1 & t . (intpos 4) = n3 + (t . (intpos 5)) & n3 = (t . (intpos 3)) - 1 & t . (intpos 2) = md & md <= n3 & n0 + 1 <= t . (intpos 3) & t . (intpos 4) <= n0 + n & g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
let n3 be Element of NAT ; ::_thesis: for g, g1 being FinSequence of INT st t . GBP = 0 & t . (intpos 5) > 0 & t . (intpos 5) <= 0 + 1 & t . (intpos 4) = n3 + (t . (intpos 5)) & n3 = (t . (intpos 3)) - 1 & t . (intpos 2) = md & md <= n3 & n0 + 1 <= t . (intpos 3) & t . (intpos 4) <= n0 + n & g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n holds
( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
let g, g1 be FinSequence of INT ; ::_thesis: ( t . GBP = 0 & t . (intpos 5) > 0 & t . (intpos 5) <= 0 + 1 & t . (intpos 4) = n3 + (t . (intpos 5)) & n3 = (t . (intpos 3)) - 1 & t . (intpos 2) = md & md <= n3 & n0 + 1 <= t . (intpos 3) & t . (intpos 4) <= n0 + n & g is_FinSequence_on t,n0 & g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 & len g = n & len g1 = n implies ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) ) )
assume that
A134: t . GBP = 0 and
A135: t . (intpos 5) > 0 and
A136: t . (intpos 5) <= 0 + 1 and
A137: t . (intpos 4) = n3 + (t . (intpos 5)) and
A138: n3 = (t . (intpos 3)) - 1 and
A139: t . (intpos 2) = md and
A140: md <= n3 and
A141: n0 + 1 <= t . (intpos 3) and
A142: t . (intpos 4) <= n0 + n ; ::_thesis: ( not g is_FinSequence_on t,n0 or not g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 or not len g = n or not len g1 = n or ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) ) )
set t2 = IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t);
set Q2 = Q;
assume A143: g is_FinSequence_on t,n0 ; ::_thesis: ( not g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 or not len g = n or not len g1 = n or ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) ) )
set t1 = IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t);
set Q1 = Q;
assume A144: g1 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t),n0 ; ::_thesis: ( not len g = n or not len g1 = n or ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) ) )
A145: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5)) by A15, A134, A135, A137, A138, A139, A140, Lm20;
consider n1, n2 being Element of NAT  such that
A146: n1 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 3)) - 1 and
A147: n2 = ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4)) + 1 and
A148: for i being Element of NAT st i >= 8 & i <> n1 & i <> n2 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) = t . (intpos i) and
A149: ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) and
A150: for i being Element of NAT st t . (intpos 3) <= i & i <= n1 holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) and
A151: for i being Element of NAT st n2 <= i & i <= t . (intpos 4) holds
(IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A15, A134, A135, A137, A138, A139, A140, Lm20;
A152: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) >= - 1 by A15, A134, A135, A137, A138, A139, A140, Lm20;
then A153: n1 <= n2 by A145, A146, A147, Lm23;
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) < t . (intpos 5) by A15, A134, A135, A137, A138, A139, A140, Lm20;
then  ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5)) + 1 <= t . (intpos 5) by INT_1:7;
then  ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5)) + 1 <= 0 + 1 by A136, XXREAL_0:2;
then A154: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 5) <= 0 by XREAL_1:6;
A155: now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_4)_=_((v_._(intpos_3))_-_1)_+_(v_._(intpos_5))_&_md_<=_(v_._(intpos_3))_-_1_&_v_._(intpos_2)_=_t_._(intpos_2)_&_v_._GBP_=_t_._GBP_&_v_._(DataLoc_((t_._GBP),5))_>_0_holds_
(_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._GBP_=_v_._GBP_&_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))_is_closed_on_v,V_&_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))_is_halting_on_v,V_&_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(DataLoc_((t_._GBP),5))_<_v_._(DataLoc_((t_._GBP),5))_&_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_4)_=_(((IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_3))_-_1)_+_((IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_5))_&_md_<=_((IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_3))_-_1_&_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
A156: DataLoc ((t . GBP),5) = intpos (0 + 5) by A134, SCMP_GCD:1;
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 4) = ((v . (intpos 3)) - 1) + (v . (intpos 5)) & md <= (v . (intpos 3)) - 1 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 holds
( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 4) = ((v . (intpos 3)) - 1) + (v . (intpos 5)) & md <= (v . (intpos 3)) - 1 & v . (intpos 2) = t . (intpos 2) & v . GBP = t . GBP & v . (DataLoc ((t . GBP),5)) > 0 implies ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A157: v . (intpos 4) = ((v . (intpos 3)) - 1) + (v . (intpos 5)) and
A158: md <= (v . (intpos 3)) - 1 and
A159: v . (intpos 2) = t . (intpos 2) and
A160: v . GBP = t . GBP and
A161: v . (DataLoc ((t . GBP),5)) > 0 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
reconsider mm = (v . (intpos 3)) - 1 as Element of NAT by A158, INT_1:3;
set Iv = IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v);
A168: v . (intpos 4) = mm + (v . (intpos 5)) by A157;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP by A15, A134, A139, A158, A159, A160, A161, A156, Lm20; ::_thesis: ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V ) by A15, A134, A139, A158, A159, A160, A161, A156, A168, Lm20; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((t . GBP),5)) < v . (DataLoc ((t . GBP),5)) by A15, A134, A139, A158, A159, A160, A161, A156, A168, Lm20; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) by A15, A134, A139, A158, A159, A160, A161, A156, A168, Lm20; ::_thesis: ( md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3) >= v . (intpos 3) by A15, A134, A139, A158, A159, A160, A161, A156, A168, Lm20;
then  ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 >= (v . (intpos 3)) - 1 by XREAL_1:9;
hence  md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 by A158, XXREAL_0:2; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2)
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) by A15, A134, A139, A158, A159, A160, A161, A156, A168, Lm20; ::_thesis: verum
end;
A169: DataLoc ((t . GBP),5) = intpos (0 + 5) by A134, SCMP_GCD:1;
assume that
A170: len g = n and
A171: len g1 = n ; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
A172: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP = 0 by A15, A134, A135, A137, A138, A139, A140, Lm20;
then A173: DataLoc (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A174: now__::_thesis:_for_x_being_Int_position_holds_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._x_=_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),Q,t))_._x
X1: (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . GBP = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . (DataLoc (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP),5)) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (DataLoc (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . GBP),5)) by SCMPDS_5:15;
let x be Int_position; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . x = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . x
thus (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . x = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,(Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))))) . x by A135, A137, A138, A140, A169, A155, Th7
.= (Initialize (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t))) . x by A173, A154, X1, X2, SCMPDS_8:23
.= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . x by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . GBP = 0 by A172; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A15, A134, A135, A137, A138, A139, A140, Lm20;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 1) = t . (intpos 1) by A174; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 2) = md by A15, A134, A135, A137, A138, A139, A140, Lm20;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 2) = md by A174; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) >= n3 by A15, A134, A135, A137, A138, A139, A140, Lm20;
then  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= n3 by A174;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) >= md by A140, XXREAL_0:2; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) & g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos 4) <= t . (intpos 4) by A15, A134, A135, A137, A138, A139, A140, Lm20;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) <= t . (intpos 4) by A174; ::_thesis: ( g,g1 are_fiberwise_equipotent & ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
A175: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_n0_+_1_&_i_<>_n1_&_i_<>_n2_holds_
(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),Q,t))_._(intpos_i)_=_t_._(intpos_i)
A176: n0 + 1 >= 7 + 1 by A12, XREAL_1:6;
let i be Element of NAT ; ::_thesis: ( i >= n0 + 1 & i <> n1 & i <> n2 implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) )
assume that
A177: i >= n0 + 1 and
A178: i <> n1 and
A179: i <> n2 ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i)
thus (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A174
.= t . (intpos i) by A148, A177, A178, A179, A176, XXREAL_0:2 ; ::_thesis: verum
end;
A180: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
proof
percases ( ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A149;
suppose ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) ; ::_thesis: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
hence  ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A174; ::_thesis: verum
end;
suppose ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ; ::_thesis: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
hence  ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A174; ::_thesis: verum
end;
end;
end;
 ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
proof
percases ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A180;
suppose ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) ; ::_thesis: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
hence  ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) ; ::_thesis: verum
end;
supposeA181: ( t . (intpos 3) <= n1 & n2 <= t . (intpos 4) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ; ::_thesis: ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) )
then A182: n2 <= n0 + n by A142, XXREAL_0:2;
 n1 >= n0 + 1 by A141, A181, XXREAL_0:2;
hence  ( ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n2) ) or ( n1 >= n0 + 1 & n2 >= n0 + 1 & n1 <= n0 + n & n2 <= n0 + n & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n1) = t . (intpos n2) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos n2) = t . (intpos n1) ) ) by A153, A181, A182, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence  g,g1 are_fiberwise_equipotent by A143, A144, A170, A171, A175, Lm24; ::_thesis: ( ( for i being Element of NAT st t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
A183: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) = n2 - 1 by A147, A174;
hereby  ::_thesis: ( ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) ) )
let i be Element of NAT ; ::_thesis: ( t . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) )
assume that
A184: t . (intpos 3) <= i and
A185: i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
 i < n2 by A183, A185, XREAL_1:146, XXREAL_0:2;
then  i <= n1 by A152, A145, A146, A147, A154, Lm22;
then  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A150, A184;
then  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A174;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) by A174; ::_thesis: verum
end;
hereby  ::_thesis: for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i)
let i be Element of NAT ; ::_thesis: ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i & i <= t . (intpos 4) implies (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) )
assume that
A186: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4) < i and
A187: i <= t . (intpos 4) ; ::_thesis: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i)
 ((IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos 4)) + 1 = n2 by A147, A174;
then  n2 <= i by A186, INT_1:7;
then  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A151, A187;
then  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),Q,t)) . (intpos i) by A174;
hence  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) by A174; ::_thesis: verum
end;
thus  for i being Element of NAT st i >= n0 + 1 & ( i < t . (intpos 3) or i > t . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),Q,t)) . (intpos i) = t . (intpos i) by A175, A153, A180, Lm25; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A133, A16);
hence  ( (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . GBP = 0 & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 1) = s . (intpos 1) & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 2) = md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) >= md & (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) <= s . (intpos 4) & f,f1 are_fiberwise_equipotent & ( for i being Element of NAT st s . (intpos 3) <= i & i <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos 4) < i & i <= s . (intpos 4) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 3) or i > s . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,s)) . (intpos i) = s . (intpos i) ) ) by A1, A3, A4, A5, A7, A8, A9, A10, A11, A13, A14, A132; ::_thesis: verum
end;

Lm27:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, m3, n0 being Element of NAT st s . GBP = 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 >= 7 holds
( while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on s,P & while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, m3, n0 being Element of NAT st s . GBP = 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 >= 7 holds
( while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on s,P & while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on s,P )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: for md, m3, n0 being Element of NAT st s . GBP = 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 >= 7 holds
( while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on s,P & while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on s,P )
let md, m3, n0 be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 4) = m3 + (s . (intpos 5)) & m3 = (s . (intpos 3)) - 1 & s . (intpos 2) = md & md >= n0 + 1 & md <= m3 & n0 >= 7 implies ( while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on s,P & while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on s,P ) )
assume that
A1: s . GBP = 0 and
A2: s . (intpos 4) = m3 + (s . (intpos 5)) and
A3: m3 = (s . (intpos 3)) - 1 and
A4: s . (intpos 2) = md and
A5: md >= n0 + 1 and
A6: md <= m3 and
A7: n0 >= 7 ; ::_thesis: ( while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on s,P & while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on s,P )
 n0 + 1 >= 7 + 1 by A7, XREAL_1:6;
then A8: md >= 8 by A5, XXREAL_0:2;
A9: DataLoc ((s . GBP),5) = intpos (0 + 5) by A1, SCMP_GCD:1;
now__::_thesis:_for_v_being_0_-started_State_of_SCMPDS
for_V_being_Instruction-Sequence_of_SCMPDS_st_v_._(intpos_4)_=_((v_._(intpos_3))_-_1)_+_(v_._(intpos_5))_&_md_<=_(v_._(intpos_3))_-_1_&_v_._(intpos_2)_=_s_._(intpos_2)_&_v_._GBP_=_s_._GBP_&_v_._(DataLoc_((s_._GBP),5))_>_0_holds_
(_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._GBP_=_v_._GBP_&_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))_is_closed_on_v,V_&_((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))_is_halting_on_v,V_&_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(DataLoc_((s_._GBP),5))_<_v_._(DataLoc_((s_._GBP),5))_&_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_4)_=_(((IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_3))_-_1)_+_((IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_5))_&_md_<=_((IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_3))_-_1_&_(IExec_((((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))),V,v))_._(intpos_2)_=_v_._(intpos_2)_)
let v be 0 -started State of SCMPDS; ::_thesis: for V being Instruction-Sequence of SCMPDS st v . (intpos 4) = ((v . (intpos 3)) - 1) + (v . (intpos 5)) & md <= (v . (intpos 3)) - 1 & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),5)) > 0 holds
( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
let V be Instruction-Sequence of SCMPDS; ::_thesis: ( v . (intpos 4) = ((v . (intpos 3)) - 1) + (v . (intpos 5)) & md <= (v . (intpos 3)) - 1 & v . (intpos 2) = s . (intpos 2) & v . GBP = s . GBP & v . (DataLoc ((s . GBP),5)) > 0 implies ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) ) )
assume that
A10: v . (intpos 4) = ((v . (intpos 3)) - 1) + (v . (intpos 5)) and
A11: md <= (v . (intpos 3)) - 1 and
A12: v . (intpos 2) = s . (intpos 2) and
A13: v . GBP = s . GBP and
A14: v . (DataLoc ((s . GBP),5)) > 0 ; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
set Iv = IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v);
reconsider mm = (v . (intpos 3)) - 1 as Element of NAT by A11, INT_1:3;
A21: v . (intpos 4) = mm + (v . (intpos 5)) by A10;
hence  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . GBP = v . GBP by A1, A4, A8, A9, A11, A12, A13, A14, Lm20; ::_thesis: ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  ( ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_closed_on v,V & ((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))) is_halting_on v,V ) by A1, A4, A8, A9, A11, A12, A13, A14, A21, Lm20; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (DataLoc ((s . GBP),5)) < v . (DataLoc ((s . GBP),5)) by A1, A4, A8, A9, A11, A12, A13, A14, A21, Lm20; ::_thesis: ( (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) & md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 4) = (((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1) + ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 5)) by A1, A4, A8, A9, A11, A12, A13, A14, A21, Lm20; ::_thesis: ( md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 & (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) )
 (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3) >= v . (intpos 3) by A1, A4, A8, A9, A11, A12, A13, A14, A21, Lm20;
then  ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 >= (v . (intpos 3)) - 1 by XREAL_1:9;
hence  md <= ((IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 3)) - 1 by A11, XXREAL_0:2; ::_thesis: (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2)
thus  (IExec ((((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))),V,v)) . (intpos 2) = v . (intpos 2) by A1, A4, A8, A9, A11, A12, A13, A14, A21, Lm20; ::_thesis: verum
end;
hence  ( while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on s,P & while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on s,P ) by A2, A3, A6, Th7; ::_thesis: verum
end;

Lm28:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS st s . GBP = 0 holds
( (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP = 0 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (s . (intpos 2)) + 1 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = s . (intpos 4) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) & ( for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS st s . GBP = 0 holds
( (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP = 0 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (s . (intpos 2)) + 1 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = s . (intpos 4) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) & ( for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
set a = GBP ;
let s be 0 -started State of SCMPDS; ::_thesis: ( s . GBP = 0 implies ( (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP = 0 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (s . (intpos 2)) + 1 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = s . (intpos 4) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) & ( for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) ) )
set t0 = Initialize s;
set t1 = IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s));
set t2 = IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s));
set t3 = IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s));
set t4 = Exec (((GBP,5) := (GBP,4)),(Initialize s));
assume  s . GBP = 0 ; ::_thesis: ( (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP = 0 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (s . (intpos 2)) + 1 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = s . (intpos 4) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) & ( for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
then A1: (Initialize s) . GBP = 0 by SCMPDS_5:15;
then A2: DataLoc (((Initialize s) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
then A3: (Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos 5) = (Initialize s) . (DataLoc (((Initialize s) . GBP),4)) by SCMPDS_2:47
.= (Initialize s) . (intpos (0 + 4)) by A1, SCMP_GCD:1
.= s . (intpos 4) by SCMPDS_5:15 ;
 (Initialize s) . (intpos 4) = s . (intpos 4) by SCMPDS_5:15;
then A4: (Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos 4) = s . (intpos 4) by A2, AMI_3:10, SCMPDS_2:47;
 (Initialize s) . (intpos 1) = s . (intpos 1) by SCMPDS_5:15;
then A5: (Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos 1) = s . (intpos 1) by A2, AMI_3:10, SCMPDS_2:47;
A6: (Exec (((GBP,5) := (GBP,4)),(Initialize s))) . GBP = 0 by A1, A2, AMI_3:10, SCMPDS_2:47;
then A7: DataLoc (((Exec (((GBP,5) := (GBP,4)),(Initialize s))) . GBP),5) = intpos (0 + 5) by SCMP_GCD:1;
A8: (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . GBP = (Exec ((SubFrom (GBP,5,GBP,2)),(Exec (((GBP,5) := (GBP,4)),(Initialize s))))) . GBP by SCMPDS_5:42
.= 0 by A6, A7, AMI_3:10, SCMPDS_2:50 ;
then A9: DataLoc (((IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . GBP),3) = intpos (0 + 3) by SCMP_GCD:1;
A10: (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . (intpos 4) = (Exec ((SubFrom (GBP,5,GBP,2)),(Exec (((GBP,5) := (GBP,4)),(Initialize s))))) . (intpos 4) by SCMPDS_5:42
.= s . (intpos 4) by A4, A7, AMI_3:10, SCMPDS_2:50 ;
A11: (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . (intpos 4) = (Exec (((GBP,3) := (GBP,2)),(IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= s . (intpos 4) by A10, A9, AMI_3:10, SCMPDS_2:47 ;
A12: (Initialize s) . (intpos 2) = s . (intpos 2) by SCMPDS_5:15;
then A13: (Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos 2) = s . (intpos 2) by A2, AMI_3:10, SCMPDS_2:47;
A14: (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . (intpos 5) = (Exec ((SubFrom (GBP,5,GBP,2)),(Exec (((GBP,5) := (GBP,4)),(Initialize s))))) . (intpos 5) by SCMPDS_5:42
.= ((Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos 5)) - ((Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (DataLoc (((Exec (((GBP,5) := (GBP,4)),(Initialize s))) . GBP),2))) by A7, SCMPDS_2:50
.= ((Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos 5)) - ((Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos (0 + 2))) by A6, SCMP_GCD:1
.= (s . (intpos 4)) - (s . (intpos 2)) by A12, A2, A3, AMI_3:10, SCMPDS_2:47 ;
A15: (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . (intpos 5) = (Exec (((GBP,3) := (GBP,2)),(IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 4)) - (s . (intpos 2)) by A14, A9, AMI_3:10, SCMPDS_2:47 ;
A16: (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . (intpos 2) = (Exec ((SubFrom (GBP,5,GBP,2)),(Exec (((GBP,5) := (GBP,4)),(Initialize s))))) . (intpos 2) by SCMPDS_5:42
.= s . (intpos 2) by A13, A7, AMI_3:10, SCMPDS_2:50 ;
A17: (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . (intpos 1) = (Exec ((SubFrom (GBP,5,GBP,2)),(Exec (((GBP,5) := (GBP,4)),(Initialize s))))) . (intpos 1) by SCMPDS_5:42
.= s . (intpos 1) by A5, A7, AMI_3:10, SCMPDS_2:50 ;
A18: (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . (intpos 1) = (Exec (((GBP,3) := (GBP,2)),(IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A17, A9, AMI_3:10, SCMPDS_2:47 ;
A19: (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . GBP = (Exec (((GBP,3) := (GBP,2)),(IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A8, A9, AMI_3:10, SCMPDS_2:47 ;
then A20: DataLoc (((IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . GBP),3) = intpos (0 + 3) by SCMP_GCD:1;
A21: (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . (intpos 3) = (Exec (((GBP,3) := (GBP,2)),(IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . (DataLoc (((IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . GBP),2)) by A9, SCMPDS_2:47
.= s . (intpos 2) by A8, A16, SCMP_GCD:1 ;
A22: (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . (intpos 2) = (Exec (((GBP,3) := (GBP,2)),(IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A16, A9, AMI_3:10, SCMPDS_2:47 ;
thus (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP = (Exec ((AddTo (GBP,3,1)),(IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))))) . GBP by SCMPDS_5:41
.= 0 by A19, A20, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 1) = s . (intpos 1) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (s . (intpos 2)) + 1 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = s . (intpos 4) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) & ( for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
thus (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 1) = (Exec ((AddTo (GBP,3,1)),(IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))))) . (intpos 1) by SCMPDS_5:41
.= s . (intpos 1) by A18, A20, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (s . (intpos 2)) + 1 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = s . (intpos 4) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) & ( for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
thus (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = (Exec ((AddTo (GBP,3,1)),(IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))))) . (intpos 2) by SCMPDS_5:41
.= s . (intpos 2) by A22, A20, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (s . (intpos 2)) + 1 & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = s . (intpos 4) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) & ( for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
thus (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (Exec ((AddTo (GBP,3,1)),(IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))))) . (intpos 3) by SCMPDS_5:41
.= (s . (intpos 2)) + 1 by A21, A20, SCMPDS_2:48 ; ::_thesis: ( (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = s . (intpos 4) & (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) & ( for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
thus (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = (Exec ((AddTo (GBP,3,1)),(IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))))) . (intpos 4) by SCMPDS_5:41
.= s . (intpos 4) by A11, A20, AMI_3:10, SCMPDS_2:48 ; ::_thesis: ( (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) & ( for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) ) )
thus (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (Exec ((AddTo (GBP,3,1)),(IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))))) . (intpos 5) by SCMPDS_5:41
.= (s . (intpos 4)) - (s . (intpos 2)) by A15, A20, AMI_3:10, SCMPDS_2:48 ; ::_thesis: for i being Element of NAT st i >= 8 holds
(IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i)
A23: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(Exec_(((GBP,5)_:=_(GBP,4)),(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos i) = s . (intpos i) )
assume  i >= 8 ; ::_thesis: (Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos i) = s . (intpos i)
then  i > 5 by XXREAL_0:2;
hence (Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos i) = (Initialize s) . (intpos i) by A2, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by SCMPDS_5:15 ;
::_thesis: verum
end;
A24: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume A25: i >= 8 ; ::_thesis: (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . (intpos i) = s . (intpos i)
then A26: i > 5 by XXREAL_0:2;
thus (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . (intpos i) = (Exec ((SubFrom (GBP,5,GBP,2)),(Exec (((GBP,5) := (GBP,4)),(Initialize s))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,5) := (GBP,4)),(Initialize s))) . (intpos i) by A7, A26, AMI_3:10, SCMPDS_2:50
.= s . (intpos i) by A23, A25 ; ::_thesis: verum
end;
A27: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2)))_';'_((GBP,3)_:=_(GBP,2))),P,(Initialize_s)))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume A28: i >= 8 ; ::_thesis: (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . (intpos i) = s . (intpos i)
then A29: i > 3 by XXREAL_0:2;
thus (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . (intpos i) = (Exec (((GBP,3) := (GBP,2)),(IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))),P,(Initialize s))) . (intpos i) by A9, A29, AMI_3:10, SCMPDS_2:47
.= s . (intpos i) by A24, A28 ; ::_thesis: verum
end;
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i) )
assume A30: i >= 8 ; ::_thesis: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = s . (intpos i)
then A31: i > 3 by XXREAL_0:2;
thus (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos i) = (Exec ((AddTo (GBP,3,1)),(IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))),P,(Initialize s))) . (intpos i) by A20, A31, AMI_3:10, SCMPDS_2:48
.= s . (intpos i) by A27, A30 ; ::_thesis: verum
end;
end;

theorem Th13: :: SCPQSORT:10
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, p0 being Element of NAT st s . GBP = 0 & s . (intpos 2) = md & md >= p0 + 1 & p0 >= 7 holds
( Partition is_closed_on s,P & Partition is_halting_on s,P )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, p0 being Element of NAT st s . GBP = 0 & s . (intpos 2) = md & md >= p0 + 1 & p0 >= 7 holds
( Partition is_closed_on s,P & Partition is_halting_on s,P )
set KW = (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))));
let s be 0 -started State of SCMPDS; ::_thesis: for md, p0 being Element of NAT st s . GBP = 0 & s . (intpos 2) = md & md >= p0 + 1 & p0 >= 7 holds
( Partition is_closed_on s,P & Partition is_halting_on s,P )
let md, n0 be Element of NAT ; ::_thesis: ( s . GBP = 0 & s . (intpos 2) = md & md >= n0 + 1 & n0 >= 7 implies ( Partition is_closed_on s,P & Partition is_halting_on s,P ) )
set s2 = IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s));
set a = GBP ;
set P2 = P;
assume that
A1: s . GBP = 0 and
A2: s . (intpos 2) = md and
A3: md >= n0 + 1 and
A4: n0 >= 7 ; ::_thesis: ( Partition is_closed_on s,P & Partition is_halting_on s,P )
A5: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) = s . (intpos 2) by A1, Lm28;
set m3 = md;
 (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) by A1, Lm28;
then A6: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) = md + ((IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5)) by A1, A2, Lm28;
 (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) = (s . (intpos 2)) + 1 by A1, Lm28;
then A7: s . (intpos 2) = ((IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3)) - 1 ;
A8: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP = 0 by A1, Lm28;
X0: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)))) . GBP = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . GBP by SCMPDS_5:15;
X1: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)))) . (intpos 2) = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 2) by SCMPDS_5:15;
X2: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)))) . (intpos 3) = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 3) by SCMPDS_5:15;
X3: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)))) . (intpos 4) = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 4) by SCMPDS_5:15;
X4: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)))) . (intpos 5) = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))) . (intpos 5) by SCMPDS_5:15;
B9: while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))),P by A2, A3, A4, A5, A7, A6, Lm27, A8, X0, X1, X2, X3, X4;
A9: while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)),P
proof
 P +* (stop (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) halts_on Initialize (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)))) by B9, SCMPDS_6:def_3;
hence  while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)),P by SCMPDS_6:def_3; ::_thesis: verum
end;
B10: while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))),P by A2, A3, A4, A8, A5, A7, A6, Lm27, X0, X1, X2, X3, X4;
A10: while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)),P
proof
 for k being Element of NAT holds IC (Comput ((P +* (stop (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))))),(Initialize (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s))))),k)) in dom (stop (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) by B10, SCMPDS_6:def_2;
hence  while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,(Initialize s)),P by SCMPDS_6:def_2; ::_thesis: verum
end;
then A11: (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))) is_closed_on s,P by A9, SCPISORT:9;
A12: (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))) is_halting_on s,P by A10, A9, SCPISORT:9;
then A13: ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0)) is_closed_on s,P by A11, SCPISORT:10;
A14: ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0)) is_halting_on s,P by A11, A12, SCPISORT:10;
then A15: (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0)) is_halting_on s,P by A13, SCPISORT:10;
 (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0)) is_closed_on s,P by A13, A14, SCPISORT:10;
hence  ( Partition is_closed_on s,P & Partition is_halting_on s,P ) by A15, SCPISORT:10; ::_thesis: verum
end;

theorem Th14: :: SCPQSORT:11
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for md, p0, n being Element of NAT
for f, f1 being FinSequence of INT st s . GBP = 0 & (s . (intpos 4)) - (s . (intpos 2)) > 0 & s . (intpos 2) = md & md >= p0 + 1 & s . (intpos 4) <= p0 + n & p0 >= 7 & f is_FinSequence_on s,p0 & len f = n & f1 is_FinSequence_on IExec (Partition,P,s),p0 & len f1 = n holds
( (IExec (Partition,P,s)) . GBP = 0 & (IExec (Partition,P,s)) . (intpos 1) = s . (intpos 1) & f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= p0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for md, p0, n being Element of NAT
for f, f1 being FinSequence of INT st s . GBP = 0 & (s . (intpos 4)) - (s . (intpos 2)) > 0 & s . (intpos 2) = md & md >= p0 + 1 & s . (intpos 4) <= p0 + n & p0 >= 7 & f is_FinSequence_on s,p0 & len f = n & f1 is_FinSequence_on IExec (Partition,P,s),p0 & len f1 = n holds
( (IExec (Partition,P,s)) . GBP = 0 & (IExec (Partition,P,s)) . (intpos 1) = s . (intpos 1) & f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= p0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) )
set KW = (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))));
let s be 0 -started State of SCMPDS; ::_thesis: for md, p0, n being Element of NAT
for f, f1 being FinSequence of INT st s . GBP = 0 & (s . (intpos 4)) - (s . (intpos 2)) > 0 & s . (intpos 2) = md & md >= p0 + 1 & s . (intpos 4) <= p0 + n & p0 >= 7 & f is_FinSequence_on s,p0 & len f = n & f1 is_FinSequence_on IExec (Partition,P,s),p0 & len f1 = n holds
( (IExec (Partition,P,s)) . GBP = 0 & (IExec (Partition,P,s)) . (intpos 1) = s . (intpos 1) & f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= p0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) )
let md, n0, n be Element of NAT ; ::_thesis: for f, f1 being FinSequence of INT st s . GBP = 0 & (s . (intpos 4)) - (s . (intpos 2)) > 0 & s . (intpos 2) = md & md >= n0 + 1 & s . (intpos 4) <= n0 + n & n0 >= 7 & f is_FinSequence_on s,n0 & len f = n & f1 is_FinSequence_on IExec (Partition,P,s),n0 & len f1 = n holds
( (IExec (Partition,P,s)) . GBP = 0 & (IExec (Partition,P,s)) . (intpos 1) = s . (intpos 1) & f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) )
let f, f1 be FinSequence of INT ; ::_thesis: ( s . GBP = 0 & (s . (intpos 4)) - (s . (intpos 2)) > 0 & s . (intpos 2) = md & md >= n0 + 1 & s . (intpos 4) <= n0 + n & n0 >= 7 & f is_FinSequence_on s,n0 & len f = n & f1 is_FinSequence_on IExec (Partition,P,s),n0 & len f1 = n implies ( (IExec (Partition,P,s)) . GBP = 0 & (IExec (Partition,P,s)) . (intpos 1) = s . (intpos 1) & f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) ) )
I: Initialize s = s by MEMSTR_0:44;
set s1 = IExec (Partition,P,s);
set s2 = IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s);
set P2 = P;
set s3 = IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s);
set a = GBP ;
A1: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . GBP = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . GBP by SCMPDS_5:15;
A2: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 1) = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 1) by SCMPDS_5:15;
A3: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 2) = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 2) by SCMPDS_5:15;
A4: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 3) by SCMPDS_5:15;
A5: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 4) = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 4) by SCMPDS_5:15;
A6: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 5) = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 5) by SCMPDS_5:15;
assume that
A7: s . GBP = 0 and
A8: (s . (intpos 4)) - (s . (intpos 2)) > 0 and
A9: s . (intpos 2) = md and
A10: md >= n0 + 1 and
A11: s . (intpos 4) <= n0 + n and
A12: n0 >= 7 ; ::_thesis: ( not f is_FinSequence_on s,n0 or not len f = n or not f1 is_FinSequence_on IExec (Partition,P,s),n0 or not len f1 = n or ( (IExec (Partition,P,s)) . GBP = 0 & (IExec (Partition,P,s)) . (intpos 1) = s . (intpos 1) & f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) ) )
A13: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . GBP = 0 by A7, Lm28, I;
A14: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 1) = s . (intpos 1) by A7, Lm28, I;
A15: n0 + 1 >= 7 + 1 by A12, XREAL_1:6;
then A16: md >= 8 by A10, XXREAL_0:2;
then A17: md > 1 by XXREAL_0:2;
A18: md > 4 by A16, XXREAL_0:2;
A19: md - n0 >= 1 by A10, XREAL_1:19;
then reconsider n1 = md - n0 as Element of NAT by INT_1:3;
A20: md = n0 + n1 ;
A21: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 4) = s . (intpos 4) by A7, Lm28, I;
set m3 = md;
 (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 5) = (s . (intpos 4)) - (s . (intpos 2)) by A7, Lm28, I;
then A22: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 4) = md + ((IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 5)) by A7, A9, Lm28, I;
set s4 = IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))));
consider f3 being FinSequence of INT  such that
A24: len f3 = n and
A25: for i being Element of NAT st 1 <= i & i <= len f3 holds
f3 . i = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos (n0 + i)) by SCPISORT:1;
A26: f3 is_FinSequence_on IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)))),n0 by A25, SCPISORT:def_1;
 md < s . (intpos 4) by A8, A9, XREAL_1:47;
then  md < n0 + n by A11, XXREAL_0:2;
then A27: n1 <= len f3 by A24, XREAL_1:20;
A28: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 2) = s . (intpos 2) by A7, Lm28, I;
A29: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) = (s . (intpos 2)) + 1 by A7, Lm28, I;
then A30: s . (intpos 2) = ((IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3)) - 1 ;
X1: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . GBP = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 2) = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 2) by SCMPDS_5:15;
X3: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 3) = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) by SCMPDS_5:15;
X4: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 4) = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 4) by SCMPDS_5:15;
X5: (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 5) = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 5) by SCMPDS_5:15;
B31: while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)),P by A9, A10, A12, A13, A28, A22, Lm27, A30, X1, X2, X3, X4, X5;
A31: while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s),P
proof
 P +* (stop (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) halts_on Initialize (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) by B31, SCMPDS_6:def_3;
hence  while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_halting_on IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s),P by SCMPDS_6:def_3; ::_thesis: verum
end;
 s . (intpos 2) < (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) by A29, XREAL_1:29;
then A32: n0 + 1 <= (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) by A9, A10, XXREAL_0:2;
B33: while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)),P by A9, A10, A12, A13, A28, A30, A22, Lm27, X1, X2, X3, X4, X5;
A33: while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s),P
proof
 for k being Element of NAT holds IC (Comput ((P +* (stop (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))))),(Initialize (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)))),k)) in dom (stop (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) by B33, SCMPDS_6:def_2;
hence  while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))) is_closed_on IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s),P by SCMPDS_6:def_2; ::_thesis: verum
end;
then A34: (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))) is_halting_on s,P by A31, I, SCPISORT:9;
consider f2 being FinSequence of INT  such that
A35: len f2 = n and
A36: for i being Element of NAT st 1 <= i & i <= len f2 holds
f2 . i = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos (n0 + i)) by SCPISORT:1;
A37: f2 is_FinSequence_on Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)),n0
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f2 or f2 . i = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos (n0 + i)) )
assume  ( 1 <= i & i <= len f2 ) ; ::_thesis: f2 . i = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos (n0 + i))
then  f2 . i = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos (n0 + i)) by A36;
hence  f2 . i = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos (n0 + i)) by SCMPDS_5:15; ::_thesis: verum
end;
then A38: f2,f3 are_fiberwise_equipotent by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A24, A26, Lm26, A1, A3, A4, A5, A6;
 (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos 4) > 0 by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A37, A24, A26, Lm26, A1, A3, A4, A5, A6;
then reconsider m4 = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos 4) as Element of NAT by INT_1:3;
A39: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos 4) >= md by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A37, A24, A26, Lm26, A1, A3, A4, A5, A6;
then A40: m4 - n0 >= md - n0 by XREAL_1:9;
then reconsider n2 = m4 - n0 as Element of NAT by A19, INT_1:3;
A41: m4 - n0 >= 1 by A19, A40, XXREAL_0:2;
 (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos 2) = md by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A37, A24, A26, Lm26, A1, A3, A4, A5, A6;
then A42: (IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)) . (intpos 2) = md by A33, A31, SCPISORT:7;
A43: (IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)) . (intpos 4) = m4 by A33, A31, SCPISORT:7;
 (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . GBP = 0 by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A37, A24, A26, Lm26, A1, A3, A4, A5, A6;
then A44: (IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)) . GBP = 0 by A33, A31, SCPISORT:7;
then A45: DataLoc (((IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)) . GBP),6) = intpos (0 + 6) by SCMP_GCD:1;
set t2 = IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s);
set t3 = IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s);
assume that
A46: f is_FinSequence_on s,n0 and
A47: len f = n ; ::_thesis: ( not f1 is_FinSequence_on IExec (Partition,P,s),n0 or not len f1 = n or ( (IExec (Partition,P,s)) . GBP = 0 & (IExec (Partition,P,s)) . (intpos 1) = s . (intpos 1) & f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) ) )
assume that
A48: f1 is_FinSequence_on IExec (Partition,P,s),n0 and
A49: len f1 = n ; ::_thesis: ( (IExec (Partition,P,s)) . GBP = 0 & (IExec (Partition,P,s)) . (intpos 1) = s . (intpos 1) & f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) )
A50: dom f2 = Seg n by A35, FINSEQ_1:def_3;
A51: now__::_thesis:_for_i_being_Nat_st_i_in_dom_f2_holds_
f2_._i_=_f_._i
let i be Nat; ::_thesis: ( i in dom f2 implies f2 . i = f . i )
reconsider a = i as Element of NAT by ORDINAL1:def_12;
assume A52: i in dom f2 ; ::_thesis: f2 . i = f . i
then A53: 1 <= i by A50, FINSEQ_1:1;
then  i + n0 >= n0 + 1 by XREAL_1:6;
then A54: n0 + i >= 8 by A15, XXREAL_0:2;
A55: i <= n by A50, A52, FINSEQ_1:1;
hence f2 . i = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos (n0 + a)) by A35, A36, A53
.= s . (intpos (n0 + a)) by A7, A54, Lm28, I
.= f . i by A46, A47, A53, A55, SCPISORT:def_1 ;
::_thesis: verum
end;
A56: (((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))) is_closed_on s,P by A33, A31, I, SCPISORT:9;
then A57: ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0)) is_closed_on s,P by A34, SCPISORT:10;
A58: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos 4) <= (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 4) by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A37, A24, A26, Lm26, A1, A3, A4, A5, A6;
then  m4 <= n0 + n by A11, A21, XXREAL_0:2;
then A59: n2 <= len f3 by A24, XREAL_1:20;
A60: (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 6) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)))) . (intpos 6) by A56, A34, SCMPDS_7:31
.= (IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)) . (DataLoc (((IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)) . (intpos 4)),0)) by A45, SCMPDS_2:47
.= (IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)) . (intpos (m4 + 0)) by A43, SCMP_GCD:1
.= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos m4) by A33, A31, SCPISORT:7 ;
A61: (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 4) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)))) . (intpos 4) by A56, A34, SCMPDS_7:31
.= m4 by A43, A45, AMI_3:10, SCMPDS_2:47 ;
then A62: DataLoc (((IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 4)),0) = intpos (m4 + 0) by SCMP_GCD:1;
A63: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos 1) = (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 1) by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A37, A24, A26, Lm26, A1, A2, A3, A4, A5, A6;
A64: (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 1) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)))) . (intpos 1) by A56, A34, SCMPDS_7:31
.= (IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)) . (intpos 1) by A45, AMI_3:10, SCMPDS_2:47
.= s . (intpos 1) by A14, A63, A33, A31, SCPISORT:7 ;
A65: (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos 4) >= 8 by A39, A16, XXREAL_0:2;
then A66: m4 > 2 by XXREAL_0:2;
A67: (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . GBP = (Exec (((GBP,6) := ((intpos 4),0)),(IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)))) . GBP by A56, A34, SCMPDS_7:31
.= 0 by A44, A45, AMI_3:10, SCMPDS_2:47 ;
A68: (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 2) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)))) . (intpos 2) by A56, A34, SCMPDS_7:31
.= md by A42, A45, AMI_3:10, SCMPDS_2:47 ;
A69: ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0)) is_halting_on s,P by A56, A34, SCPISORT:10;
then A70: (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0)) is_halting_on s,P by A57, SCPISORT:10;
(IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos 2) = (Exec ((((intpos 4),0) := ((intpos 2),0)),(IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 2) by A57, A69, SCMPDS_7:31
.= md by A68, A62, A66, AMI_3:10, SCMPDS_2:47 ;
then A71: DataLoc (((IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos 2)),0) = intpos (md + 0) by SCMP_GCD:1;
A72: m4 > 6 by A65, XXREAL_0:2;
A73: m4 > 1 by A65, XXREAL_0:2;
A74: (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos 6) = (Exec ((((intpos 4),0) := ((intpos 2),0)),(IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 6) by A57, A69, SCMPDS_7:31
.= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos m4) by A60, A62, A72, AMI_3:10, SCMPDS_2:47 ;
A75: (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . GBP = (Exec ((((intpos 4),0) := ((intpos 2),0)),(IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . GBP by A57, A69, SCMPDS_7:31
.= 0 by A10, A39, A67, A62, AMI_3:10, SCMPDS_2:47 ;
A76: (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0)) is_closed_on s,P by A57, A69, SCPISORT:10;
then A77: (IExec (Partition,P,s)) . (intpos md) = (Exec ((((intpos 2),0) := (GBP,6)),(IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)))) . (intpos md) by A70, SCMPDS_7:31
.= (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (DataLoc (((IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . GBP),6)) by A71, SCMPDS_2:47
.= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos m4) by A75, A74, SCMP_GCD:1 ;
A78: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_((((((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2)))_';'_((GBP,3)_:=_(GBP,2)))_';'_(AddTo_(GBP,3,1)))_';'_(while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))))_';'_((GBP,6)_:=_((intpos_4),0))),P,s))_._(intpos_i)_=_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2)))_';'_((GBP,3)_:=_(GBP,2)))_';'_(AddTo_(GBP,3,1))),P,s)))))_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) )
assume  i >= 8 ; ::_thesis: (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i)
then A79: i > 6 by XXREAL_0:2;
thus (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) = (Exec (((GBP,6) := ((intpos 4),0)),(IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)))) . (intpos i) by A56, A34, SCMPDS_7:31
.= (IExec (((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))),P,s)) . (intpos i) by A45, A79, AMI_3:10, SCMPDS_2:47
.= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) by A33, A31, SCPISORT:7 ; ::_thesis: verum
end;
A80: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m4_holds_
(IExec_(((((((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2)))_';'_((GBP,3)_:=_(GBP,2)))_';'_(AddTo_(GBP,3,1)))_';'_(while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(((intpos_4),0)_:=_((intpos_2),0))),P,s))_._(intpos_i)_=_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2)))_';'_((GBP,3)_:=_(GBP,2)))_';'_(AddTo_(GBP,3,1))),P,s)))))_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m4 implies (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) )
assume that
A81: i >= 8 and
A82: i <> m4 ; ::_thesis: (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos i) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i)
thus (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos i) = (Exec ((((intpos 4),0) := ((intpos 2),0)),(IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos i) by A57, A69, SCMPDS_7:31
.= (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos i) by A62, A82, AMI_3:10, SCMPDS_2:47
.= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) by A78, A81 ; ::_thesis: verum
end;
A83: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<>_m4_&_i_<>_md_holds_
(IExec_(Partition,P,s))_._(intpos_i)_=_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2)))_';'_((GBP,3)_:=_(GBP,2)))_';'_(AddTo_(GBP,3,1))),P,s)))))_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <> m4 & i <> md implies (IExec (Partition,P,s)) . (intpos i) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) )
assume that
A84: i >= 8 and
A85: i <> m4 and
A86: i <> md ; ::_thesis: (IExec (Partition,P,s)) . (intpos i) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i)
thus (IExec (Partition,P,s)) . (intpos i) = (Exec ((((intpos 2),0) := (GBP,6)),(IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)))) . (intpos i) by A76, A70, SCMPDS_7:31
.= (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos i) by A71, A86, AMI_3:10, SCMPDS_2:47
.= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) by A80, A84, A85 ; ::_thesis: verum
end;
A87: now__::_thesis:_for_k_being_Element_of_NAT_st_k_<>_n1_&_k_<>_n2_&_1_<=_k_&_k_<=_len_f3_holds_
f3_._k_=_f1_._k
let k be Element of NAT ; ::_thesis: ( k <> n1 & k <> n2 & 1 <= k & k <= len f3 implies f3 . k = f1 . k )
assume that
A88: k <> n1 and
A89: k <> n2 and
A90: 1 <= k and
A91: k <= len f3 ; ::_thesis: f3 . k = f1 . k
A92: k + n0 <> md by A88;
A93: k + n0 >= n0 + 1 by A90, XREAL_1:6;
A94: k + n0 <> m4 by A89;
thus f3 . k = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos (n0 + k)) by A25, A90, A91
.= (IExec (Partition,P,s)) . (intpos (n0 + k)) by A15, A83, A92, A94, A93, XXREAL_0:2
.= f1 . k by A48, A49, A24, A90, A91, SCPISORT:def_1 ; ::_thesis: verum
end;
A95: m4 = n0 + n2 ;
then A96: f3 . n2 = (IExec (Partition,P,s)) . (intpos md) by A25, A77, A41, A59
.= f1 . n1 by A48, A49, A24, A19, A27, A20, SCPISORT:def_1 ;
A97: (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos 1) = (Exec ((((intpos 4),0) := ((intpos 2),0)),(IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 1) by A57, A69, SCMPDS_7:31
.= s . (intpos 1) by A64, A62, A73, AMI_3:10, SCMPDS_2:47 ;
thus (IExec (Partition,P,s)) . GBP = (Exec ((((intpos 2),0) := (GBP,6)),(IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)))) . GBP by A76, A70, SCMPDS_7:31
.= 0 by A10, A75, A71, AMI_3:10, SCMPDS_2:47 ; ::_thesis: ( (IExec (Partition,P,s)) . (intpos 1) = s . (intpos 1) & f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) )
thus (IExec (Partition,P,s)) . (intpos 1) = (Exec ((((intpos 2),0) := (GBP,6)),(IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)))) . (intpos 1) by A76, A70, SCMPDS_7:31
.= s . (intpos 1) by A97, A71, A17, AMI_3:10, SCMPDS_2:47 ; ::_thesis: ( f,f1 are_fiberwise_equipotent & ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) ) )
A98: m4 > 4 by A65, XXREAL_0:2;
A99: (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos 4) = (Exec ((((intpos 4),0) := ((intpos 2),0)),(IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos 4) by A57, A69, SCMPDS_7:31
.= m4 by A61, A62, A98, AMI_3:10, SCMPDS_2:47 ;
A100: (IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)) . (intpos m4) = (Exec ((((intpos 4),0) := ((intpos 2),0)),(IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)))) . (intpos m4) by A57, A69, SCMPDS_7:31
.= (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (DataLoc (((IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos 2)),0)) by A62, SCMPDS_2:47
.= (IExec ((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))),P,s)) . (intpos (md + 0)) by A68, SCMP_GCD:1
.= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos md) by A10, A15, A78, XXREAL_0:2 ;
A101: now__::_thesis:_(IExec_(Partition,P,s))_._(intpos_m4)_=_(IExec_((while>0_(GBP,5,(((while>0_(GBP,5,((((((GBP,7)_:=_(GBP,5))_';'_(AddTo_(GBP,5,(-_1))))_';'_((GBP,6)_:=_((intpos_4),0)))_';'_(SubFrom_(GBP,6,(intpos_2),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,4,(-_1)))_';'_(AddTo_(GBP,7,(-_1)))),(Load_((GBP,5)_:=_0)))))))_';'_(while>0_(GBP,7,((((((GBP,5)_:=_(GBP,7))_';'_(AddTo_(GBP,7,(-_1))))_';'_((GBP,6)_:=_((intpos_2),0)))_';'_(SubFrom_(GBP,6,(intpos_3),0)))_';'_(if>0_(GBP,6,((AddTo_(GBP,3,1))_';'_(AddTo_(GBP,5,(-_1)))),(Load_((GBP,7)_:=_0))))))))_';'_(if>0_(GBP,5,(((((((GBP,6)_:=_((intpos_4),0))_';'_(((intpos_4),0)_:=_((intpos_3),0)))_';'_(((intpos_3),0)_:=_(GBP,6)))_';'_(AddTo_(GBP,5,(-_2))))_';'_(AddTo_(GBP,3,1)))_';'_(AddTo_(GBP,4,(-_1))))))))),P,(Initialize_(IExec_((((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2)))_';'_((GBP,3)_:=_(GBP,2)))_';'_(AddTo_(GBP,3,1))),P,s)))))_._(intpos_md)
percases ( m4 = md or m4 <> md ) ;
suppose m4 = md ; ::_thesis: (IExec (Partition,P,s)) . (intpos m4) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos md)
hence  (IExec (Partition,P,s)) . (intpos m4) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos md) by A77; ::_thesis: verum
end;
supposeA102: m4 <> md ; ::_thesis: (IExec (Partition,P,s)) . (intpos m4) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos md)
thus (IExec (Partition,P,s)) . (intpos m4) = (Exec ((((intpos 2),0) := (GBP,6)),(IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)))) . (intpos m4) by A76, A70, SCMPDS_7:31
.= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos md) by A100, A71, A102, AMI_3:10, SCMPDS_2:47 ; ::_thesis: verum
end;
end;
end;
then f3 . n1 = (IExec (Partition,P,s)) . (intpos m4) by A25, A19, A27, A20
.= f1 . n2 by A48, A49, A24, A41, A59, A95, SCPISORT:def_1 ;
then  f3,f1 are_fiberwise_equipotent by A49, A24, A19, A27, A41, A59, A96, A87, SCPISORT:3;
then  f2,f1 are_fiberwise_equipotent by A38, CLASSES1:76;
hence  f,f1 are_fiberwise_equipotent by A47, A35, A51, FINSEQ_2:9; ::_thesis: ex m4 being Element of NAT st
( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) )
take  m4 ; ::_thesis: ( m4 = (IExec (Partition,P,s)) . (intpos 4) & md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec (Partition,P,s)) . (intpos 4) = (Exec ((((intpos 2),0) := (GBP,6)),(IExec (((((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))) ';' (while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1)))))))))) ';' ((GBP,6) := ((intpos 4),0))) ';' (((intpos 4),0) := ((intpos 2),0))),P,s)))) . (intpos 4) by A76, A70, SCMPDS_7:31
.= m4 by A99, A71, A18, AMI_3:10, SCMPDS_2:47 ; ::_thesis: ( md <= m4 & m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) )
A103: md = ((IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3)) - 1 by A9, A29;
hence  md <= m4 by A8, A9, A10, A11, A12, A13, A28, A21, A22, A32, A35, A37, A24, A26, Lm26, A1, A3, A4, A5, A6; ::_thesis: ( m4 <= s . (intpos 4) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) )
thus  m4 <= s . (intpos 4) by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A37, A24, A26, Lm26, A1, A3, A4, A5, A6; ::_thesis: ( ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) )
A104: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) = md + 1 by A7, A9, Lm28, I;
hereby  ::_thesis: ( ( for i being Element of NAT st m4 < i & i <= s . (intpos 4) holds
(IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) ) & ( for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i) ) )
let i be Element of NAT ; ::_thesis: ( md <= i & i < m4 implies (IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos b1) )
assume that
A105: md <= i and
A106: i < m4 ; ::_thesis: (IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos b1)
percases ( i = md or i <> md ) ;
supposeA107: i = md ; ::_thesis: (IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos b1)
then  (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) <= m4 by A104, A106, INT_1:7;
hence  (IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) by A107, A77, A8, A9, A10, A11, A12, A13, A28, A21, A103, A22, A32, A35, A37, A24, A26, A101, Lm26, A1, A3, A4, A5, A6; ::_thesis: verum
end;
supposeA108: i <> md ; ::_thesis: (IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos b1)
then  md < i by A105, XXREAL_0:1;
then  (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) <= i by A9, A29, INT_1:7;
then  (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos md) >= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) by A8, A9, A10, A11, A12, A13, A28, A21, A103, A22, A32, A35, A37, A24, A26, A106, Lm26, A1, A3, A4, A5, A6;
hence  (IExec (Partition,P,s)) . (intpos m4) >= (IExec (Partition,P,s)) . (intpos i) by A16, A101, A83, A105, A108, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hereby  ::_thesis: for i being Element of NAT st i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) holds
(IExec (Partition,P,s)) . (intpos i) = s . (intpos i)
let i be Element of NAT ; ::_thesis: ( m4 < i & i <= s . (intpos 4) implies (IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) )
assume that
A109: m4 < i and
A110: i <= s . (intpos 4) ; ::_thesis: (IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i)
A111: md < i by A39, A109, XXREAL_0:2;
 (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos md) <= (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A37, A24, A26, A109, A110, Lm26, A1, A3, A4, A5, A6;
hence  (IExec (Partition,P,s)) . (intpos m4) <= (IExec (Partition,P,s)) . (intpos i) by A16, A101, A83, A111, XXREAL_0:2; ::_thesis: verum
end;
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i >= n0 + 1 & ( i < s . (intpos 2) or i > s . (intpos 4) ) implies (IExec (Partition,P,s)) . (intpos i) = s . (intpos i) )
assume that
A112: i >= n0 + 1 and
A113: ( i < s . (intpos 2) or i > s . (intpos 4) ) ; ::_thesis: (IExec (Partition,P,s)) . (intpos i) = s . (intpos i)
A114: i >= 8 by A15, A112, XXREAL_0:2;
A115: now__::_thesis:_(_(_i_<_s_._(intpos_2)_&_i_<_(IExec_((((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2)))_';'_((GBP,3)_:=_(GBP,2)))_';'_(AddTo_(GBP,3,1))),P,s))_._(intpos_3)_)_or_(_i_>_s_._(intpos_4)_&_i_>_(IExec_((((((GBP,5)_:=_(GBP,4))_';'_(SubFrom_(GBP,5,GBP,2)))_';'_((GBP,3)_:=_(GBP,2)))_';'_(AddTo_(GBP,3,1))),P,s))_._(intpos_4)_)_)
percases ( i < s . (intpos 2) or i > s . (intpos 4) ) by A113;
caseA116: i < s . (intpos 2) ; ::_thesis: i < (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3)
 s . (intpos 2) < (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) by A29, XREAL_1:29;
hence  i < (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 3) by A116, XXREAL_0:2; ::_thesis: verum
end;
case i > s . (intpos 4) ; ::_thesis: i > (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 4)
hence  i > (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos 4) by A7, Lm28, I; ::_thesis: verum
end;
end;
end;
A117: (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos i) = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos i) by SCMPDS_5:15;
A118: for i being Element of NAT st i >= n0 + 1 & ( i < (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 3) or i > (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos 4) ) holds
(IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) = (Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))) . (intpos i) by A8, A9, A10, A11, A12, A13, A28, A21, A103, A22, A32, A35, A37, A24, A26, Lm26, A1, A3, A4, A5, A6;
 ( i <> md & i <> m4 )
proof
percases ( i < s . (intpos 2) or i > s . (intpos 4) ) by A113;
suppose i < s . (intpos 2) ; ::_thesis: ( i <> md & i <> m4 )
hence  ( i <> md & i <> m4 ) by A8, A9, A10, A11, A12, A13, A28, A21, A30, A22, A32, A35, A37, A24, A26, Lm26, A1, A3, A4, A5, A6; ::_thesis: verum
end;
suppose i > s . (intpos 4) ; ::_thesis: ( i <> md & i <> m4 )
hence  ( i <> md & i <> m4 ) by A21, A39, A58, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
hence (IExec (Partition,P,s)) . (intpos i) = (IExec ((while>0 (GBP,5,(((while>0 (GBP,5,((((((GBP,7) := (GBP,5)) ';' (AddTo (GBP,5,(- 1)))) ';' ((GBP,6) := ((intpos 4),0))) ';' (SubFrom (GBP,6,(intpos 2),0))) ';' (if>0 (GBP,6,((AddTo (GBP,4,(- 1))) ';' (AddTo (GBP,7,(- 1)))),(Load ((GBP,5) := 0))))))) ';' (while>0 (GBP,7,((((((GBP,5) := (GBP,7)) ';' (AddTo (GBP,7,(- 1)))) ';' ((GBP,6) := ((intpos 2),0))) ';' (SubFrom (GBP,6,(intpos 3),0))) ';' (if>0 (GBP,6,((AddTo (GBP,3,1)) ';' (AddTo (GBP,5,(- 1)))),(Load ((GBP,7) := 0)))))))) ';' (if>0 (GBP,5,(((((((GBP,6) := ((intpos 4),0)) ';' (((intpos 4),0) := ((intpos 3),0))) ';' (((intpos 3),0) := (GBP,6))) ';' (AddTo (GBP,5,(- 2)))) ';' (AddTo (GBP,3,1))) ';' (AddTo (GBP,4,(- 1))))))))),P,(Initialize (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s))))) . (intpos i) by A15, A83, A112, XXREAL_0:2
.= (IExec ((((((GBP,5) := (GBP,4)) ';' (SubFrom (GBP,5,GBP,2))) ';' ((GBP,3) := (GBP,2))) ';' (AddTo (GBP,3,1))),P,s)) . (intpos i) by A112, A115, A118, A4, A5, A117
.= s . (intpos i) by A7, A114, Lm28, I ;
::_thesis: verum
end;
end;

theorem :: SCPQSORT:12
( Partition is halt-free & Partition is shiftable ) ;

Lm29:  for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for p0, n being Element of NAT holds
( card (QuickSort (n,p0)) = 57 & ( p0 >= 7 implies ( QuickSort (n,p0) is_halting_on s,P & ex f, g being FinSequence of INT st
( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) ) ) )
proof
let P be Instruction-Sequence of SCMPDS; ::_thesis: for s being 0 -started State of SCMPDS
for p0, n being Element of NAT holds
( card (QuickSort (n,p0)) = 57 & ( p0 >= 7 implies ( QuickSort (n,p0) is_halting_on s,P & ex f, g being FinSequence of INT st
( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) ) ) )
set b1 = DataLoc (0,1);
reconsider Pt = Partition as halt-free shiftable Program of SCMPDS ;
let s be 0 -started State of SCMPDS; ::_thesis: for p0, n being Element of NAT holds
( card (QuickSort (n,p0)) = 57 & ( p0 >= 7 implies ( QuickSort (n,p0) is_halting_on s,P & ex f, g being FinSequence of INT st
( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) ) ) )
let p0, n be Element of NAT ; ::_thesis: ( card (QuickSort (n,p0)) = 57 & ( p0 >= 7 implies ( QuickSort (n,p0) is_halting_on s,P & ex f, g being FinSequence of INT st
( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) ) ) )
I: Initialize s = s by MEMSTR_0:44;
set pn = p0 + n;
set l1 = (GBP,2) := (SBP,((p0 + n) + 1));
set l2 = SubFrom (GBP,2,SBP,(p0 + n));
set l3 = (GBP,2) := (SBP,(p0 + n));
set l4 = (GBP,4) := (SBP,((p0 + n) + 1));
set l5 = (SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1));
set l6 = (SBP,((p0 + n) + 1)) := (GBP,4);
set l7 = (SBP,((p0 + n) + 2)) := (GBP,4);
set l8 = AddTo (SBP,((p0 + n) + 1),(- 1));
set l9 = AddTo (SBP,((p0 + n) + 2),1);
set l0 = AddTo (GBP,1,2);
set lb = Load (AddTo (GBP,1,(- 2)));
set L5 = ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2));
set TR = ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)));
set IF4 = if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))));
set WB4 = (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))));
set WH4 = while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))));
set m1 = GBP := 0;
set m2 = SBP := 1;
set m3 = (SBP,(p0 + n)) := (p0 + 1);
set m4 = (SBP,((p0 + n) + 1)) := (p0 + n);
set M4 = (((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n));
set s1 = IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s);
set a = GBP ;
set P1 = P;
A1: 7 + n >= 7 by NAT_1:11;
A2: card (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) = (card ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1)))) + 1 by SCMP_GCD:4
.= ((card (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1))))) + 1) + 1 by SCMP_GCD:4
.= (4 + 1) + 1 by Th4 ;
A3: card ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) = (card (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n))))) + (card (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) by AFINSQ_1:17
.= 2 + (card (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) by SCMP_GCD:5
.= 2 + (((card (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + (card (Load (AddTo (GBP,1,(- 2)))))) + 2) by SCMPDS_6:65
.= 2 + (((card (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + 1) + 2) by COMPOS_1:54
.= 2 + ((((card ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt)) + (card (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + 1) + 2) by AFINSQ_1:17
.= 2 + (((((card (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1))))) + (card Pt)) + (card (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + 1) + 2) by AFINSQ_1:17
.= 2 + ((((2 + 38) + (card (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) + 1) + 2) by Th12, SCMP_GCD:5
.= 51 by A2 ;
thus  card (QuickSort (n,p0)) = (card ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n)))) + (card (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))) by AFINSQ_1:17
.= 4 + (card (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))) by Th4
.= 4 + (51 + 2) by A3, SCMPDS_8:17
.= 57 ; ::_thesis: ( p0 >= 7 implies ( QuickSort (n,p0) is_halting_on s,P & ex f, g being FinSequence of INT st
( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) ) )
assume A4: p0 >= 7 ; ::_thesis: ( QuickSort (n,p0) is_halting_on s,P & ex f, g being FinSequence of INT st
( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) )
then  p0 + n >= 7 + n by XREAL_1:6;
then A5: p0 + n >= 7 by A1, XXREAL_0:2;
A6: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m1 being Element of NAT st t . GBP = 0 & t . SBP = m1 holds
( (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = t . (intpos (m1 + (p0 + n))) & (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 4) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i <> 2 & i <> 4 holds
(IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
proof
let t be State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for m1 being Element of NAT st t . GBP = 0 & t . SBP = m1 holds
( (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = t . (intpos (m1 + (p0 + n))) & (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 4) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i <> 2 & i <> 4 holds
(IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for m1 being Element of NAT st t . GBP = 0 & t . SBP = m1 holds
( (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = t . (intpos (m1 + (p0 + n))) & (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 4) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i <> 2 & i <> 4 holds
(IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
let m1 be Element of NAT ; ::_thesis: ( t . GBP = 0 & t . SBP = m1 implies ( (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = t . (intpos (m1 + (p0 + n))) & (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 4) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i <> 2 & i <> 4 holds
(IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) ) )
set t0 = Initialize t;
set t2 = IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t));
set t3 = Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t));
set mp = m1 + (p0 + n);
set Q0 = Q;
assume that
A8: t . GBP = 0 and
A9: t . SBP = m1 ; ::_thesis: ( (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = t . (intpos (m1 + (p0 + n))) & (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 4) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i <> 2 & i <> 4 holds
(IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
A10: (Initialize t) . GBP = 0 by A8, SCMPDS_5:15;
then A11: DataLoc (((Initialize t) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
then  (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . GBP = 0 by A10, AMI_3:10, SCMPDS_2:47;
then A12: DataLoc (((Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . GBP),4) = intpos (0 + 4) by SCMP_GCD:1;
A13: (Initialize t) . SBP = m1 by A9, SCMPDS_5:15;
then A14: (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . SBP = m1 by A11, AMI_3:10, SCMPDS_2:47;
 m1 + (p0 + n) >= 0 + 7 by A5, XREAL_1:7;
then  (m1 + (p0 + n)) + 1 >= 7 + 1 by XREAL_1:6;
then  (m1 + (p0 + n)) + 1 > 2 by XXREAL_0:2;
then A15: (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (Initialize t) . (intpos ((m1 + (p0 + n)) + 1)) by A11, AMI_3:10, SCMPDS_2:47
.= t . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:15 ;
A16: (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . (intpos 2) = (Initialize t) . (DataLoc (((Initialize t) . SBP),(p0 + n))) by A11, SCMPDS_2:47
.= (Initialize t) . (intpos (m1 + (p0 + n))) by A13, SCMP_GCD:1
.= t . (intpos (m1 + (p0 + n))) by SCMPDS_5:15 ;
thus (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = (Exec (((GBP,4) := (SBP,((p0 + n) + 1))),(Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))))) . (intpos 2) by SCMPDS_5:42
.= t . (intpos (m1 + (p0 + n))) by A16, A12, AMI_3:10, SCMPDS_2:47 ; ::_thesis: ( (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 4) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i <> 2 & i <> 4 holds
(IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
thus (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 4) = (Exec (((GBP,4) := (SBP,((p0 + n) + 1))),(Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))))) . (intpos 4) by SCMPDS_5:42
.= (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . (DataLoc (((Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . SBP),((p0 + n) + 1))) by A12, SCMPDS_2:47
.= t . (intpos ((m1 + (p0 + n)) + 1)) by A14, A15, SCMP_GCD:1 ; ::_thesis: for i being Element of NAT st i <> 2 & i <> 4 holds
(IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i)
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i <> 2 & i <> 4 implies (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) )
assume that
A17: i <> 2 and
A18: i <> 4 ; ::_thesis: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = t . (intpos i)
thus (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos i) = (Exec (((GBP,4) := (SBP,((p0 + n) + 1))),(Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,2) := (SBP,(p0 + n))),(Initialize t))) . (intpos i) by A12, A18, AMI_3:10, SCMPDS_2:47
.= (Initialize t) . (intpos i) by A11, A17, AMI_3:10, SCMPDS_2:47
.= t . (intpos i) by SCMPDS_5:15 ; ::_thesis: verum
end;
end;
A19: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m, m1 being Element of NAT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 holds
( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . GBP = 0 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . SBP = m1 + 2 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
proof
let t be State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for m, m1 being Element of NAT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 holds
( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . GBP = 0 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . SBP = m1 + 2 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for m, m1 being Element of NAT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 holds
( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . GBP = 0 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . SBP = m1 + 2 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
let m, m1 be Element of NAT ; ::_thesis: ( t . GBP = 0 & t . SBP = m1 & m1 = m + 1 implies ( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . GBP = 0 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . SBP = m1 + 2 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) ) )
assume that
A20: t . GBP = 0 and
A21: t . SBP = m1 and
A22: m1 = m + 1 ; ::_thesis: ( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . GBP = 0 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . SBP = m1 + 2 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
set t0 = Initialize t;
set t1 = IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t));
set Q0 = Q;
set t2 = IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t));
set t3 = IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t));
set t4 = IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t));
set t5 = IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t));
set t6 = Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t));
set mp = m1 + (p0 + n);
A28: (m1 + (p0 + n)) + 3 <> (m1 + (p0 + n)) + 1 ;
 m1 >= 1 by A22, NAT_1:11;
then A29: m1 + (p0 + n) >= 1 + 7 by A5, XREAL_1:7;
then A30: (m1 + (p0 + n)) + 1 >= 8 + 1 by XREAL_1:6;
then A31: (m1 + (p0 + n)) + 1 > 4 by XXREAL_0:2;
A32: (m1 + (p0 + n)) + 1 > 1 by A30, XXREAL_0:2;
A33: (m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 2 ;
A34: m1 + (p0 + n) > 1 by A29, XXREAL_0:2;
A35: (m1 + (p0 + n)) + 1 <> (m1 + (p0 + n)) + 2 ;
A36: (Initialize t) . SBP = m1 by A21, SCMPDS_5:15;
then A37: DataLoc (((Initialize t) . SBP),((p0 + n) + 3)) = intpos ((m1 + (p0 + n)) + 3) by SCMP_GCD:1;
A38: now__::_thesis:_for_i_being_Element_of_NAT_st_i_<>_(m1_+_(p0_+_n))_+_3_holds_
(Exec_(((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1))),(Initialize_t)))_._(intpos_i)_=_t_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i <> (m1 + (p0 + n)) + 3 implies (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) = t . (intpos i) )
assume  i <> (m1 + (p0 + n)) + 3 ; ::_thesis: (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) = t . (intpos i)
hence (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) = (Initialize t) . (intpos i) by A37, AMI_3:10, SCMPDS_2:47
.= t . (intpos i) by SCMPDS_5:15 ;
::_thesis: verum
end;
 (m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 3 ;
then A39: (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A38;
A40: (m1 + (p0 + n)) + 1 <> (m1 + (p0 + n)) + 2 ;
A41: (m1 + (p0 + n)) + 3 <> (m1 + (p0 + n)) + 2 ;
A42: (m1 + (p0 + n)) + 3 >= 8 + 3 by A29, XREAL_1:6;
then A43: (m1 + (p0 + n)) + 3 > 1 by XXREAL_0:2;
 (m1 + (p0 + n)) + 3 > 1 by A42, XXREAL_0:2;
then A44: (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . SBP = m1 by A21, A38;
then A45: DataLoc (((Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . SBP),((p0 + n) + 1)) = intpos ((m1 + (p0 + n)) + 1) by SCMP_GCD:1;
A46: now__::_thesis:_for_i_being_Element_of_NAT_st_i_<>_(m1_+_(p0_+_n))_+_1_holds_
(IExec_((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4))),Q,(Initialize_t)))_._(intpos_i)_=_(Exec_(((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1))),(Initialize_t)))_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i <> (m1 + (p0 + n)) + 1 implies (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i) = (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) )
assume A47: i <> (m1 + (p0 + n)) + 1 ; ::_thesis: (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i) = (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i)
thus (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i) = (Exec (((SBP,((p0 + n) + 1)) := (GBP,4)),(Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) by A45, A47, AMI_3:10, SCMPDS_2:47 ; ::_thesis: verum
end;
 (m1 + (p0 + n)) + 1 > 1 by A30, XXREAL_0:2;
then A48: (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . SBP = m1 by A44, A46;
then A49: DataLoc (((IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . SBP),((p0 + n) + 2)) = intpos ((m1 + (p0 + n)) + 2) by SCMP_GCD:1;
A50: now__::_thesis:_for_i_being_Element_of_NAT_st_i_<>_(m1_+_(p0_+_n))_+_2_holds_
(IExec_(((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4))),Q,(Initialize_t)))_._(intpos_i)_=_(IExec_((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4))),Q,(Initialize_t)))_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i <> (m1 + (p0 + n)) + 2 implies (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i) = (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i) )
assume A51: i <> (m1 + (p0 + n)) + 2 ; ::_thesis: (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i) = (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i)
thus (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i) = (Exec (((SBP,((p0 + n) + 2)) := (GBP,4)),(IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i) by A49, A51, AMI_3:10, SCMPDS_2:47 ; ::_thesis: verum
end;
(Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = (Initialize t) . (DataLoc (((Initialize t) . SBP),((p0 + n) + 1))) by A37, SCMPDS_2:47
.= (Initialize t) . (intpos (m1 + ((p0 + n) + 1))) by A36, SCMP_GCD:1
.= t . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:15 ;
then  (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A46, A28;
then A52: (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A50, A41;
A53: (m1 + (p0 + n)) + 3 > 4 by A42, XXREAL_0:2;
then A54: (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos 4) = t . (intpos 4) by A38;
A55: (m1 + (p0 + n)) + 2 >= 8 + 2 by A29, XREAL_1:6;
then A56: (m1 + (p0 + n)) + 2 > 1 by XXREAL_0:2;
 (m1 + (p0 + n)) + 2 > 1 by A55, XXREAL_0:2;
then A57: (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . SBP = m1 by A48, A50;
then A58: DataLoc (((IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . SBP),((p0 + n) + 1)) = intpos ((m1 + (p0 + n)) + 1) by SCMP_GCD:1;
A59: now__::_thesis:_for_i_being_Element_of_NAT_st_i_<>_(m1_+_(p0_+_n))_+_1_holds_
(IExec_((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1)))),Q,(Initialize_t)))_._(intpos_i)_=_(IExec_(((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4))),Q,(Initialize_t)))_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i <> (m1 + (p0 + n)) + 1 implies (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i) = (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i) )
assume A60: i <> (m1 + (p0 + n)) + 1 ; ::_thesis: (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i) = (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i)
thus (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i) = (Exec ((AddTo (SBP,((p0 + n) + 1),(- 1))),(IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i) by A58, A60, AMI_3:10, SCMPDS_2:48 ; ::_thesis: verum
end;
 (m1 + (p0 + n)) + 1 > 1 by A30, XXREAL_0:2;
then A61: (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . SBP = m1 by A57, A59;
then A62: DataLoc (((IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . SBP),((p0 + n) + 2)) = intpos ((m1 + (p0 + n)) + 2) by SCMP_GCD:1;
A63: now__::_thesis:_for_i_being_Element_of_NAT_st_i_<>_(m1_+_(p0_+_n))_+_2_holds_
(IExec_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1))),Q,(Initialize_t)))_._(intpos_i)_=_(IExec_((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1)))),Q,(Initialize_t)))_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i <> (m1 + (p0 + n)) + 2 implies (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i) = (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i) )
assume A64: i <> (m1 + (p0 + n)) + 2 ; ::_thesis: (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i) = (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i)
thus (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i) = (Exec ((AddTo (SBP,((p0 + n) + 2),1)),(IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i) by A62, A64, AMI_3:10, SCMPDS_2:48 ; ::_thesis: verum
end;
A65: (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . GBP = 0 by A20, A38;
then A66: (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . GBP = 0 by A46;
then  (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . GBP = 0 by A50;
then  (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . GBP = 0 by A59;
then A67: (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . GBP = 0 by A63;
then A68: DataLoc (((IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . GBP),1) = intpos (0 + 1) by SCMP_GCD:1;
 (m1 + (p0 + n)) + 2 > 1 by A55, XXREAL_0:2;
then A69: (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . SBP = m1 by A61, A63;
A70: now__::_thesis:_for_i_being_Element_of_NAT_st_i_<>_1_holds_
(IExec_((((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2))),Q,(Initialize_t)))_._(intpos_i)_=_(IExec_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1))),Q,(Initialize_t)))_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i <> 1 implies (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i) )
assume  i <> 1 ; ::_thesis: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i)
then A71: intpos i <> DataLoc (((IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . GBP),1) by A67, AMI_3:10, SCMP_GCD:1;
thus (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = (Exec ((AddTo (GBP,1,2)),(IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))))) . (intpos i) by SCMPDS_5:41
.= (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i) by A71, SCMPDS_2:48 ; ::_thesis: verum
end;
hence  (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . GBP = 0 by A67; ::_thesis: ( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . SBP = m1 + 2 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
A72: (m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 2 ;
 (m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 1 ;
then  (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A39, A46;
then A73: (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A50, A72;
thus (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . SBP = (Exec ((AddTo (GBP,1,2)),(IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))))) . SBP by SCMPDS_5:41
.= m1 + 2 by A69, A68, SCMPDS_2:48 ; ::_thesis: ( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
 (m1 + (p0 + n)) + 0 <> (m1 + (p0 + n)) + 1 ;
then  (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A73, A59;
then  (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A63, A33;
hence  (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos (m1 + (p0 + n))) = t . (intpos (m1 + (p0 + n))) by A70, A34; ::_thesis: ( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
A74: (m1 + (p0 + n)) + 1 <> (m1 + (p0 + n)) + 2 ;
(IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (Exec (((SBP,((p0 + n) + 2)) := (GBP,4)),(IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))))) . (intpos ((m1 + (p0 + n)) + 2)) by SCMPDS_5:41
.= (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (DataLoc (((IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . GBP),4)) by A49, SCMPDS_2:47
.= (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos (0 + 4)) by A66, SCMP_GCD:1
.= t . (intpos 4) by A54, A46, A31 ;
then A75: (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = t . (intpos 4) by A59, A35;
(IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (Exec (((SBP,((p0 + n) + 1)) := (GBP,4)),(Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))))) . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:42
.= (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (DataLoc (((Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . GBP),4)) by A45, SCMPDS_2:47
.= (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos (0 + 4)) by A65, SCMP_GCD:1
.= t . (intpos 4) by A38, A53 ;
then A76: (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = t . (intpos 4) by A50, A74;
(IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (Exec ((AddTo (SBP,((p0 + n) + 1),(- 1))),(IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))))) . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:41
.= (t . (intpos 4)) + (- 1) by A76, A58, SCMPDS_2:48
.= (t . (intpos 4)) - 1 ;
then  (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 by A63, A40;
hence  (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 1)) = (t . (intpos 4)) - 1 by A70, A32; ::_thesis: ( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 & (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
(IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (Exec ((AddTo (SBP,((p0 + n) + 2),1)),(IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))))) . (intpos ((m1 + (p0 + n)) + 2)) by SCMPDS_5:41
.= (t . (intpos 4)) + 1 by A75, A62, SCMPDS_2:48 ;
hence  (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 2)) = (t . (intpos 4)) + 1 by A70, A56; ::_thesis: ( (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) & ( for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
A77: (m1 + (p0 + n)) + 3 <> (m1 + (p0 + n)) + 2 ;
 (m1 + (p0 + n)) + 3 <> (m1 + (p0 + n)) + 1 ;
then  (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A52, A59;
then  (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A63, A77;
hence  (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos ((m1 + (p0 + n)) + 3)) = t . (intpos ((m1 + (p0 + n)) + 1)) by A70, A43; ::_thesis: for i being Element of NAT st i >= 8 & i < m1 + (p0 + n) holds
(IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i)
hereby  ::_thesis: verum
A78: m1 + (p0 + n) <= (m1 + (p0 + n)) + 3 by NAT_1:11;
A79: m1 + (p0 + n) <= (m1 + (p0 + n)) + 2 by NAT_1:11;
A80: m1 + (p0 + n) <= (m1 + (p0 + n)) + 1 by NAT_1:11;
let i be Element of NAT ; ::_thesis: ( i >= 8 & i < m1 + (p0 + n) implies (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i) )
assume that
A81: i >= 8 and
A82: i < m1 + (p0 + n) ; ::_thesis: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = t . (intpos i)
 i > 1 by A81, XXREAL_0:2;
hence (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize t))) . (intpos i) = (IExec (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))),Q,(Initialize t))) . (intpos i) by A70
.= (IExec ((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))),Q,(Initialize t))) . (intpos i) by A63, A82, A79
.= (IExec (((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))),Q,(Initialize t))) . (intpos i) by A59, A82, A80
.= (IExec ((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))),Q,(Initialize t))) . (intpos i) by A50, A82, A79
.= (Exec (((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) by A46, A82, A80
.= t . (intpos i) by A38, A82, A78 ;
::_thesis: verum
end;
end;
A83: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m, m1, md being Element of NAT
for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 & len f2 = n holds
( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for m, m1, md being Element of NAT
for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 & len f2 = n holds
( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for m, m1, md being Element of NAT
for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 & len f2 = n holds
( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let m, m1, md be Element of NAT ; ::_thesis: for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 & len f2 = n holds
( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let n4 be Integer; ::_thesis: for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 & len f2 = n holds
( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let f1, f2 be FinSequence of INT ; ::_thesis: ( t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 & len f2 = n implies ( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) ) )
I: Initialize t = t by MEMSTR_0:44;
set t1 = IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t);
set lPt = (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt;
set t2 = IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t);
set Q2 = Q;
set t4 = IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t);
set mp = m1 + (p0 + n);
set Q4 = Q;
A84: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . GBP = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . GBP by SCMPDS_5:15;
A85: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos 2) by SCMPDS_5:15;
A86: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos 4) by SCMPDS_5:15;
A87: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 1) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos 1) by SCMPDS_5:15;
set tp = IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))));
assume that
A89: t . GBP = 0 and
A90: t . SBP = m1 and
A91: m1 = m + 1 and
A92: t . (intpos (m1 + (p0 + n))) = md and
A93: md >= p0 + 1 and
A94: n4 = t . (intpos ((m1 + (p0 + n)) + 1)) and
A95: n4 - md > 0 and
A96: n4 <= p0 + n ; ::_thesis: ( not f1 is_FinSequence_on t,p0 or not len f1 = n or not f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 or not len f2 = n or ( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) ) )
A97: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . GBP = 0 by A6, A89, A90, I;
assume that
A98: f1 is_FinSequence_on t,p0 and
A99: len f1 = n ; ::_thesis: ( not f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 or not len f2 = n or ( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) ) )
A100: p0 + 1 >= 7 + 1 by A4, XREAL_1:6;
A101: now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f1_holds_
f1_._i_=_(IExec_((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1)))),Q,t))_._(intpos_(p0_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f1 implies f1 . i = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (p0 + i)) )
assume that
A102: 1 <= i and
A103: i <= len f1 ; ::_thesis: f1 . i = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (p0 + i))
 p0 + i >= p0 + 1 by A102, XREAL_1:6;
then A104: p0 + i >= 8 by A100, XXREAL_0:2;
then A105: p0 + i > 2 by XXREAL_0:2;
A106: p0 + i > 4 by A104, XXREAL_0:2;
thus f1 . i = t . (intpos (p0 + i)) by A98, A102, A103, SCPISORT:def_1
.= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (p0 + i)) by A6, A89, A90, A105, A106, I ; ::_thesis: verum
end;
A107: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) = n4 by A6, A89, A90, A94, I;
A108: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) = t . (intpos ((m1 + (p0 + n)) + 1)) by A6, A89, A90, I;
A109: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) = t . (intpos (m1 + (p0 + n))) by A6, A89, A90, I;
then  Pt is_closed_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)),Q by A4, A92, A93, A97, Th13, A84, A85;
then A110: Pt is_closed_on IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t),Q by SCMPDS_6:125;
 Pt is_halting_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)),Q by A4, A92, A93, A109, A97, Th13, A84, A85;
then A111: Pt is_halting_on IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t),Q by SCMPDS_6:126;
then A112: (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt is_closed_on t,Q by A110, I, SCPISORT:9;
assume that
A113: f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t),p0 and
A114: len f2 = n ; ::_thesis: ( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 & (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
consider f4 being FinSequence of INT  such that
A115: len f4 = n and
A116: for i being Element of NAT st 1 <= i & i <= len f4 holds
f4 . i = (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos (p0 + i)) by SCPISORT:1;
A117: f1 is_FinSequence_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)),p0
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f1 or f1 . i = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos (p0 + i)) )
assume  ( 1 <= i & i <= len f1 ) ; ::_thesis: f1 . i = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos (p0 + i))
then  f1 . i = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (p0 + i)) by A101;
hence  f1 . i = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos (p0 + i)) by SCMPDS_5:15; ::_thesis: verum
end;
A118: f4 is_FinSequence_on IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)))),p0 by A116, SCPISORT:def_1;
then consider m4 being Element of NAT  such that
A119: m4 = (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos 4) and
A120: md <= m4 and
A121: m4 <= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) and
A122: for i being Element of NAT st md <= i & i < m4 holds
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos m4) >= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) and
A123: for i being Element of NAT st m4 < i & i <= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) holds
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos m4) <= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) and
A124: for i being Element of NAT st i >= p0 + 1 & ( i < (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) or i > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) ) holds
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos i) by A4, A92, A93, A94, A95, A96, A99, A109, A108, A97, A117, A115, Th14, A84, A85, A86;
 (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . GBP = 0 by A4, A92, A93, A94, A95, A96, A99, A109, A108, A97, A117, A115, A118, Th14, A84, A85, A86;
then A125: (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . GBP = 0 by A110, A111, SCPISORT:7;
A126: (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt is_halting_on t,Q by A110, A111, I, SCPISORT:9;
 (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos 1) = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 1) by A4, A92, A93, A94, A95, A96, A99, A109, A108, A97, A117, A115, A118, Th14, A84, A85, A86, A87;
then A127: (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . SBP = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 1) by A110, A111, SCPISORT:7
.= m1 by A6, A89, A90, I ;
then A128: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos (m1 + (p0 + n))) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos (m1 + (p0 + n))) by A19, A91, A125;
 (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . GBP = 0 by A19, A91, A125, A127;
hence  (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . GBP = 0 by A112, A126, SCPISORT:6; ::_thesis: ( (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
 (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . SBP = m1 + 2 by A19, A91, A125, A127;
hence  (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos 1) = m1 + 2 by A112, A126, SCPISORT:6; ::_thesis: ( md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
A129: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((m1 + (p0 + n)) + 3)) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A19, A91, A125, A127;
A130: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((m1 + (p0 + n)) + 2)) = ((IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos 4)) + 1 by A19, A91, A125, A127;
A131: (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((m1 + (p0 + n)) + 1)) = ((IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos 4)) - 1 by A19, A91, A125, A127;
A132: 1 + (p0 + n) > p0 + n by XREAL_1:29;
A133: m1 >= 1 by A91, NAT_1:11;
then  m1 + (p0 + n) >= 1 + (p0 + n) by XREAL_1:6;
then A134: m1 + (p0 + n) > p0 + n by A132, XXREAL_0:2;
then A135: m1 + (p0 + n) > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) by A96, A107, XXREAL_0:2;
then A136: m4 < m1 + (p0 + n) by A121, XXREAL_0:2;
A137: dom f2 = Seg n by A114, FINSEQ_1:def_3;
A138: now__::_thesis:_for_i_being_Nat_st_i_in_dom_f2_holds_
f2_._i_=_f4_._i
let i be Nat; ::_thesis: ( i in dom f2 implies f2 . i = f4 . i )
reconsider a = i as Element of NAT by ORDINAL1:def_12;
assume A139: i in dom f2 ; ::_thesis: f2 . i = f4 . i
then A140: 1 <= i by A137, FINSEQ_1:1;
then  p0 + i >= p0 + 1 by XREAL_1:6;
then A141: p0 + i >= 8 by A100, XXREAL_0:2;
A142: i <= n by A137, A139, FINSEQ_1:1;
then  p0 + i <= p0 + n by XREAL_1:6;
then A143: p0 + i < m1 + (p0 + n) by A134, XXREAL_0:2;
thus f2 . i = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (p0 + a)) by A113, A114, A140, A142, SCPISORT:def_1
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos (p0 + a)) by A112, A126, SCPISORT:6
.= (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos (p0 + a)) by A19, A91, A125, A127, A141, A143
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos (p0 + a)) by A110, A111, SCPISORT:7
.= f4 . i by A115, A116, A140, A142 ; ::_thesis: verum
end;
A144: m1 + (p0 + n) >= 1 + 7 by A5, A133, XREAL_1:7;
then A145: m1 + (p0 + n) > 4 by XXREAL_0:2;
A146: (m1 + (p0 + n)) + 1 > m1 + (p0 + n) by XREAL_1:29;
then A147: (m1 + (p0 + n)) + 1 > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) by A135, XXREAL_0:2;
A148: p0 + n >= p0 by NAT_1:11;
then A149: m1 + (p0 + n) >= p0 + 1 by A133, XREAL_1:7;
then A150: (m1 + (p0 + n)) + 1 >= p0 + 1 by A146, XXREAL_0:2;
A151: (m1 + (p0 + n)) + 1 >= 8 + 1 by A144, XREAL_1:6;
then A152: (m1 + (p0 + n)) + 1 > 4 by XXREAL_0:2;
A153: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (m1 + (p0 + n))) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos (m1 + (p0 + n))) by SCMPDS_5:15;
A154: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:15;
 m1 + (p0 + n) > 2 by A144, XXREAL_0:2;
hence md = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos (m1 + (p0 + n))) by A6, A89, A90, A92, A145, I
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos (m1 + (p0 + n))) by A124, A149, A135, A153
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos (m1 + (p0 + n))) by A110, A111, A128, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos (m1 + (p0 + n))) by A112, A126, SCPISORT:6 ;
::_thesis: ( n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
 (m1 + (p0 + n)) + 1 > 2 by A151, XXREAL_0:2;
hence n4 = (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A6, A89, A90, A94, A152, I
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos ((m1 + (p0 + n)) + 1)) by A124, A150, A147, A154
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((m1 + (p0 + n)) + 3)) by A110, A111, A129, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) by A112, A126, SCPISORT:6 ;
::_thesis: ( ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
hereby  ::_thesis: ( f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < m1 implies (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) )
assume that
A155: 1 <= j and
A156: j < m1 ; ::_thesis: (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j))
A157: (p0 + n) + j < m1 + (p0 + n) by A156, XREAL_1:6;
 (p0 + n) + j >= (p0 + n) + 1 by A155, XREAL_1:6;
then  (p0 + n) + j > p0 + n by A132, XXREAL_0:2;
then A158: (p0 + n) + j > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) by A96, A107, XXREAL_0:2;
A159: (p0 + n) + j >= p0 + 1 by A148, A155, XREAL_1:7;
A160: (p0 + n) + j >= 1 + 7 by A5, A155, XREAL_1:7;
then A161: (p0 + n) + j > 2 by XXREAL_0:2;
A162: (p0 + n) + j > 4 by A160, XXREAL_0:2;
A163: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos ((p0 + n) + j)) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos ((p0 + n) + j)) by SCMPDS_5:15;
thus (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((p0 + n) + j)) = (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos ((p0 + n) + j)) by A112, A126, SCPISORT:6
.= (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos ((p0 + n) + j)) by A19, A91, A125, A127, A160, A157
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos ((p0 + n) + j)) by A110, A111, SCPISORT:7
.= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos ((p0 + n) + j)) by A124, A159, A158, A163
.= t . (intpos ((p0 + n) + j)) by A6, A89, A90, A161, A162, I ; ::_thesis: verum
end;
 f1,f4 are_fiberwise_equipotent by A4, A92, A93, A94, A95, A96, A99, A109, A108, A97, A117, A115, A118, Th14, A84, A85, A86;
hence  f1,f2 are_fiberwise_equipotent by A114, A115, A138, FINSEQ_2:9; ::_thesis: ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) )
take  m4 ; ::_thesis: ( md <= m4 & m4 <= n4 & m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) )
thus  ( md <= m4 & m4 <= n4 ) by A6, A89, A90, A94, A120, A121, I; ::_thesis: ( m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) )
thus m4 - 1 = ((IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos 4)) - 1 by A119, A110, A111, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A112, A126, A131, SCPISORT:6 ; ::_thesis: ( m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) )
thus m4 + 1 = ((IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos 4)) + 1 by A119, A110, A111, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) by A112, A126, A130, SCPISORT:6 ; ::_thesis: ( ( for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) )
A164: md >= 8 by A93, A100, XXREAL_0:2;
then A165: m4 >= 8 by A120, XXREAL_0:2;
hereby  ::_thesis: ( ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) ) )
A166: (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos m4) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos m4) by A110, A111, SCPISORT:7
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos m4) by A19, A91, A125, A127, A136, A165
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) by A112, A126, SCPISORT:6 ;
let i be Element of NAT ; ::_thesis: ( md <= i & i < m4 implies (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) )
assume that
A167: md <= i and
A168: i < m4 ; ::_thesis: (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i)
A169: i < m1 + (p0 + n) by A136, A168, XXREAL_0:2;
A170: i >= 8 by A164, A167, XXREAL_0:2;
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos i) by A110, A111, SCPISORT:7
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos i) by A19, A91, A125, A127, A169, A170
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) by A112, A126, SCPISORT:6 ;
hence  (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) by A122, A167, A168, A166; ::_thesis: verum
end;
hereby  ::_thesis: for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j)
A171: (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos m4) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos m4) by A110, A111, SCPISORT:7
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos m4) by A19, A91, A125, A127, A136, A165
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) by A112, A126, SCPISORT:6 ;
let i be Element of NAT ; ::_thesis: ( m4 < i & i <= n4 implies (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) )
assume that
A172: m4 < i and
A173: i <= n4 ; ::_thesis: (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i)
A174: i < m1 + (p0 + n) by A94, A108, A135, A173, XXREAL_0:2;
 i >= md by A120, A172, XXREAL_0:2;
then A175: i >= 8 by A164, XXREAL_0:2;
(IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos i) = (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos i) by A110, A111, SCPISORT:7
.= (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos i) by A19, A91, A125, A127, A174, A175
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) by A112, A126, SCPISORT:6 ;
hence  (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos i) by A107, A123, A172, A173, A171; ::_thesis: verum
end;
hereby  ::_thesis: verum
let j be Element of NAT ; ::_thesis: ( ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) implies (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j) )
assume A176: ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) ; ::_thesis: (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = t . (intpos j)
A177: now__::_thesis:_(_j_>=_p0_+_1_&_(_j_<_(IExec_((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1)))),Q,t))_._(intpos_2)_or_j_>_(IExec_((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1)))),Q,t))_._(intpos_4)_)_&_j_<_m1_+_(p0_+_n)_)
percases ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) by A176;
supposeA178: ( p0 + 1 <= j & j < md ) ; ::_thesis: ( j >= p0 + 1 & ( j < (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) or j > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) ) & j < m1 + (p0 + n) )
hence  ( j >= p0 + 1 & ( j < (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) or j > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) ) ) by A6, A89, A90, A92, I; ::_thesis: j < m1 + (p0 + n)
 md < m1 + (p0 + n) by A120, A136, XXREAL_0:2;
hence  j < m1 + (p0 + n) by A178, XXREAL_0:2; ::_thesis: verum
end;
supposeA179: ( n4 < j & j <= p0 + n ) ; ::_thesis: ( j >= p0 + 1 & ( j < (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) or j > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) ) & j < m1 + (p0 + n) )
 n4 >= md by A107, A120, A121, XXREAL_0:2;
then  j >= md by A179, XXREAL_0:2;
hence  j >= p0 + 1 by A93, XXREAL_0:2; ::_thesis: ( ( j < (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) or j > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) ) & j < m1 + (p0 + n) )
thus  ( j < (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 2) or j > (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos 4) ) by A6, A89, A90, A94, A179, I; ::_thesis: j < m1 + (p0 + n)
thus  j < m1 + (p0 + n) by A134, A179, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
then A180: j >= 8 by A100, XXREAL_0:2;
then A181: j > 2 by XXREAL_0:2;
A182: j > 4 by A180, XXREAL_0:2;
A183: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos j) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))) . (intpos j) by SCMPDS_5:15;
thus (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,t)) . (intpos j) = (IExec ((((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))),Q,(Initialize (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t))))) . (intpos j) by A112, A126, SCPISORT:6
.= (IExec (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt),Q,t)) . (intpos j) by A19, A91, A125, A127, A177, A180
.= (IExec (Pt,Q,(Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t))))) . (intpos j) by A110, A111, SCPISORT:7
.= (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,t)) . (intpos j) by A124, A177, A183
.= t . (intpos j) by A6, A89, A90, A181, A182, I ; ::_thesis: verum
end;
end;
A184: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m being Element of NAT st t . GBP = 0 & t . SBP = m holds
( (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) & ( for i being Element of NAT st i <> 2 holds
(IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
proof
let t be State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for m being Element of NAT st t . GBP = 0 & t . SBP = m holds
( (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) & ( for i being Element of NAT st i <> 2 holds
(IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for m being Element of NAT st t . GBP = 0 & t . SBP = m holds
( (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) & ( for i being Element of NAT st i <> 2 holds
(IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
let m be Element of NAT ; ::_thesis: ( t . GBP = 0 & t . SBP = m implies ( (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) & ( for i being Element of NAT st i <> 2 holds
(IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) ) )
assume that
A185: t . GBP = 0 and
A186: t . SBP = m ; ::_thesis: ( (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) & ( for i being Element of NAT st i <> 2 holds
(IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) ) )
set t0 = Initialize t;
set t1 = IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t));
set t2 = Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t));
set mp = m + (p0 + n);
A188: (Initialize t) . GBP = 0 by A185, SCMPDS_5:15;
then A189: DataLoc (((Initialize t) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
then  (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . GBP = 0 by A188, AMI_3:10, SCMPDS_2:47;
then A190: DataLoc (((Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
A191: (Initialize t) . SBP = m by A186, SCMPDS_5:15;
then A192: (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . SBP = m by A189, AMI_3:10, SCMPDS_2:47;
 m + (p0 + n) >= 0 + 7 by A5, XREAL_1:7;
then  m + (p0 + n) > 2 by XXREAL_0:2;
then A193: (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos (m + (p0 + n))) = (Initialize t) . (intpos (m + (p0 + n))) by A189, AMI_3:10, SCMPDS_2:47
.= t . (intpos (m + (p0 + n))) by SCMPDS_5:15 ;
A194: (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos 2) = (Initialize t) . (DataLoc (((Initialize t) . SBP),((p0 + n) + 1))) by A189, SCMPDS_2:47
.= (Initialize t) . (intpos (m + ((p0 + n) + 1))) by A191, SCMP_GCD:1
.= t . (intpos ((m + (p0 + n)) + 1)) by SCMPDS_5:15 ;
thus (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (Exec ((SubFrom (GBP,2,SBP,(p0 + n))),(Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))))) . (intpos 2) by SCMPDS_5:42
.= ((Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos 2)) - ((Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (DataLoc (((Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . SBP),(p0 + n)))) by A190, SCMPDS_2:50
.= (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) by A192, A194, A193, SCMP_GCD:1 ; ::_thesis: for i being Element of NAT st i <> 2 holds
(IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = t . (intpos i)
hereby  ::_thesis: verum
let i be Element of NAT ; ::_thesis: ( i <> 2 implies (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = t . (intpos i) )
assume A195: i <> 2 ; ::_thesis: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = t . (intpos i)
thus (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos i) = (Exec ((SubFrom (GBP,2,SBP,(p0 + n))),(Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))))) . (intpos i) by SCMPDS_5:42
.= (Exec (((GBP,2) := (SBP,((p0 + n) + 1))),(Initialize t))) . (intpos i) by A190, A195, AMI_3:10, SCMPDS_2:50
.= (Initialize t) . (intpos i) by A189, A195, AMI_3:10, SCMPDS_2:47
.= t . (intpos i) by SCMPDS_5:15 ; ::_thesis: verum
end;
end;
A196: (p0 + n) + 1 >= 7 + 1 by A5, XREAL_1:6;
A197: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m being Element of NAT st t . GBP = 0 & t . SBP = m & t . (intpos ((m + (p0 + n)) + 1)) <= t . (intpos (m + (p0 + n))) holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m - 2 & ( for j being Element of NAT st 1 <= j & j < m holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) )
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for m being Element of NAT st t . GBP = 0 & t . SBP = m & t . (intpos ((m + (p0 + n)) + 1)) <= t . (intpos (m + (p0 + n))) holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m - 2 & ( for j being Element of NAT st 1 <= j & j < m holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for m being Element of NAT st t . GBP = 0 & t . SBP = m & t . (intpos ((m + (p0 + n)) + 1)) <= t . (intpos (m + (p0 + n))) holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m - 2 & ( for j being Element of NAT st 1 <= j & j < m holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) )
let m be Element of NAT ; ::_thesis: ( t . GBP = 0 & t . SBP = m & t . (intpos ((m + (p0 + n)) + 1)) <= t . (intpos (m + (p0 + n))) implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m - 2 & ( for j being Element of NAT st 1 <= j & j < m holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) ) )
set mp = m + (p0 + n);
set t1 = IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t);
set Q1 = Q;
I: Initialize t = t by MEMSTR_0:44;
assume that
A198: t . GBP = 0 and
A199: t . SBP = m and
A200: t . (intpos ((m + (p0 + n)) + 1)) <= t . (intpos (m + (p0 + n))) ; ::_thesis: ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m - 2 & ( for j being Element of NAT st 1 <= j & j < m holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) )
A201: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos 1) = m by A184, A198, A199, I;
A202: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) by A184, A198, A199, I;
then A203: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos 2) <= 0 by A200, XREAL_1:47;
set li = AddTo (GBP,1,(- 2));
set t0 = Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t));
A204: Load (AddTo (GBP,1,(- 2))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by SCMPDS_6:21;
A205: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP = 0 by A184, A198, A199, I;
then A206: DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
A207: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP = 0 by A205, SCMPDS_5:15;
then A208: DataLoc (((Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP),1) = intpos (0 + 1) by SCMP_GCD:1;
A209: Load (AddTo (GBP,1,(- 2))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by SCMPDS_6:20;
then A210: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by A206, A203, A204, SCMPDS_6:69;
A211: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by A206, A203, A209, A204, SCMPDS_6:69;
A212: now__::_thesis:_for_x_being_Int_position_holds_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._x_=_(Exec_((AddTo_(GBP,1,(-_2))),(Initialize_(IExec_((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n)))),Q,t)))))_._x
let x be Int_position; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . x = (Exec ((AddTo (GBP,1,(- 2))),(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x
X1: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2)) by SCMPDS_5:15;
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . x = (IExec ((if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by A211, A210, SCPISORT:7
.= (IExec ((Load (AddTo (GBP,1,(- 2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by A200, A202, A206, X1, X2, SCMPDS_6:74, XREAL_1:47
.= (Exec ((AddTo (GBP,1,(- 2))),(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by SCMPDS_5:40 ; ::_thesis: verum
end;
A213: now__::_thesis:_for_i_being_Element_of_NAT_st_i_<>_1_holds_
(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_i)_=_(IExec_((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n)))),Q,t))_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i <> 1 implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos i) )
assume  i <> 1 ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos i)
then A214: intpos i <> DataLoc (((Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP),1) by A207, AMI_3:10, SCMP_GCD:1;
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) = (Exec ((AddTo (GBP,1,(- 2))),(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A212
.= (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos i) by A214, SCMPDS_2:48
.= (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos i) by SCMPDS_5:15 ; ::_thesis: verum
end;
hence  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 by A205; ::_thesis: ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m - 2 & ( for j being Element of NAT st 1 <= j & j < m holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) )
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = (Exec ((AddTo (GBP,1,(- 2))),(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos 1) by A212
.= ((Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos 1)) + (- 2) by A208, SCMPDS_2:48
.= ((Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos 1)) - 2
.= m - 2 by A201, SCMPDS_5:15 ; ::_thesis: ( ( for j being Element of NAT st 1 <= j & j < m holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) )
hereby  ::_thesis: for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j))
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < m implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) )
assume that
A215: 1 <= j and
 j < m ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j))
 (p0 + n) + j >= (p0 + n) + 1 by A215, XREAL_1:6;
then A216: (p0 + n) + j >= 8 by A196, XXREAL_0:2;
then A217: (p0 + n) + j > 2 by XXREAL_0:2;
 (p0 + n) + j > 1 by A216, XXREAL_0:2;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos ((p0 + n) + j)) by A213
.= t . (intpos ((p0 + n) + j)) by A184, A198, A199, A217, I ;
::_thesis: verum
end;
hereby  ::_thesis: verum
let j be Element of NAT ; ::_thesis: ( 1 <= j & j <= n implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) )
assume that
A218: 1 <= j and
 j <= n ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j))
A219: p0 + 1 >= 7 + 1 by A4, XREAL_1:6;
 p0 + j >= p0 + 1 by A218, XREAL_1:6;
then A220: p0 + j >= 8 by A219, XXREAL_0:2;
then A221: p0 + j > 2 by XXREAL_0:2;
 p0 + j > 1 by A220, XXREAL_0:2;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (p0 + j)) by A213
.= t . (intpos (p0 + j)) by A184, A198, A199, A221, I ;
::_thesis: verum
end;
end;
A222: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m1, md being Element of NAT st t . GBP = 0 & t . SBP = m1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & (t . (intpos ((m1 + (p0 + n)) + 1))) - md > 0 holds
( ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on t,Q & ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on t,Q )
proof
let t be State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for m1, md being Element of NAT st t . GBP = 0 & t . SBP = m1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & (t . (intpos ((m1 + (p0 + n)) + 1))) - md > 0 holds
( ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on t,Q & ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on t,Q )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for m1, md being Element of NAT st t . GBP = 0 & t . SBP = m1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & (t . (intpos ((m1 + (p0 + n)) + 1))) - md > 0 holds
( ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on t,Q & ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on t,Q )
let m1, md be Element of NAT ; ::_thesis: ( t . GBP = 0 & t . SBP = m1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & (t . (intpos ((m1 + (p0 + n)) + 1))) - md > 0 implies ( ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on t,Q & ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on t,Q ) )
set lPt = (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt;
set t2 = IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t));
set mp = m1 + (p0 + n);
set Q2 = Q;
A223: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . GBP = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t)))) . GBP by SCMPDS_5:15;
A224: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = (Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t)))) . (intpos 2) by SCMPDS_5:15;
assume that
A225: t . GBP = 0 and
A226: t . SBP = m1 and
A227: t . (intpos (m1 + (p0 + n))) = md and
A228: md >= p0 + 1 and
 (t . (intpos ((m1 + (p0 + n)) + 1))) - md > 0 ; ::_thesis: ( ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on t,Q & ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on t,Q )
A229: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . GBP = 0 by A6, A225, A226;
A230: (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))) . (intpos 2) = t . (intpos (m1 + (p0 + n))) by A6, A225, A226;
then  Pt is_halting_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))),Q by A4, A227, A228, A229, Th13, A223, A224;
then A231: Pt is_halting_on IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t)),Q by SCMPDS_6:126;
 Pt is_closed_on Initialize (IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t))),Q by A4, A227, A228, A230, A229, Th13, A223, A224;
then A232: Pt is_closed_on IExec ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))),Q,(Initialize t)),Q by SCMPDS_6:125;
then A233: (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt is_halting_on t,Q by A231, SCPISORT:9;
 (((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt is_closed_on t,Q by A232, A231, SCPISORT:9;
hence  ( ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on t,Q & ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on t,Q ) by A233, SCPISORT:8; ::_thesis: verum
end;
A234: for t being 0 -started State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m, m1, md being Element of NAT
for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f2 = n holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
proof
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for m, m1, md being Element of NAT
for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f2 = n holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for m, m1, md being Element of NAT
for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f2 = n holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let m, m1, md be Element of NAT ; ::_thesis: for n4 being Integer
for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f2 = n holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let n4 be Integer; ::_thesis: for f1, f2 being FinSequence of INT st t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f2 = n holds
( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let f1, f2 be FinSequence of INT ; ::_thesis: ( t . GBP = 0 & t . SBP = m1 & m1 = m + 1 & t . (intpos (m1 + (p0 + n))) = md & md >= p0 + 1 & n4 = t . (intpos ((m1 + (p0 + n)) + 1)) & n4 - md > 0 & n4 <= p0 + n & f1 is_FinSequence_on t,p0 & len f1 = n & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f2 = n implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) ) )
I: Initialize t = t by MEMSTR_0:44;
set mp = m1 + (p0 + n);
set t1 = IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t);
set Q1 = Q;
assume that
A235: t . GBP = 0 and
A236: t . SBP = m1 and
A237: m1 = m + 1 and
A238: t . (intpos (m1 + (p0 + n))) = md and
A239: md >= p0 + 1 and
A240: n4 = t . (intpos ((m1 + (p0 + n)) + 1)) and
A241: n4 - md > 0 and
A242: n4 <= p0 + n ; ::_thesis: ( not f1 is_FinSequence_on t,p0 or not len f1 = n or not f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 or not len f2 = n or ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) ) )
A243: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos 2) > 0 by A184, A235, A236, A238, A240, A241, I;
A244: m1 + (p0 + n) >= 0 + 7 by A5, XREAL_1:7;
then  m1 + (p0 + n) > 2 by XXREAL_0:2;
then A245: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (m1 + (p0 + n))) = md by A184, A235, A236, A238, I;
assume that
A246: f1 is_FinSequence_on t,p0 and
A247: len f1 = n ; ::_thesis: ( not f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 or not len f2 = n or ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) ) )
A248: p0 + 1 >= 7 + 1 by A4, XREAL_1:6;
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f1_holds_
f1_._i_=_(IExec_((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n)))),Q,t))_._(intpos_(p0_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f1 implies f1 . i = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (p0 + i)) )
assume that
A249: 1 <= i and
A250: i <= len f1 ; ::_thesis: f1 . i = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (p0 + i))
 p0 + i >= p0 + 1 by A249, XREAL_1:6;
then  p0 + i >= 8 by A248, XXREAL_0:2;
then A251: p0 + i > 2 by XXREAL_0:2;
thus f1 . i = t . (intpos (p0 + i)) by A246, A249, A250, SCPISORT:def_1
.= (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (p0 + i)) by A184, A235, A236, A251, I ; ::_thesis: verum
end;
then A252: f1 is_FinSequence_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),p0 by SCPISORT:def_1;
 (m1 + (p0 + n)) + 1 >= 7 + 1 by A244, XREAL_1:6;
then  (m1 + (p0 + n)) + 1 > 2 by XXREAL_0:2;
then A253: n4 = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A184, A235, A236, A240, I;
A254: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP = 0 by A184, A235, A236, I;
then A255: DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
A256: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . SBP = m1 by A184, A235, A236, I;
then A257: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by A222, A239, A241, A245, A253, A254;
B257: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q
proof
 Q +* (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) halts_on Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) by A257, SCMPDS_6:def_3;
hence  ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q by SCMPDS_6:def_3; ::_thesis: verum
end;
X1: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP),2)) by SCMPDS_5:15;
A258: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by A222, A239, A241, A256, A245, A253, A254;
B258: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q
proof
 for k being Element of NAT holds IC (Comput ((Q +* (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))))),(Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)))),k)) in dom (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) by A258, SCMPDS_6:def_2;
hence  ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q by SCMPDS_6:def_2; ::_thesis: verum
end;
then A259: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q by A255, A243, X1, X2, B257, SCMPDS_6:68;
B259: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q
proof
 Q +* (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) halts_on Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) by A259, SCMPDS_6:def_3;
hence  if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by SCMPDS_6:def_3; ::_thesis: verum
end;
assume that
A260: f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 and
A261: len f2 = n ; ::_thesis: ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
A262: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),Q by A255, A243, X1, X2, B257, B258, SCMPDS_6:68;
B262: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q
proof
 for k being Element of NAT holds IC (Comput ((Q +* (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))),(Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)))),k)) in dom (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) by A262, SCMPDS_6:def_2;
hence  if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t),Q by SCMPDS_6:def_2; ::_thesis: verum
end;
A263: now__::_thesis:_for_x_being_Int_position_holds_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._x_=_(IExec_((((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),Q,(Initialize_(IExec_((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n)))),Q,t)))))_._x
let x be Int_position; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . x = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . x = (IExec ((if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by B262, B259, SCPISORT:7
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . x by A255, A243, Th1, X1, X2, B258, B257 ; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f2_holds_
f2_._i_=_(IExec_((((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),Q,(Initialize_(IExec_((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n)))),Q,t)))))_._(intpos_(p0_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f2 implies f2 . i = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos (p0 + i)) )
assume that
A264: 1 <= i and
A265: i <= len f2 ; ::_thesis: f2 . i = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos (p0 + i))
thus f2 . i = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + i)) by A260, A264, A265, SCPISORT:def_1
.= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos (p0 + i)) by A263 ; ::_thesis: verum
end;
then A266: f2 is_FinSequence_on IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)))),p0 by SCPISORT:def_1;
X1: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . SBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . SBP by SCMPDS_5:15;
X2: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . GBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . GBP by SCMPDS_5:15;
X3: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos (m1 + (p0 + n))) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (m1 + (p0 + n))) by SCMPDS_5:15;
X4: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos ((m1 + (p0 + n)) + 1)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by SCMPDS_5:15;
X5: f1 is_FinSequence_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)),p0
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f1 or f1 . i = (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos (p0 + i)) )
assume  ( 1 <= i & i <= len f1 ) ; ::_thesis: f1 . i = (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos (p0 + i))
then  f1 . i = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos (p0 + i)) by A252, SCPISORT:def_1;
hence  f1 . i = (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos (p0 + i)) by SCMPDS_5:15; ::_thesis: verum
end;
 (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . GBP = 0 by A83, A237, A239, A241, A242, A247, A261, A256, A245, A253, A254, A266, X1, X2, X3, X4, X5;
hence  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = 0 by A263; ::_thesis: ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 & md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
consider m4 being Element of NAT  such that
A267: md <= m4 and
A268: m4 <= n4 and
A269: m4 - 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos ((m1 + (p0 + n)) + 1)) and
A270: m4 + 1 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos ((m1 + (p0 + n)) + 2)) and
A271: for i being Element of NAT st md <= i & i < m4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) and
A272: for i being Element of NAT st m4 < i & i <= n4 holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) and
B273: for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos j) by A83, A237, A239, A241, A242, A247, A261, A256, A245, A253, A254, A266, X1, X2, X3, X4, X5;
A273: for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos j)
proof
let j be Element of NAT ; ::_thesis: ( ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) implies (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos j) )
assume  ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) ; ::_thesis: (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos j)
then  (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos j) by B273;
hence  (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos j) by SCMPDS_5:15; ::_thesis: verum
end;
 (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos 1) = m1 + 2 by A83, A237, A239, A241, A242, A247, A261, A256, A245, A253, A254, A266, X1, X2, X3, X4, X5;
hence  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos 1) = m1 + 2 by A263; ::_thesis: ( md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) & n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
 md = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos (m1 + (p0 + n))) by A83, A237, A239, A241, A242, A247, A261, A256, A245, A253, A254, A266, X1, X2, X3, X4, X5;
hence  md = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (m1 + (p0 + n))) by A263; ::_thesis: ( n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) & ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
 n4 = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos ((m1 + (p0 + n)) + 3)) by A83, A237, A239, A241, A242, A247, A261, A256, A245, A253, A254, A266, X1, X2, X3, X4, X5;
hence  n4 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 3)) by A263; ::_thesis: ( ( for j being Element of NAT st 1 <= j & j < m1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
hereby  ::_thesis: ( f1,f2 are_fiberwise_equipotent & ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) ) )
let j be Element of NAT ; ::_thesis: ( 1 <= j & j < m1 implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) )
assume that
A274: 1 <= j and
A275: j < m1 ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j))
 (p0 + n) + j >= (p0 + n) + 1 by A274, XREAL_1:6;
then  (p0 + n) + j >= 8 by A196, XXREAL_0:2;
then A276: (p0 + n) + j > 2 by XXREAL_0:2;
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos ((p0 + n) + j)) by A263
.= (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))) . (intpos ((p0 + n) + j)) by A83, A237, A239, A241, A242, A247, A261, A256, A245, A253, A254, A266, A274, A275, X1, X2, X3, X4, X5
.= (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos ((p0 + n) + j)) by SCMPDS_5:15
.= t . (intpos ((p0 + n) + j)) by A184, A235, A236, A276, I ; ::_thesis: verum
end;
thus  f1,f2 are_fiberwise_equipotent by A83, A237, A239, A241, A242, A247, A261, A256, A245, A253, A254, A266, X1, X2, X3, X4, X5; ::_thesis: ex m4 being Element of NAT st
( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) )
take  m4 ; ::_thesis: ( md <= m4 & m4 <= n4 & m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) )
thus  ( md <= m4 & m4 <= n4 ) by A267, A268; ::_thesis: ( m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) & m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) )
thus  m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 1)) by A263, A269; ::_thesis: ( m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) & ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) )
thus  m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((m1 + (p0 + n)) + 2)) by A263, A270; ::_thesis: ( ( for i being Element of NAT st md <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) )
hereby  ::_thesis: ( ( for i being Element of NAT st m4 < i & i <= n4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) ) & ( for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) ) )
let i be Element of NAT ; ::_thesis: ( md <= i & i < m4 implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) )
assume that
A277: md <= i and
A278: i < m4 ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i)
 (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A271, A277, A278;
then  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A263;
hence  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) by A263; ::_thesis: verum
end;
hereby  ::_thesis: for j being Element of NAT st ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j)
let i be Element of NAT ; ::_thesis: ( m4 < i & i <= n4 implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) )
assume that
A279: m4 < i and
A280: i <= n4 ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i)
 (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A272, A279, A280;
then  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos i) by A263;
hence  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) by A263; ::_thesis: verum
end;
hereby  ::_thesis: verum
let j be Element of NAT ; ::_thesis: ( ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) )
assume A281: ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j)
now__::_thesis:_p0_+_1_<=_j
percases ( ( p0 + 1 <= j & j < md ) or ( n4 < j & j <= p0 + n ) ) by A281;
suppose ( p0 + 1 <= j & j < md ) ; ::_thesis: p0 + 1 <= j
hence  p0 + 1 <= j ; ::_thesis: verum
end;
suppose ( n4 < j & j <= p0 + n ) ; ::_thesis: j >= p0 + 1
then  j >= m4 by A268, XXREAL_0:2;
then  j >= md by A267, XXREAL_0:2;
hence  j >= p0 + 1 by A239, XXREAL_0:2; ::_thesis: verum
end;
end;
end;
then  j >= 8 by A248, XXREAL_0:2;
then A282: j > 2 by XXREAL_0:2;
thus (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = (IExec ((((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),Q,(Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t))))) . (intpos j) by A263
.= (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,t)) . (intpos j) by A273, A281
.= t . (intpos j) by A184, A235, A236, A282, I ; ::_thesis: verum
end;
end;
consider f being FinSequence of INT  such that
A283: len f = n and
A284: f is_FinSequence_on s,p0 by SCPISORT:2;
A285: (p0 + n) + 2 >= 7 + 2 by A5, XREAL_1:6;
A286: ( (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . GBP = 0 & (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos 1) = 1 & (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) = p0 + 1 & (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 2)) = p0 + n & ( for i being Element of NAT st i >= 8 & i <= p0 + n holds
(IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = s . (intpos i) ) )
proof
set t2 = IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s);
set t3 = IExec (((GBP := 0) ';' (SBP := 1)),P,s);
set t4 = Exec ((GBP := 0),s);
A287: (Exec ((GBP := 0),s)) . GBP = 0 by SCMPDS_2:45;
A288: (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . GBP = (Exec ((SBP := 1),(Exec ((GBP := 0),s)))) . GBP by SCMPDS_5:42
.= 0 by A287, SCMPDS_2:45, SCMP_GCD:3 ;
A289: (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . SBP = (Exec ((SBP := 1),(Exec ((GBP := 0),s)))) . SBP by SCMPDS_5:42
.= 1 by SCMPDS_2:45 ;
then A290: DataLoc (((IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . SBP),(p0 + n)) = intpos ((p0 + n) + 1) by SCMP_GCD:1;
A291: (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . GBP = (Exec (((SBP,(p0 + n)) := (p0 + 1)),(IExec (((GBP := 0) ';' (SBP := 1)),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A288, A290, AMI_3:10, SCMPDS_2:46 ;
A292: (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . (intpos ((p0 + n) + 1)) = (Exec (((SBP,(p0 + n)) := (p0 + 1)),(IExec (((GBP := 0) ';' (SBP := 1)),P,s)))) . (intpos ((p0 + n) + 1)) by SCMPDS_5:41
.= p0 + 1 by A290, SCMPDS_2:46 ;
A293: (p0 + n) + 2 > 1 by A285, XXREAL_0:2;
A294: (p0 + n) + 1 > 1 by A196, XXREAL_0:2;
A295: (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . SBP = (Exec (((SBP,(p0 + n)) := (p0 + 1)),(IExec (((GBP := 0) ';' (SBP := 1)),P,s)))) . SBP by SCMPDS_5:41
.= 1 by A289, A290, A294, AMI_3:10, SCMPDS_2:46 ;
then A296: DataLoc (((IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . SBP),((p0 + n) + 1)) = intpos ((p0 + n) + (1 + 1)) by SCMP_GCD:1;
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . GBP = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . GBP by SCMPDS_5:41
.= 0 by A291, A296, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos 1) = 1 & (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) = p0 + 1 & (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 2)) = p0 + n & ( for i being Element of NAT st i >= 8 & i <= p0 + n holds
(IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos 1) = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . SBP by SCMPDS_5:41
.= 1 by A295, A296, A293, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) = p0 + 1 & (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 2)) = p0 + n & ( for i being Element of NAT st i >= 8 & i <= p0 + n holds
(IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = s . (intpos i) ) )
A297: (p0 + n) + 2 > (p0 + n) + 1 by XREAL_1:6;
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . (intpos ((p0 + n) + 1)) by SCMPDS_5:41
.= p0 + 1 by A292, A296, A297, AMI_3:10, SCMPDS_2:46 ; ::_thesis: ( (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 2)) = p0 + n & ( for i being Element of NAT st i >= 8 & i <= p0 + n holds
(IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = s . (intpos i) ) )
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 2)) = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . (intpos ((p0 + n) + 2)) by SCMPDS_5:41
.= p0 + n by A296, SCMPDS_2:46 ; ::_thesis: for i being Element of NAT st i >= 8 & i <= p0 + n holds
(IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = s . (intpos i)
A298: for i being Element of NAT st i >= 8 holds
(Exec ((GBP := 0),s)) . (intpos i) = s . (intpos i) by AMI_3:10, SCMPDS_2:45;
A299: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_holds_
(IExec_(((GBP_:=_0)_';'_(SBP_:=_1)),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 implies (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . (intpos i) = s . (intpos i) )
assume A300: i >= 8 ; ::_thesis: (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . (intpos i) = s . (intpos i)
then A301: i > 1 by XXREAL_0:2;
thus (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . (intpos i) = (Exec ((SBP := 1),(Exec ((GBP := 0),s)))) . (intpos i) by SCMPDS_5:42
.= (Exec ((GBP := 0),s)) . (intpos i) by A301, AMI_3:10, SCMPDS_2:45
.= s . (intpos i) by A298, A300 ; ::_thesis: verum
end;
A302: now__::_thesis:_for_i_being_Element_of_NAT_st_i_>=_8_&_i_<=_p0_+_n_holds_
(IExec_((((GBP_:=_0)_';'_(SBP_:=_1))_';'_((SBP,(p0_+_n))_:=_(p0_+_1))),P,s))_._(intpos_i)_=_s_._(intpos_i)
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <= p0 + n implies (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . (intpos i) = s . (intpos i) )
assume that
A303: i >= 8 and
A304: i <= p0 + n ; ::_thesis: (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . (intpos i) = s . (intpos i)
A305: (p0 + n) + 1 > p0 + n by XREAL_1:29;
thus (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . (intpos i) = (Exec (((SBP,(p0 + n)) := (p0 + 1)),(IExec (((GBP := 0) ';' (SBP := 1)),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec (((GBP := 0) ';' (SBP := 1)),P,s)) . (intpos i) by A290, A304, A305, AMI_3:10, SCMPDS_2:46
.= s . (intpos i) by A299, A303 ; ::_thesis: verum
end;
A306: (p0 + n) + 2 > (p0 + n) + 0 by XREAL_1:6;
let i be Element of NAT ; ::_thesis: ( i >= 8 & i <= p0 + n implies (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = s . (intpos i) )
assume that
A307: i >= 8 and
A308: i <= p0 + n ; ::_thesis: (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = s . (intpos i)
thus (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos i) = (Exec (((SBP,((p0 + n) + 1)) := (p0 + n)),(IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)))) . (intpos i) by SCMPDS_5:41
.= (IExec ((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))),P,s)) . (intpos i) by A296, A308, A306, AMI_3:10, SCMPDS_2:46
.= s . (intpos i) by A302, A307, A308 ; ::_thesis: verum
end;
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_f_holds_
f_._i_=_(IExec_(((((GBP_:=_0)_';'_(SBP_:=_1))_';'_((SBP,(p0_+_n))_:=_(p0_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(p0_+_n))),P,s))_._(intpos_(p0_+_i))
A309: p0 + 1 >= 7 + 1 by A4, XREAL_1:6;
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len f implies f . i = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (p0 + i)) )
assume that
A310: 1 <= i and
A311: i <= len f ; ::_thesis: f . i = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (p0 + i))
A312: p0 + i <= p0 + n by A283, A311, XREAL_1:6;
 p0 + i >= p0 + 1 by A310, XREAL_1:6;
then A313: p0 + i >= 8 by A309, XXREAL_0:2;
thus f . i = s . (intpos (p0 + i)) by A284, A310, A311, SCPISORT:def_1
.= (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (p0 + i)) by A286, A313, A312 ; ::_thesis: verum
end;
then A314: f is_FinSequence_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),p0 by SCPISORT:def_1;
consider g being FinSequence of INT  such that
A315: len g = n and
A316: g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 by SCPISORT:2;
A317: DataLoc (0,1) = intpos (0 + 1) by SCMP_GCD:1;
A318: for t being State of SCMPDS
for Q being Instruction-Sequence of SCMPDS
for m, md being Element of NAT st t . GBP = 0 & t . SBP = m & t . (intpos (m + (p0 + n))) = md & ( md >= p0 + 1 or (t . (intpos ((m + (p0 + n)) + 1))) - md <= 0 ) holds
( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q )
proof
let t be State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for m, md being Element of NAT st t . GBP = 0 & t . SBP = m & t . (intpos (m + (p0 + n))) = md & ( md >= p0 + 1 or (t . (intpos ((m + (p0 + n)) + 1))) - md <= 0 ) holds
( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for m, md being Element of NAT st t . GBP = 0 & t . SBP = m & t . (intpos (m + (p0 + n))) = md & ( md >= p0 + 1 or (t . (intpos ((m + (p0 + n)) + 1))) - md <= 0 ) holds
( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q )
let m, md be Element of NAT ; ::_thesis: ( t . GBP = 0 & t . SBP = m & t . (intpos (m + (p0 + n))) = md & ( md >= p0 + 1 or (t . (intpos ((m + (p0 + n)) + 1))) - md <= 0 ) implies ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q ) )
set mp = m + (p0 + n);
set t1 = IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t));
set Q1 = Q;
assume that
A319: t . GBP = 0 and
A320: t . SBP = m and
A321: t . (intpos (m + (p0 + n))) = md and
A322: ( md >= p0 + 1 or (t . (intpos ((m + (p0 + n)) + 1))) - md <= 0 ) ; ::_thesis: ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q )
A323: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos 2) = (t . (intpos ((m + (p0 + n)) + 1))) - (t . (intpos (m + (p0 + n)))) by A184, A319, A320;
A324: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP = 0 by A184, A319, A320;
then A325: DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2) = intpos (0 + 2) by SCMP_GCD:1;
percases ( (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) > 0 or (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) <= 0 ) ;
supposeA326: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) > 0 ; ::_thesis: ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q )
A327: m + (p0 + n) >= 0 + 7 by A5, XREAL_1:7;
then  (m + (p0 + n)) + 1 >= 7 + 1 by XREAL_1:6;
then  (m + (p0 + n)) + 1 > 2 by XXREAL_0:2;
then A328: ((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos ((m + (p0 + n)) + 1))) - md > 0 by A184, A319, A320, A321, A323, A325, A326;
A329: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . SBP = m by A184, A319, A320;
 m + (p0 + n) > 2 by A327, XXREAL_0:2;
then A330: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (intpos (m + (p0 + n))) = md by A184, A319, A320, A321;
then A331: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by A222, A184, A319, A320, A321, A322, A324, A325, A326, A329, A328;
X1: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) by SCMPDS_5:15;
X2: (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)))) . GBP = (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP by SCMPDS_5:15;
A332: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by A222, A184, A319, A320, A321, A322, A324, A325, A326, A329, A330, A328;
B331: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q
proof
 Q +* (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) halts_on Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)))) by A331, SCMPDS_6:def_3;
hence  ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q by SCMPDS_6:def_3; ::_thesis: verum
end;
B332: ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q
proof
 for k being Element of NAT holds IC (Comput ((Q +* (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))))),(Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))))),k)) in dom (stop (((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))))) by A332, SCMPDS_6:def_2;
hence  ((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q by SCMPDS_6:def_2; ::_thesis: verum
end;
then A333: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q by A326, B331, X1, X2, SCMPDS_6:68;
B333: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q
proof
 Q +* (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) halts_on Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)))) by A333, SCMPDS_6:def_3;
hence  if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by SCMPDS_6:def_3; ::_thesis: verum
end;
XX: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))),Q by A326, B332, B331, X1, X2, SCMPDS_6:68;
 if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q
proof
 for k being Element of NAT holds IC (Comput ((Q +* (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))),(Initialize (Initialize (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))))),k)) in dom (stop (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))) by XX, SCMPDS_6:def_2;
hence  if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by SCMPDS_6:def_2; ::_thesis: verum
end;
hence  ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q ) by B333, SCPISORT:9; ::_thesis: verum
end;
supposeA334: (IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . (DataLoc (((IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t))) . GBP),2)) <= 0 ; ::_thesis: ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q )
A335: Load (AddTo (GBP,1,(- 2))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by SCMPDS_6:21;
A336: Load (AddTo (GBP,1,(- 2))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by SCMPDS_6:20;
then A337: if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_halting_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by A334, A335, SCMPDS_6:69;
 if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))) is_closed_on IExec ((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))),Q,(Initialize t)),Q by A334, A336, A335, SCMPDS_6:69;
hence  ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q ) by A337, SCPISORT:9; ::_thesis: verum
end;
end;
end;
A338: now__::_thesis:_for_t_being_0_-started_State_of_SCMPDS
for_Q_being_Instruction-Sequence_of_SCMPDS
for_f1,_f2_being_FinSequence_of_INT_
for_k1,_k2,_y1,_yn_being_Element_of_NAT_st_t_._GBP_=_0_&_(2_*_k1)_+_1_=_t_._(DataLoc_(0,1))_&_k2_=_((p0_+_n)_+_(2_*_k1))_+_1_&_p0_+_y1_=_t_._(intpos_k2)_&_p0_+_yn_=_t_._(intpos_(k2_+_1))_&_(_(_1_<=_y1_&_yn_<=_n_)_or_y1_>=_yn_)_holds_
(_(((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))_is_closed_on_t,Q_&_(((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))_is_halting_on_t,Q_&_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._GBP_=_t_._GBP_&_(_for_j_being_Element_of_NAT_st_1_<=_j_&_j_<_(2_*_k1)_+_1_holds_
(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_((p0_+_n)_+_j))_=_t_._(intpos_((p0_+_n)_+_j))_)_&_(_y1_>=_yn_implies_(_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(DataLoc_(0,1))_=_(2_*_k1)_-_1_&_(_for_j_being_Element_of_NAT_st_1_<=_j_&_j_<=_n_holds_
(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_(p0_+_j))_=_t_._(intpos_(p0_+_j))_)_)_)_&_(_y1_<_yn_implies_(_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(DataLoc_(0,1))_=_(2_*_k1)_+_3_&_(_for_j_being_Element_of_NAT_st_(_(_1_<=_j_&_j_<_y1_)_or_(_yn_<_j_&_j_<=_n_)_)_holds_
(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_(p0_+_j))_=_t_._(intpos_(p0_+_j))_)_&_ex_ym_being_Element_of_NAT_st_
(_y1_<=_ym_&_ym_<=_yn_&_p0_+_y1_=_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_k2)_&_(p0_+_ym)_-_1_=_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_(k2_+_1))_&_(p0_+_ym)_+_1_=_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_(k2_+_2))_&_p0_+_yn_=_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_(k2_+_3))_&_(_for_j_being_Element_of_NAT_st_y1_<=_j_&_j_<_ym_holds_
(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_(p0_+_j))_<=_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_(p0_+_ym))_)_&_(_for_j_being_Element_of_NAT_st_ym_<_j_&_j_<=_yn_holds_
(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_(p0_+_j))_>=_(IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t))_._(intpos_(p0_+_ym))_)_)_)_)_&_(_f1_is_FinSequence_on_t,p0_&_f2_is_FinSequence_on_IExec_(((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))),Q,t),p0_&_len_f1_=_n_&_len_f2_=_n_implies_f1,f2_are_fiberwise_equipotent_)_)
let t be 0 -started State of SCMPDS; ::_thesis: for Q being Instruction-Sequence of SCMPDS
for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . GBP = 0 & (2 * k1) + 1 = t . (DataLoc (0,1)) & k2 = ((p0 + n) + (2 * k1)) + 1 & p0 + y1 = t . (intpos k2) & p0 + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( y1 >= yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) ) ) & ( y1 < yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) )
let Q be Instruction-Sequence of SCMPDS; ::_thesis: for f1, f2 being FinSequence of INT
for k1, k2, y1, yn being Element of NAT st t . GBP = 0 & (2 * k1) + 1 = t . (DataLoc (0,1)) & k2 = ((p0 + n) + (2 * k1)) + 1 & p0 + y1 = t . (intpos k2) & p0 + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( y1 >= yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) ) ) & ( y1 < yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) )
let f1, f2 be FinSequence of INT ; ::_thesis: for k1, k2, y1, yn being Element of NAT st t . GBP = 0 & (2 * k1) + 1 = t . (DataLoc (0,1)) & k2 = ((p0 + n) + (2 * k1)) + 1 & p0 + y1 = t . (intpos k2) & p0 + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) holds
( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( y1 >= yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) ) ) & ( y1 < yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) )
let k1, k2, y1, yn be Element of NAT ; ::_thesis: ( t . GBP = 0 & (2 * k1) + 1 = t . (DataLoc (0,1)) & k2 = ((p0 + n) + (2 * k1)) + 1 & p0 + y1 = t . (intpos k2) & p0 + yn = t . (intpos (k2 + 1)) & ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) implies ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( y1 >= yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) ) ) & ( y1 < yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) ) )
set mm = (2 * k1) + 1;
set md = p0 + y1;
set n4 = p0 + yn;
assume that
A339: t . GBP = 0 and
A340: (2 * k1) + 1 = t . (DataLoc (0,1)) and
A341: k2 = ((p0 + n) + (2 * k1)) + 1 and
A342: p0 + y1 = t . (intpos k2) and
A343: p0 + yn = t . (intpos (k2 + 1)) and
A344: ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) ; ::_thesis: ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q & (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( y1 >= yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) ) ) & ( y1 < yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) )
set mp = ((2 * k1) + 1) + (p0 + n);
A345: p0 + y1 = t . (intpos (((2 * k1) + 1) + (p0 + n))) by A341, A342;
now__::_thesis:_(_(_1_<=_y1_&_yn_<=_n_&_p0_+_1_<=_p0_+_y1_)_or_(_y1_>=_yn_&_(t_._(intpos_((((2_*_k1)_+_1)_+_(p0_+_n))_+_1)))_-_(p0_+_y1)_<=_0_)_)
percases ( ( 1 <= y1 & yn <= n ) or y1 >= yn ) by A344;
case ( 1 <= y1 & yn <= n ) ; ::_thesis: p0 + 1 <= p0 + y1
hence  p0 + 1 <= p0 + y1 by XREAL_1:6; ::_thesis: verum
end;
case y1 >= yn ; ::_thesis: (t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1))) - (p0 + y1) <= 0
then  p0 + y1 >= t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) by A341, A343, XREAL_1:6;
hence  (t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1))) - (p0 + y1) <= 0 by XREAL_1:47; ::_thesis: verum
end;
end;
end;
hence  ( (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_closed_on t,Q & (((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))) is_halting_on t,Q ) by A318, A317, A339, A340, A345; ::_thesis: ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) & ( y1 >= yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) ) ) & ( y1 < yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) )
consider f3 being FinSequence of INT  such that
A346: len f3 = n and
A347: for i being Element of NAT st 1 <= i & i <= len f3 holds
f3 . i = t . (intpos (p0 + i)) by SCPISORT:1;
consider f4 being FinSequence of INT  such that
A348: len f4 = n and
A349: for i being Element of NAT st 1 <= i & i <= len f4 holds
f4 . i = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + i)) by SCPISORT:1;
A350: f4 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 by A349, SCPISORT:def_1;
A351: f3 is_FinSequence_on t,p0 by A347, SCPISORT:def_1;
hereby  ::_thesis: ( ( y1 >= yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) ) ) & ( y1 < yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) )
percases ( t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) <= t . (intpos (((2 * k1) + 1) + (p0 + n))) or t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) > t . (intpos (((2 * k1) + 1) + (p0 + n))) ) ;
suppose t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) <= t . (intpos (((2 * k1) + 1) + (p0 + n))) ; ::_thesis: ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) )
hence  ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) ) by A197, A317, A339, A340; ::_thesis: verum
end;
supposeA352: t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) > t . (intpos (((2 * k1) + 1) + (p0 + n))) ; ::_thesis: ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) )
then A353: (p0 + yn) - (p0 + y1) > 0 by A341, A342, A343, XREAL_1:50;
A354: p0 + yn <= p0 + n by A341, A342, A343, A344, A352, XREAL_1:6;
 p0 + 1 <= p0 + y1 by A341, A342, A343, A344, A352, XREAL_1:6;
hence  ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . GBP = t . GBP & ( for j being Element of NAT st 1 <= j & j < (2 * k1) + 1 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((p0 + n) + j)) = t . (intpos ((p0 + n) + j)) ) ) by A234, A317, A339, A340, A341, A343, A345, A346, A351, A348, A350, A353, A354; ::_thesis: verum
end;
end;
end;
hereby  ::_thesis: ( ( y1 < yn implies ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) ) ) & ( f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent ) )
assume  y1 >= yn ; ::_thesis: ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) - 1 & ( for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) )
then A355: t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) <= t . (intpos (((2 * k1) + 1) + (p0 + n))) by A341, A342, A343, XREAL_1:6;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = ((2 * k1) + 1) - 2 by A197, A317, A339, A340
.= (2 * k1) - 1 ;
::_thesis: for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j))
thus  for j being Element of NAT st 1 <= j & j <= n holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) by A197, A317, A339, A340, A355; ::_thesis: verum
end;
A356: k2 = ((2 * k1) + 1) + (p0 + n) by A341;
hereby  ::_thesis: ( f1 is_FinSequence_on t,p0 & f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 & len f1 = n & len f2 = n implies f1,f2 are_fiberwise_equipotent )
assume A357: y1 < yn ; ::_thesis: ( (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = (2 * k1) + 3 & ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) )
then A358: p0 + 1 <= p0 + y1 by A344, XREAL_1:6;
 p0 + yn > p0 + y1 by A357, XREAL_1:6;
then A359: (p0 + yn) - (p0 + y1) > 0 by XREAL_1:50;
A360: p0 + yn <= p0 + n by A344, A357, XREAL_1:6;
hence (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (DataLoc (0,1)) = ((2 * k1) + 1) + 2 by A234, A317, A339, A340, A341, A343, A345, A346, A351, A348, A350, A359, A358
.= (2 * k1) + 3 ;
::_thesis: ( ( for j being Element of NAT st ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) ) & ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) ) )
consider m4 being Element of NAT  such that
A361: p0 + y1 <= m4 and
A362: m4 <= p0 + yn and
A363: m4 - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) and
A364: m4 + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos ((((2 * k1) + 1) + (p0 + n)) + 2)) and
A365: for i being Element of NAT st p0 + y1 <= i & i < m4 holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) and
A366: for i being Element of NAT st m4 < i & i <= p0 + yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos m4) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos i) and
A367: for j being Element of NAT st ( ( p0 + 1 <= j & j < p0 + y1 ) or ( p0 + yn < j & j <= p0 + n ) ) holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos j) = t . (intpos j) by A234, A317, A339, A340, A341, A342, A343, A346, A351, A348, A350, A359, A358, A360;
hereby  ::_thesis: ex ym being Element of NAT st
( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) )
let j be Element of NAT ; ::_thesis: ( ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) )
assume  ( ( 1 <= j & j < y1 ) or ( yn < j & j <= n ) ) ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j))
then  ( ( p0 + 1 <= p0 + j & p0 + j < p0 + y1 ) or ( p0 + yn < p0 + j & p0 + j <= p0 + n ) ) by XREAL_1:6;
hence  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) = t . (intpos (p0 + j)) by A367; ::_thesis: verum
end;
 m4 - p0 >= y1 by A361, XREAL_1:19;
then reconsider ym = m4 - p0 as Element of NAT by INT_1:3;
take ym = ym; ::_thesis: ( y1 <= ym & ym <= yn & p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) )
 m4 = p0 + ym ;
hence  ( y1 <= ym & ym <= yn ) by A361, A362, XREAL_1:6; ::_thesis: ( p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) & (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) )
thus  p0 + y1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos k2) by A234, A317, A339, A340, A342, A343, A356, A346, A351, A348, A350, A359, A358, A360; ::_thesis: ( (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) & (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) )
thus  (p0 + ym) - 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 1)) by A341, A363; ::_thesis: ( (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) & p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) )
thus  (p0 + ym) + 1 = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 2)) by A341, A364; ::_thesis: ( p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) & ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) )
thus  p0 + yn = (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (k2 + 3)) by A234, A317, A339, A340, A342, A343, A356, A346, A351, A348, A350, A359, A358, A360; ::_thesis: ( ( for j being Element of NAT st y1 <= j & j < ym holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) & ( for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) ) )
hereby  ::_thesis: for j being Element of NAT st ym < j & j <= yn holds
(IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym))
let j be Element of NAT ; ::_thesis: ( y1 <= j & j < ym implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) )
assume that
A368: y1 <= j and
A369: j < ym ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym))
A370: p0 + j < p0 + ym by A369, XREAL_1:6;
 p0 + y1 <= p0 + j by A368, XREAL_1:6;
hence  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) <= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) by A365, A370; ::_thesis: verum
end;
hereby  ::_thesis: verum
let j be Element of NAT ; ::_thesis: ( ym < j & j <= yn implies (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) )
assume that
A371: ym < j and
A372: j <= yn ; ::_thesis: (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym))
A373: p0 + j <= p0 + yn by A372, XREAL_1:6;
 p0 + ym < p0 + j by A371, XREAL_1:6;
hence  (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + j)) >= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + ym)) by A366, A373; ::_thesis: verum
end;
end;
hereby  ::_thesis: verum
assume that
A374: f1 is_FinSequence_on t,p0 and
A375: f2 is_FinSequence_on IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t),p0 and
A376: len f1 = n and
A377: len f2 = n ; ::_thesis: f1,f2 are_fiberwise_equipotent
percases ( t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) <= t . (intpos (((2 * k1) + 1) + (p0 + n))) or t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) > t . (intpos (((2 * k1) + 1) + (p0 + n))) ) ;
supposeA378: t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) <= t . (intpos (((2 * k1) + 1) + (p0 + n))) ; ::_thesis: f1,f2 are_fiberwise_equipotent
A379: dom f1 = Seg n by A376, FINSEQ_1:def_3;
now__::_thesis:_for_i_being_Nat_st_i_in_dom_f1_holds_
f1_._i_=_f2_._i
let i be Nat; ::_thesis: ( i in dom f1 implies f1 . i = f2 . i )
reconsider a = i as Element of NAT by ORDINAL1:def_12;
assume A380: i in dom f1 ; ::_thesis: f1 . i = f2 . i
then A381: 1 <= i by A379, FINSEQ_1:1;
A382: i <= n by A379, A380, FINSEQ_1:1;
hence f1 . i = t . (intpos (p0 + a)) by A374, A376, A381, SCPISORT:def_1
.= (IExec (((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))),Q,t)) . (intpos (p0 + a)) by A197, A317, A339, A340, A378, A381, A382
.= f2 . i by A375, A377, A381, A382, SCPISORT:def_1 ;
::_thesis: verum
end;
hence  f1,f2 are_fiberwise_equipotent by A376, A377, FINSEQ_2:9; ::_thesis: verum
end;
supposeA383: t . (intpos ((((2 * k1) + 1) + (p0 + n)) + 1)) > t . (intpos (((2 * k1) + 1) + (p0 + n))) ; ::_thesis: f1,f2 are_fiberwise_equipotent
then A384: (p0 + yn) - (p0 + y1) > 0 by A341, A342, A343, XREAL_1:50;
A385: p0 + yn <= p0 + n by A341, A342, A343, A344, A383, XREAL_1:6;
 p0 + 1 <= p0 + y1 by A341, A342, A343, A344, A383, XREAL_1:6;
hence  f1,f2 are_fiberwise_equipotent by A234, A317, A339, A340, A341, A343, A345, A374, A375, A376, A377, A384, A385; ::_thesis: verum
end;
end;
end;
end;
X1: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . GBP = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos ((p0 + n) + 1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) by SCMPDS_5:15;
X3: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos (((p0 + n) + 1) + 1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (((p0 + n) + 1) + 1)) by SCMPDS_5:15;
X4: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (DataLoc (0,1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (DataLoc (0,1)) by SCMPDS_5:15;
B387: while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_closed_on Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)),P by A317, A338, A286, Lm6, X1, X2, X3, X4;
A387: while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_closed_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),P
proof
 for k being Element of NAT holds IC (Comput ((P +* (stop (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))))),(Initialize (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)))),k)) in dom (stop (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))) by B387, SCMPDS_6:def_2;
hence  while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_closed_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),P by SCMPDS_6:def_2; ::_thesis: verum
end;
B388: while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_halting_on Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)),P by A317, A338, A286, Lm6, X1, X2, X3, X4;
A388: while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_halting_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),P
proof
 P +* (stop (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))) halts_on Initialize (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) by B388, SCMPDS_6:def_3;
hence  while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))) is_halting_on IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s),P by SCMPDS_6:def_3; ::_thesis: verum
end;
hence  QuickSort (n,p0) is_halting_on s,P by A387, I, SCPISORT:9; ::_thesis: ex f, g being FinSequence of INT st
( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
take  f ; ::_thesis: ex g being FinSequence of INT st
( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
take  g ; ::_thesis: ( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
thus  ( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 ) by A283, A284, A315, A316; ::_thesis: ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n )
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<=_len_g_holds_
g_._i_=_(IExec_((while>0_(GBP,1,((((GBP,2)_:=_(SBP,((p0_+_n)_+_1)))_';'_(SubFrom_(GBP,2,SBP,(p0_+_n))))_';'_(if>0_(GBP,2,(((((GBP,2)_:=_(SBP,(p0_+_n)))_';'_((GBP,4)_:=_(SBP,((p0_+_n)_+_1))))_';'_Pt)_';'_(((((((SBP,((p0_+_n)_+_3))_:=_(SBP,((p0_+_n)_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(GBP,4)))_';'_((SBP,((p0_+_n)_+_2))_:=_(GBP,4)))_';'_(AddTo_(SBP,((p0_+_n)_+_1),(-_1))))_';'_(AddTo_(SBP,((p0_+_n)_+_2),1)))_';'_(AddTo_(GBP,1,2)))),(Load_(AddTo_(GBP,1,(-_2))))))))),P,(Initialize_(IExec_(((((GBP_:=_0)_';'_(SBP_:=_1))_';'_((SBP,(p0_+_n))_:=_(p0_+_1)))_';'_((SBP,((p0_+_n)_+_1))_:=_(p0_+_n))),P,s)))))_._(intpos_(p0_+_i))
let i be Element of NAT ; ::_thesis: ( 1 <= i & i <= len g implies g . i = (IExec ((while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))),P,(Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))))) . (intpos (p0 + i)) )
assume that
A389: 1 <= i and
A390: i <= len g ; ::_thesis: g . i = (IExec ((while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))),P,(Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))))) . (intpos (p0 + i))
thus g . i = (IExec ((((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))) ';' (while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2)))))))))),P,s)) . (intpos (p0 + i)) by A316, A389, A390, SCPISORT:def_1
.= (IExec ((while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))),P,(Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))))) . (intpos (p0 + i)) by A388, A387, SCPISORT:7 ; ::_thesis: verum
end;
then XX: g is_FinSequence_on IExec ((while>0 (GBP,1,((((GBP,2) := (SBP,((p0 + n) + 1))) ';' (SubFrom (GBP,2,SBP,(p0 + n)))) ';' (if>0 (GBP,2,(((((GBP,2) := (SBP,(p0 + n))) ';' ((GBP,4) := (SBP,((p0 + n) + 1)))) ';' Pt) ';' (((((((SBP,((p0 + n) + 3)) := (SBP,((p0 + n) + 1))) ';' ((SBP,((p0 + n) + 1)) := (GBP,4))) ';' ((SBP,((p0 + n) + 2)) := (GBP,4))) ';' (AddTo (SBP,((p0 + n) + 1),(- 1)))) ';' (AddTo (SBP,((p0 + n) + 2),1))) ';' (AddTo (GBP,1,2)))),(Load (AddTo (GBP,1,(- 2))))))))),P,(Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)))),p0 by SCPISORT:def_1;
X1: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . GBP = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . GBP by SCMPDS_5:15;
X2: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (DataLoc (0,1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (DataLoc (0,1)) by SCMPDS_5:15;
X4: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos ((p0 + n) + 1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos ((p0 + n) + 1)) by SCMPDS_5:15;
X5: (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos (((p0 + n) + 1) + 1)) = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (((p0 + n) + 1) + 1)) by SCMPDS_5:15;
 f is_FinSequence_on Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)),p0
proof
let i be Element of NAT ; :: according to SCPISORT:def_1 ::_thesis: ( not 1 <= i or not i <= len f or f . i = (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos (p0 + i)) )
assume  ( 1 <= i & i <= len f ) ; ::_thesis: f . i = (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos (p0 + i))
then  f . i = (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s)) . (intpos (p0 + i)) by A314, SCPISORT:def_1;
hence  f . i = (Initialize (IExec (((((GBP := 0) ';' (SBP := 1)) ';' ((SBP,(p0 + n)) := (p0 + 1))) ';' ((SBP,((p0 + n) + 1)) := (p0 + n))),P,s))) . (intpos (p0 + i)) by SCMPDS_5:15; ::_thesis: verum
end;
hence  ( f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) by A317, A338, A286, A283, A315, Lm4, XX, X1, X2, X4, X5; ::_thesis: verum
end;

begin

theorem :: SCPQSORT:13
for n, p0 being Element of NAT holds card (QuickSort (n,p0)) = 57 by Lm29;

theorem :: SCPQSORT:14
for p0, n being Element of NAT st p0 >= 7 holds
QuickSort (n,p0) is parahalting
proof
let p0, n be Element of NAT ; ::_thesis: ( p0 >= 7 implies QuickSort (n,p0) is parahalting )
assume A1: p0 >= 7 ; ::_thesis: QuickSort (n,p0) is parahalting
 for s being State of SCMPDS
for P being Instruction-Sequence of SCMPDS holds QuickSort (n,p0) is_halting_on s,P
proof
let s be State of SCMPDS; ::_thesis: for P being Instruction-Sequence of SCMPDS holds QuickSort (n,p0) is_halting_on s,P
let P be Instruction-Sequence of SCMPDS; ::_thesis: QuickSort (n,p0) is_halting_on s,P
 QuickSort (n,p0) is_halting_on Initialize s,P by A1, Lm29;
hence  QuickSort (n,p0) is_halting_on s,P by SCMPDS_6:126; ::_thesis: verum
end;
hence  QuickSort (n,p0) is parahalting by SCMPDS_6:21; ::_thesis: verum
end;

theorem :: SCPQSORT:15
for P being Instruction-Sequence of SCMPDS
for s being 0 -started State of SCMPDS
for p0, n being Element of NAT st p0 >= 7 holds
ex f, g being FinSequence of INT st
( len f = n & f is_FinSequence_on s,p0 & len g = n & g is_FinSequence_on IExec ((QuickSort (n,p0)),P,s),p0 & f,g are_fiberwise_equipotent & g is_non_decreasing_on 1,n ) by Lm29;