:: SCMBSORT semantic presentation

begin

                set SA0 = Start-At (0,SCM+FSA);

theorem Th1: :: SCMBSORT:1
for I being Program of
for a, b being Int-Location st not I destroys b & a <> b holds
not Times (a,I) destroys b
proof
let I be Program of ; ::_thesis: for a, b being Int-Location st not I destroys b & a <> b holds
not Times (a,I) destroys b
let a, b be Int-Location; ::_thesis: ( not I destroys b & a <> b implies not Times (a,I) destroys b )
assume that
A1: not I destroys b and
A2: a <> b ; ::_thesis: not Times (a,I) destroys b
set Gi = Goto 2;
set Si = SubFrom (a,(intloc 0));
set SS = Stop SCM+FSA;
set if0 = if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))));
A3: not Goto 2 destroys b by SCMFSA8C:57;
 not I ";" (SubFrom (a,(intloc 0))) destroys b by A1, A2, SCMFSA7B:8, SCMFSA8C:54;
then  not if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))) destroys b by A3, SCMFSA8C:88;
then A4: not loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))) destroys b by SCMFSA8C:74;
 not Stop SCM+FSA destroys b by SCMFSA8C:56;
hence  not Times (a,I) destroys b by A4, SCMFSA8C:88; ::_thesis: verum
end;

theorem Th2: :: SCMBSORT:2
for s being State of SCM+FSA
for f being FinSeq-Location
for a, b being Int-Location holds (Exec ((b := (f,a)),s)) . b = (s . f) /. (abs (s . a))
proof
let s be State of SCM+FSA; ::_thesis: for f being FinSeq-Location
for a, b being Int-Location holds (Exec ((b := (f,a)),s)) . b = (s . f) /. (abs (s . a))
let f be FinSeq-Location ; ::_thesis: for a, b being Int-Location holds (Exec ((b := (f,a)),s)) . b = (s . f) /. (abs (s . a))
let a, b be Int-Location; ::_thesis: (Exec ((b := (f,a)),s)) . b = (s . f) /. (abs (s . a))
 ex k being Element of NAT st
( k = abs (s . a) & (Exec ((b := (f,a)),s)) . b = (s . f) /. k ) by SCMFSA_2:72;
hence  (Exec ((b := (f,a)),s)) . b = (s . f) /. (abs (s . a)) ; ::_thesis: verum
end;

theorem Th3: :: SCMBSORT:3
for s being State of SCM+FSA
for f being FinSeq-Location
for a, b being Int-Location holds (Exec (((f,a) := b),s)) . f = (s . f) +* ((abs (s . a)),(s . b))
proof
let s be State of SCM+FSA; ::_thesis: for f being FinSeq-Location
for a, b being Int-Location holds (Exec (((f,a) := b),s)) . f = (s . f) +* ((abs (s . a)),(s . b))
let f be FinSeq-Location ; ::_thesis: for a, b being Int-Location holds (Exec (((f,a) := b),s)) . f = (s . f) +* ((abs (s . a)),(s . b))
let a, b be Int-Location; ::_thesis: (Exec (((f,a) := b),s)) . f = (s . f) +* ((abs (s . a)),(s . b))
 ex k being Element of NAT st
( k = abs (s . a) & (Exec (((f,a) := b),s)) . f = (s . f) +* (k,(s . b)) ) by SCMFSA_2:73;
hence  (Exec (((f,a) := b),s)) . f = (s . f) +* ((abs (s . a)),(s . b)) ; ::_thesis: verum
end;

theorem Th4: :: SCMBSORT:4
for s being State of SCM+FSA
for f being FinSeq-Location
for m, n being Element of NAT
for a being Int-Location st m <> n + 1 holds
(Exec (((intloc m) := (f,a)),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1))
proof
let s be State of SCM+FSA; ::_thesis: for f being FinSeq-Location
for m, n being Element of NAT
for a being Int-Location st m <> n + 1 holds
(Exec (((intloc m) := (f,a)),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1))
let f be FinSeq-Location ; ::_thesis: for m, n being Element of NAT
for a being Int-Location st m <> n + 1 holds
(Exec (((intloc m) := (f,a)),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1))
let m, n be Element of NAT ; ::_thesis: for a being Int-Location st m <> n + 1 holds
(Exec (((intloc m) := (f,a)),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1))
let a be Int-Location; ::_thesis: ( m <> n + 1 implies (Exec (((intloc m) := (f,a)),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1)) )
assume  m <> n + 1 ; ::_thesis: (Exec (((intloc m) := (f,a)),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1))
then  intloc m <> intloc (n + 1) by SCMFSA_2:101;
hence (Exec (((intloc m) := (f,a)),(Initialized s))) . (intloc (n + 1)) = (Initialized s) . (intloc (n + 1)) by SCMFSA_2:72
.= s . (intloc (n + 1)) by SCMFSA_M:37 ;
::_thesis: verum
end;

theorem Th5: :: SCMBSORT:5
for s being State of SCM+FSA
for m, n being Element of NAT
for a being Int-Location st m <> n + 1 holds
(Exec (((intloc m) := a),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1))
proof
let s be State of SCM+FSA; ::_thesis: for m, n being Element of NAT
for a being Int-Location st m <> n + 1 holds
(Exec (((intloc m) := a),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1))
let m, n be Element of NAT ; ::_thesis: for a being Int-Location st m <> n + 1 holds
(Exec (((intloc m) := a),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1))
let a be Int-Location; ::_thesis: ( m <> n + 1 implies (Exec (((intloc m) := a),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1)) )
assume  m <> n + 1 ; ::_thesis: (Exec (((intloc m) := a),(Initialized s))) . (intloc (n + 1)) = s . (intloc (n + 1))
then  intloc m <> intloc (n + 1) by SCMFSA_2:101;
hence (Exec (((intloc m) := a),(Initialized s))) . (intloc (n + 1)) = (Initialized s) . (intloc (n + 1)) by SCMFSA_2:63
.= s . (intloc (n + 1)) by SCMFSA_M:37 ;
::_thesis: verum
end;

theorem Th6: :: SCMBSORT:6
for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for f being FinSeq-Location
for a being read-write Int-Location holds
( (IExec ((Stop SCM+FSA),p,s)) . a = s . a & (IExec ((Stop SCM+FSA),p,s)) . f = s . f )
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for f being FinSeq-Location
for a being read-write Int-Location holds
( (IExec ((Stop SCM+FSA),p,s)) . a = s . a & (IExec ((Stop SCM+FSA),p,s)) . f = s . f )
let s be State of SCM+FSA; ::_thesis: for f being FinSeq-Location
for a being read-write Int-Location holds
( (IExec ((Stop SCM+FSA),p,s)) . a = s . a & (IExec ((Stop SCM+FSA),p,s)) . f = s . f )
let f be FinSeq-Location ; ::_thesis: for a being read-write Int-Location holds
( (IExec ((Stop SCM+FSA),p,s)) . a = s . a & (IExec ((Stop SCM+FSA),p,s)) . f = s . f )
let a be read-write Int-Location; ::_thesis: ( (IExec ((Stop SCM+FSA),p,s)) . a = s . a & (IExec ((Stop SCM+FSA),p,s)) . f = s . f )
A1: Initialized s = s +* (((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA)))
.= Initialize (s +* ((intloc 0) .--> 1)) by FUNCT_4:14 ;
A2: IExec ((Stop SCM+FSA),p,s) = Initialize (s +* ((intloc 0) .--> 1)) by A1, SCMFSA8C:14
.= Initialized s by A1 ;
hence  (IExec ((Stop SCM+FSA),p,s)) . a = s . a by SCMFSA_M:37; ::_thesis: (IExec ((Stop SCM+FSA),p,s)) . f = s . f
thus  (IExec ((Stop SCM+FSA),p,s)) . f = s . f by A2, SCMFSA_M:37; ::_thesis: verum
end;

theorem Th7: :: SCMBSORT:7
for p being preProgram of SCM+FSA
for ic being Instruction of SCM+FSA
for a, b being Int-Location st ic in rng p & ( ic = a := b or ic = AddTo (a,b) or ic = SubFrom (a,b) or ic = MultBy (a,b) or ic = Divide (a,b) ) holds
( a in UsedIntLoc p & b in UsedIntLoc p )
proof
let p be preProgram of SCM+FSA; ::_thesis: for ic being Instruction of SCM+FSA
for a, b being Int-Location st ic in rng p & ( ic = a := b or ic = AddTo (a,b) or ic = SubFrom (a,b) or ic = MultBy (a,b) or ic = Divide (a,b) ) holds
( a in UsedIntLoc p & b in UsedIntLoc p )
let ic be Instruction of SCM+FSA; ::_thesis: for a, b being Int-Location st ic in rng p & ( ic = a := b or ic = AddTo (a,b) or ic = SubFrom (a,b) or ic = MultBy (a,b) or ic = Divide (a,b) ) holds
( a in UsedIntLoc p & b in UsedIntLoc p )
let a, b be Int-Location; ::_thesis: ( ic in rng p & ( ic = a := b or ic = AddTo (a,b) or ic = SubFrom (a,b) or ic = MultBy (a,b) or ic = Divide (a,b) ) implies ( a in UsedIntLoc p & b in UsedIntLoc p ) )
assume that
A1: ic in rng p and
A2: ( ic = a := b or ic = AddTo (a,b) or ic = SubFrom (a,b) or ic = MultBy (a,b) or ic = Divide (a,b) ) ; ::_thesis: ( a in UsedIntLoc p & b in UsedIntLoc p )
A3: UsedIntLoc ic = {a,b} by A2, SF_MASTR:14;
 UsedIntLoc ic c= UsedIntLoc p by A1, SF_MASTR:19;
hence  ( a in UsedIntLoc p & b in UsedIntLoc p ) by A3, ZFMISC_1:32; ::_thesis: verum
end;

theorem Th8: :: SCMBSORT:8
for p being preProgram of SCM+FSA
for ic being Instruction of SCM+FSA
for a being Int-Location
for la being Element of NAT st ic in rng p & ( ic = a =0_goto la or ic = a >0_goto la ) holds
a in UsedIntLoc p
proof
let p be preProgram of SCM+FSA; ::_thesis: for ic being Instruction of SCM+FSA
for a being Int-Location
for la being Element of NAT st ic in rng p & ( ic = a =0_goto la or ic = a >0_goto la ) holds
a in UsedIntLoc p
let ic be Instruction of SCM+FSA; ::_thesis: for a being Int-Location
for la being Element of NAT st ic in rng p & ( ic = a =0_goto la or ic = a >0_goto la ) holds
a in UsedIntLoc p
let a be Int-Location; ::_thesis: for la being Element of NAT st ic in rng p & ( ic = a =0_goto la or ic = a >0_goto la ) holds
a in UsedIntLoc p
let la be Element of NAT ; ::_thesis: ( ic in rng p & ( ic = a =0_goto la or ic = a >0_goto la ) implies a in UsedIntLoc p )
assume that
A1: ic in rng p and
A2: ( ic = a =0_goto la or ic = a >0_goto la ) ; ::_thesis: a in UsedIntLoc p
A3: UsedIntLoc ic = {a} by A2, SF_MASTR:16;
 UsedIntLoc ic c= UsedIntLoc p by A1, SF_MASTR:19;
hence  a in UsedIntLoc p by A3, ZFMISC_1:31; ::_thesis: verum
end;

theorem Th9: :: SCMBSORT:9
for p being preProgram of SCM+FSA
for ic being Instruction of SCM+FSA
for fa being FinSeq-Location
for b, a being Int-Location st ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) holds
( a in UsedIntLoc p & b in UsedIntLoc p )
proof
let p be preProgram of SCM+FSA; ::_thesis: for ic being Instruction of SCM+FSA
for fa being FinSeq-Location
for b, a being Int-Location st ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) holds
( a in UsedIntLoc p & b in UsedIntLoc p )
let ic be Instruction of SCM+FSA; ::_thesis: for fa being FinSeq-Location
for b, a being Int-Location st ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) holds
( a in UsedIntLoc p & b in UsedIntLoc p )
let fa be FinSeq-Location ; ::_thesis: for b, a being Int-Location st ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) holds
( a in UsedIntLoc p & b in UsedIntLoc p )
let b, a be Int-Location; ::_thesis: ( ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) implies ( a in UsedIntLoc p & b in UsedIntLoc p ) )
assume that
A1: ic in rng p and
A2: ( ic = b := (fa,a) or ic = (fa,a) := b ) ; ::_thesis: ( a in UsedIntLoc p & b in UsedIntLoc p )
A3: UsedIntLoc ic = {a,b} by A2, SF_MASTR:17;
 UsedIntLoc ic c= UsedIntLoc p by A1, SF_MASTR:19;
hence  ( a in UsedIntLoc p & b in UsedIntLoc p ) by A3, ZFMISC_1:32; ::_thesis: verum
end;

theorem Th10: :: SCMBSORT:10
for p being preProgram of SCM+FSA
for ic being Instruction of SCM+FSA
for fa being FinSeq-Location
for b, a being Int-Location st ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) holds
fa in UsedInt*Loc p
proof
let p be preProgram of SCM+FSA; ::_thesis: for ic being Instruction of SCM+FSA
for fa being FinSeq-Location
for b, a being Int-Location st ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) holds
fa in UsedInt*Loc p
let ic be Instruction of SCM+FSA; ::_thesis: for fa being FinSeq-Location
for b, a being Int-Location st ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) holds
fa in UsedInt*Loc p
let fa be FinSeq-Location ; ::_thesis: for b, a being Int-Location st ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) holds
fa in UsedInt*Loc p
let b, a be Int-Location; ::_thesis: ( ic in rng p & ( ic = b := (fa,a) or ic = (fa,a) := b ) implies fa in UsedInt*Loc p )
assume that
A1: ic in rng p and
A2: ( ic = b := (fa,a) or ic = (fa,a) := b ) ; ::_thesis: fa in UsedInt*Loc p
A3: UsedInt*Loc ic = {fa} by A2, SF_MASTR:33;
 UsedInt*Loc ic c= UsedInt*Loc p by A1, SF_MASTR:35;
hence  fa in UsedInt*Loc p by A3, ZFMISC_1:31; ::_thesis: verum
end;

theorem Th11: :: SCMBSORT:11
for p being preProgram of SCM+FSA
for ic being Instruction of SCM+FSA
for fa being FinSeq-Location
for a being Int-Location st ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) holds
a in UsedIntLoc p
proof
let p be preProgram of SCM+FSA; ::_thesis: for ic being Instruction of SCM+FSA
for fa being FinSeq-Location
for a being Int-Location st ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) holds
a in UsedIntLoc p
let ic be Instruction of SCM+FSA; ::_thesis: for fa being FinSeq-Location
for a being Int-Location st ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) holds
a in UsedIntLoc p
let fa be FinSeq-Location ; ::_thesis: for a being Int-Location st ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) holds
a in UsedIntLoc p
let a be Int-Location; ::_thesis: ( ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) implies a in UsedIntLoc p )
assume that
A1: ic in rng p and
A2: ( ic = a :=len fa or ic = fa :=<0,...,0> a ) ; ::_thesis: a in UsedIntLoc p
A3: UsedIntLoc ic = {a} by A2, SF_MASTR:18;
 UsedIntLoc ic c= UsedIntLoc p by A1, SF_MASTR:19;
hence  a in UsedIntLoc p by A3, ZFMISC_1:31; ::_thesis: verum
end;

theorem Th12: :: SCMBSORT:12
for p being preProgram of SCM+FSA
for ic being Instruction of SCM+FSA
for fa being FinSeq-Location
for a being Int-Location st ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) holds
fa in UsedInt*Loc p
proof
let p be preProgram of SCM+FSA; ::_thesis: for ic being Instruction of SCM+FSA
for fa being FinSeq-Location
for a being Int-Location st ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) holds
fa in UsedInt*Loc p
let ic be Instruction of SCM+FSA; ::_thesis: for fa being FinSeq-Location
for a being Int-Location st ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) holds
fa in UsedInt*Loc p
let fa be FinSeq-Location ; ::_thesis: for a being Int-Location st ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) holds
fa in UsedInt*Loc p
let a be Int-Location; ::_thesis: ( ic in rng p & ( ic = a :=len fa or ic = fa :=<0,...,0> a ) implies fa in UsedInt*Loc p )
assume that
A1: ic in rng p and
A2: ( ic = a :=len fa or ic = fa :=<0,...,0> a ) ; ::_thesis: fa in UsedInt*Loc p
A3: UsedInt*Loc ic = {fa} by A2, SF_MASTR:34;
 UsedInt*Loc ic c= UsedInt*Loc p by A1, SF_MASTR:35;
hence  fa in UsedInt*Loc p by A3, ZFMISC_1:31; ::_thesis: verum
end;

theorem Th13: :: SCMBSORT:13
for t being FinPartState of SCM+FSA
for p being Program of
for x being set st dom t c= Int-Locations \/ FinSeq-Locations & x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) & x is not Int-Location holds
x is FinSeq-Location
proof
let t be FinPartState of SCM+FSA; ::_thesis: for p being Program of
for x being set st dom t c= Int-Locations \/ FinSeq-Locations & x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) & x is not Int-Location holds
x is FinSeq-Location
let p be Program of ; ::_thesis: for x being set st dom t c= Int-Locations \/ FinSeq-Locations & x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) & x is not Int-Location holds
x is FinSeq-Location
let x be set ; ::_thesis: ( dom t c= Int-Locations \/ FinSeq-Locations & x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) & x is not Int-Location implies x is FinSeq-Location )
set D1 = UsedInt*Loc p;
set D2 = UsedIntLoc p;
assume that
A1: dom t c= Int-Locations \/ FinSeq-Locations and
A2: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: ( x is Int-Location or x is FinSeq-Location )
 ( x in (dom t) \/ (UsedInt*Loc p) or x in UsedIntLoc p ) by A2, XBOOLE_0:def_3;
then A3: ( x in dom t or x in UsedInt*Loc p or x in UsedIntLoc p ) by XBOOLE_0:def_3;
percases ( x in Int-Locations or x in FinSeq-Locations or x in UsedInt*Loc p or x in UsedIntLoc p ) by A1, A3, XBOOLE_0:def_3;
suppose x in Int-Locations ; ::_thesis: ( x is Int-Location or x is FinSeq-Location )
hence  ( x is Int-Location or x is FinSeq-Location ) by AMI_2:def_16; ::_thesis: verum
end;
suppose x in FinSeq-Locations ; ::_thesis: ( x is Int-Location or x is FinSeq-Location )
hence  ( x is Int-Location or x is FinSeq-Location ) by SCMFSA_2:def_5; ::_thesis: verum
end;
suppose x in UsedInt*Loc p ; ::_thesis: ( x is Int-Location or x is FinSeq-Location )
then  x in FinSeq-Locations ;
hence  ( x is Int-Location or x is FinSeq-Location ) by SCMFSA_2:def_5; ::_thesis: verum
end;
suppose x in UsedIntLoc p ; ::_thesis: ( x is Int-Location or x is FinSeq-Location )
then  x in Int-Locations ;
hence  ( x is Int-Location or x is FinSeq-Location ) by AMI_2:def_16; ::_thesis: verum
end;
end;
end;

theorem Th14: :: SCMBSORT:14
for p1, p2 being Instruction-Sequence of SCM+FSA
for i, k being Element of NAT
for t being FinPartState of SCM+FSA
for p being Program of
for s1, s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 & dom t c= Int-Locations \/ FinSeq-Locations & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) holds
( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) )
proof
let p1, p2 be Instruction-Sequence of SCM+FSA; ::_thesis: for i, k being Element of NAT
for t being FinPartState of SCM+FSA
for p being Program of
for s1, s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 & dom t c= Int-Locations \/ FinSeq-Locations & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) holds
( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) )
let i, k be Element of NAT ; ::_thesis: for t being FinPartState of SCM+FSA
for p being Program of
for s1, s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 & dom t c= Int-Locations \/ FinSeq-Locations & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) holds
( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) )
let t be FinPartState of SCM+FSA; ::_thesis: for p being Program of
for s1, s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 & dom t c= Int-Locations \/ FinSeq-Locations & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) holds
( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) )
let p be Program of ; ::_thesis: for s1, s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 & dom t c= Int-Locations \/ FinSeq-Locations & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) holds
( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) )
let s1, s2 be State of SCM+FSA; ::_thesis: ( k <= i & p c= p1 & p c= p2 & dom t c= Int-Locations \/ FinSeq-Locations & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) implies ( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) ) )
set Dloc = ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p);
assume that
A1: k <= i and
A2: p c= p1 and
A3: p c= p2 and
A4: dom t c= Int-Locations \/ FinSeq-Locations and
A5: for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) and
A6: (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) and
A7: (Comput (p1,s1,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) ; ::_thesis: ( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) )
consider m being Nat such that
A8: i = k + m by A1, NAT_1:10;
reconsider m = m as Element of NAT by ORDINAL1:def_12;
A9: i = k + m by A8;
A10: UsedIntLoc p c= ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) by XBOOLE_1:7;
 ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) = ((dom t) \/ (UsedIntLoc p)) \/ (UsedInt*Loc p) by XBOOLE_1:4;
then A11: UsedInt*Loc p c= ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) by XBOOLE_1:7;
defpred S1[ Nat] means ( (Comput (p1,s1,(k + $1))) . (IC ) = (Comput (p2,s2,(k + $1))) . (IC ) & (Comput (p1,s1,(k + $1))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,(k + $1))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) );
A12: S1[ 0 ] by A6, A7;
A13: now__::_thesis:_for_m_being_Element_of_NAT_st_S1[m]_holds_
S1[m_+_1]
let m be Element of NAT ; ::_thesis: ( S1[m] implies S1[b1 + 1] )
assume A14: S1[m] ; ::_thesis: S1[b1 + 1]
set sk1 = Comput (p1,s1,(k + m));
set sk11 = Comput (p1,s1,(k + (m + 1)));
set i1 = CurInstr (p1,(Comput (p1,s1,(k + m))));
set sk2 = Comput (p2,s2,(k + m));
set sk12 = Comput (p2,s2,(k + (m + 1)));
set i2 = CurInstr (p2,(Comput (p2,s2,(k + m))));
A15: IC (Comput (p1,s1,(k + m))) in dom p by A5;
A16: p2 /. (IC (Comput (p2,s2,(k + m)))) = p2 . (IC (Comput (p2,s2,(k + m)))) by PBOOLE:143;
A17: p1 /. (IC (Comput (p1,s1,(k + m)))) = p1 . (IC (Comput (p1,s1,(k + m)))) by PBOOLE:143;
 CurInstr (p1,(Comput (p1,s1,(k + m)))) = p . (IC (Comput (p1,s1,(k + m)))) by A2, A15, A17, GRFUNC_1:2;
then A18: CurInstr (p1,(Comput (p1,s1,(k + m)))) in rng p by A15, FUNCT_1:def_3;
A19: CurInstr (p2,(Comput (p2,s2,(k + m)))) = (p2 | (dom p)) . (IC (Comput (p2,s2,(k + m)))) by A16, A5, FUNCT_1:49
.= (p1 | (dom p)) . (IC (Comput (p1,s1,(k + m)))) by A2, A3, A14, GRFUNC_1:33
.= CurInstr (p1,(Comput (p1,s1,(k + m)))) by A17, A5, FUNCT_1:49 ;
A20: Comput (p1,s1,(k + (m + 1))) = Comput (p1,s1,((k + m) + 1))
.= Following (p1,(Comput (p1,s1,(k + m)))) by EXTPRO_1:3
.= Exec ((CurInstr (p1,(Comput (p1,s1,(k + m))))),(Comput (p1,s1,(k + m)))) ;
A21: Comput (p2,s2,(k + (m + 1))) = Comput (p2,s2,((k + m) + 1))
.= Following (p2,(Comput (p2,s2,(k + m)))) by EXTPRO_1:3
.= Exec ((CurInstr (p2,(Comput (p2,s2,(k + m))))),(Comput (p2,s2,(k + m)))) ;
A22: dom (Comput (p1,s1,(k + (m + 1)))) = the carrier of SCM+FSA by PARTFUN1:def_2
.= dom (Comput (p2,s2,(k + (m + 1)))) by PARTFUN1:def_2 ;
percases ( InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 0 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 1 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 2 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 3 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 4 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 5 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 6 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 7 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 8 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 9 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 10 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 11 or InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 12 ) by NAT_1:36, SCMFSA_2:16;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 0 ; ::_thesis: S1[b1 + 1]
then A23: CurInstr (p1,(Comput (p1,s1,(k + m)))) = halt SCM+FSA by SCMFSA_2:95;
then  Comput (p1,s1,(k + (m + 1))) = Comput (p1,s1,(k + m)) by A20, EXTPRO_1:def_3;
hence  S1[m + 1] by A14, A19, A21, A23, EXTPRO_1:def_3; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 1 ; ::_thesis: S1[b1 + 1]
then consider da, db being Int-Location such that
A24: CurInstr (p1,(Comput (p1,s1,(k + m)))) = da := db by SCMFSA_2:30;
A25: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p1,s1,(k + m)))) by A20, A24, SCMFSA_2:63
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A14, A19, A21, A24, SCMFSA_2:63 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A26: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A26, Th13;
supposeA27: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x = da or x <> da ) ;
supposeA28: x = da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A29: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . db by A19, A21, A24, SCMFSA_2:63;
A30: db in UsedIntLoc p by A18, A24, Th7;
then (Comput (p1,s1,(k + m))) . db = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . db by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . db by A10, A30, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A24, A28, A29, SCMFSA_2:63; ::_thesis: verum
end;
supposeA31: x <> da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A32: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A24, A27, SCMFSA_2:63;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A26, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A26, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A24, A27, A31, A32, SCMFSA_2:63; ::_thesis: verum
end;
end;
end;
supposeA33: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A34: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A24, SCMFSA_2:63;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A26, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A26, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A24, A33, A34, SCMFSA_2:63; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A25, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 2 ; ::_thesis: S1[b1 + 1]
then consider da, db being Int-Location such that
A35: CurInstr (p1,(Comput (p1,s1,(k + m)))) = AddTo (da,db) by SCMFSA_2:31;
A36: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p1,s1,(k + m)))) by A20, A35, SCMFSA_2:64
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A14, A19, A21, A35, SCMFSA_2:64 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A37: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A37, Th13;
supposeA38: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x = da or x <> da ) ;
supposeA39: x = da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A40: (Comput (p2,s2,(k + (m + 1)))) . x = ((Comput (p2,s2,(k + m))) . da) + ((Comput (p2,s2,(k + m))) . db) by A19, A21, A35, SCMFSA_2:64;
A41: da in UsedIntLoc p by A18, A35, Th7;
then A42: (Comput (p1,s1,(k + m))) . da = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . da by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . da by A10, A41, FUNCT_1:49 ;
A43: db in UsedIntLoc p by A18, A35, Th7;
then (Comput (p1,s1,(k + m))) . db = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . db by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . db by A10, A43, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A35, A39, A40, A42, SCMFSA_2:64; ::_thesis: verum
end;
supposeA44: x <> da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A45: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A35, A38, SCMFSA_2:64;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A37, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A37, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A35, A38, A44, A45, SCMFSA_2:64; ::_thesis: verum
end;
end;
end;
supposeA46: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A47: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A35, SCMFSA_2:64;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A37, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A37, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A35, A46, A47, SCMFSA_2:64; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A36, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 3 ; ::_thesis: S1[b1 + 1]
then consider da, db being Int-Location such that
A48: CurInstr (p1,(Comput (p1,s1,(k + m)))) = SubFrom (da,db) by SCMFSA_2:32;
A49: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p1,s1,(k + m)))) by A20, A48, SCMFSA_2:65
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A14, A19, A21, A48, SCMFSA_2:65 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A50: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A50, Th13;
supposeA51: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x = da or x <> da ) ;
supposeA52: x = da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A53: (Comput (p2,s2,(k + (m + 1)))) . x = ((Comput (p2,s2,(k + m))) . da) - ((Comput (p2,s2,(k + m))) . db) by A19, A21, A48, SCMFSA_2:65;
A54: da in UsedIntLoc p by A18, A48, Th7;
then A55: (Comput (p1,s1,(k + m))) . da = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . da by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . da by A10, A54, FUNCT_1:49 ;
A56: db in UsedIntLoc p by A18, A48, Th7;
then (Comput (p1,s1,(k + m))) . db = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . db by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . db by A10, A56, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A48, A52, A53, A55, SCMFSA_2:65; ::_thesis: verum
end;
supposeA57: x <> da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A58: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A48, A51, SCMFSA_2:65;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A50, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A50, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A48, A51, A57, A58, SCMFSA_2:65; ::_thesis: verum
end;
end;
end;
supposeA59: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A60: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A48, SCMFSA_2:65;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A50, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A50, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A48, A59, A60, SCMFSA_2:65; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A49, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 4 ; ::_thesis: S1[b1 + 1]
then consider da, db being Int-Location such that
A61: CurInstr (p1,(Comput (p1,s1,(k + m)))) = MultBy (da,db) by SCMFSA_2:33;
A62: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p1,s1,(k + m)))) by A20, A61, SCMFSA_2:66
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A14, A19, A21, A61, SCMFSA_2:66 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A63: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A63, Th13;
supposeA64: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x = da or x <> da ) ;
supposeA65: x = da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A66: (Comput (p2,s2,(k + (m + 1)))) . x = ((Comput (p2,s2,(k + m))) . da) * ((Comput (p2,s2,(k + m))) . db) by A19, A21, A61, SCMFSA_2:66;
A67: da in UsedIntLoc p by A18, A61, Th7;
then A68: (Comput (p1,s1,(k + m))) . da = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . da by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . da by A10, A67, FUNCT_1:49 ;
A69: db in UsedIntLoc p by A18, A61, Th7;
then (Comput (p1,s1,(k + m))) . db = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . db by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . db by A10, A69, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A61, A65, A66, A68, SCMFSA_2:66; ::_thesis: verum
end;
supposeA70: x <> da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A71: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A61, A64, SCMFSA_2:66;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A63, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A63, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A61, A64, A70, A71, SCMFSA_2:66; ::_thesis: verum
end;
end;
end;
supposeA72: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A73: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A61, SCMFSA_2:66;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A63, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A63, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A61, A72, A73, SCMFSA_2:66; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A62, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 5 ; ::_thesis: S1[b1 + 1]
then consider da, db being Int-Location such that
A74: CurInstr (p1,(Comput (p1,s1,(k + m)))) = Divide (da,db) by SCMFSA_2:34;
A75: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p1,s1,(k + m)))) by A20, A74, SCMFSA_2:67
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A14, A19, A21, A74, SCMFSA_2:67 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A76: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A76, Th13;
supposeA77: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
A78: da in UsedIntLoc p by A18, A74, Th7;
then A79: (Comput (p1,s1,(k + m))) . da = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . da by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . da by A10, A78, FUNCT_1:49 ;
A80: db in UsedIntLoc p by A18, A74, Th7;
then A81: (Comput (p1,s1,(k + m))) . db = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . db by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . db by A10, A80, FUNCT_1:49 ;
A82: (Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A76, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A76, FUNCT_1:49 ;
now__::_thesis:_(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
percases ( da <> db or da = db ) ;
supposeA83: da <> db ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x
percases ( x = da or x = db or ( x <> da & x <> db ) ) ;
supposeA84: x = da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x
then  (Comput (p1,s1,(k + (m + 1)))) . x = ((Comput (p1,s1,(k + m))) . da) div ((Comput (p1,s1,(k + m))) . db) by A20, A74, A83, SCMFSA_2:67;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A74, A79, A81, A83, A84, SCMFSA_2:67; ::_thesis: verum
end;
supposeA85: x = db ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x
then  (Comput (p1,s1,(k + (m + 1)))) . x = ((Comput (p1,s1,(k + m))) . da) mod ((Comput (p1,s1,(k + m))) . db) by A20, A74, SCMFSA_2:67;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A74, A79, A81, A85, SCMFSA_2:67; ::_thesis: verum
end;
supposeA86: ( x <> da & x <> db ) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x
then  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p1,s1,(k + m))) . x by A20, A74, A77, SCMFSA_2:67;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A74, A77, A82, A86, SCMFSA_2:67; ::_thesis: verum
end;
end;
end;
supposeA87: da = db ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x
now__::_thesis:_(_(_x_=_da_&_(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x_)_or_(_x_<>_da_&_(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x_)_)
percases ( x = da or x <> da ) ;
caseA88: x = da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x
then  (Comput (p1,s1,(k + (m + 1)))) . x = ((Comput (p1,s1,(k + m))) . da) mod ((Comput (p1,s1,(k + m))) . da) by A20, A74, A87, SCMFSA_2:68;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A74, A79, A87, A88, SCMFSA_2:68; ::_thesis: verum
end;
caseA89: x <> da ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x
then  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p1,s1,(k + m))) . x by A20, A74, A77, A87, SCMFSA_2:68;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A74, A77, A82, A87, A89, SCMFSA_2:68; ::_thesis: verum
end;
end;
end;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x ; ::_thesis: verum
end;
end;
end;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x ; ::_thesis: verum
end;
supposeA90: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A91: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A74, SCMFSA_2:67;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A76, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A76, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A74, A90, A91, SCMFSA_2:67; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A75, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 6 ; ::_thesis: S1[b1 + 1]
then consider lb being Element of NAT  such that
A92: CurInstr (p1,(Comput (p1,s1,(k + m)))) = goto lb by SCMFSA_2:35;
A93: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = lb by A20, A92, SCMFSA_2:69
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A19, A21, A92, SCMFSA_2:69 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A94: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A95: (Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A94, FUNCT_1:49 ;
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A94, Th13;
supposeA96: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p1,s1,(k + m))) . x by A20, A92, SCMFSA_2:69;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A92, A95, A96, SCMFSA_2:69; ::_thesis: verum
end;
supposeA97: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p1,s1,(k + m))) . x by A20, A92, SCMFSA_2:69;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A92, A95, A97, SCMFSA_2:69; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A93, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 7 ; ::_thesis: S1[b1 + 1]
then consider lb being Element of NAT , da being Int-Location such that
A98: CurInstr (p1,(Comput (p1,s1,(k + m)))) = da =0_goto lb by SCMFSA_2:36;
A99: da in UsedIntLoc p by A18, A98, Th8;
then A100: (Comput (p1,s1,(k + m))) . da = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . da by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . da by A10, A99, FUNCT_1:49 ;
A101: now__::_thesis:_(Comput_(p1,s1,(k_+_(m_+_1))))_._(IC_)_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._(IC_)
percases ( (Comput (p1,s1,(k + m))) . da = 0 or (Comput (p1,s1,(k + m))) . da <> 0 ) ;
supposeA102: (Comput (p1,s1,(k + m))) . da = 0 ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = (Comput (p2,s2,(k + (m + 1)))) . (IC )
hence (Comput (p1,s1,(k + (m + 1)))) . (IC ) = lb by A20, A98, SCMFSA_2:70
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A19, A21, A98, A100, A102, SCMFSA_2:70 ;
::_thesis: verum
end;
supposeA103: (Comput (p1,s1,(k + m))) . da <> 0 ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = (Comput (p2,s2,(k + (m + 1)))) . (IC )
hence (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p2,s2,(k + m)))) by A14, A20, A98, SCMFSA_2:70
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A19, A21, A98, A100, A103, SCMFSA_2:70 ;
::_thesis: verum
end;
end;
end;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A104: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A105: (Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A104, FUNCT_1:49 ;
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A104, Th13;
supposeA106: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p1,s1,(k + m))) . x by A20, A98, SCMFSA_2:70;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A98, A105, A106, SCMFSA_2:70; ::_thesis: verum
end;
supposeA107: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p1,s1,(k + m))) . x by A20, A98, SCMFSA_2:70;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A98, A105, A107, SCMFSA_2:70; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A101, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 8 ; ::_thesis: S1[b1 + 1]
then consider lb being Element of NAT , da being Int-Location such that
A108: CurInstr (p1,(Comput (p1,s1,(k + m)))) = da >0_goto lb by SCMFSA_2:37;
A109: da in UsedIntLoc p by A18, A108, Th8;
then A110: (Comput (p1,s1,(k + m))) . da = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . da by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . da by A10, A109, FUNCT_1:49 ;
A111: now__::_thesis:_(Comput_(p1,s1,(k_+_(m_+_1))))_._(IC_)_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._(IC_)
percases ( (Comput (p1,s1,(k + m))) . da > 0 or (Comput (p1,s1,(k + m))) . da <= 0 ) ;
supposeA112: (Comput (p1,s1,(k + m))) . da > 0 ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = (Comput (p2,s2,(k + (m + 1)))) . (IC )
hence (Comput (p1,s1,(k + (m + 1)))) . (IC ) = lb by A20, A108, SCMFSA_2:71
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A19, A21, A108, A110, A112, SCMFSA_2:71 ;
::_thesis: verum
end;
supposeA113: (Comput (p1,s1,(k + m))) . da <= 0 ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = (Comput (p2,s2,(k + (m + 1)))) . (IC )
hence (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p2,s2,(k + m)))) by A14, A20, A108, SCMFSA_2:71
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A19, A21, A108, A110, A113, SCMFSA_2:71 ;
::_thesis: verum
end;
end;
end;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A114: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A115: (Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A114, FUNCT_1:49 ;
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A114, Th13;
supposeA116: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p1,s1,(k + m))) . x by A20, A108, SCMFSA_2:71;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A108, A115, A116, SCMFSA_2:71; ::_thesis: verum
end;
supposeA117: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p1,s1,(k + m))) . x by A20, A108, SCMFSA_2:71;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A21, A108, A115, A117, SCMFSA_2:71; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A111, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 9 ; ::_thesis: S1[b1 + 1]
then consider a, b being Int-Location, fa being FinSeq-Location  such that
A118: CurInstr (p1,(Comput (p1,s1,(k + m)))) = b := (fa,a) by SCMFSA_2:38;
A119: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p2,s2,(k + m)))) by A14, A20, A118, SCMFSA_2:72
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A19, A21, A118, SCMFSA_2:72 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A120: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A120, Th13;
supposeA121: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x = b or x <> b ) ;
supposeA122: x = b ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
A123: ex k1 being Element of NAT st
( k1 = abs ((Comput (p1,s1,(k + m))) . a) & (Exec ((b := (fa,a)),(Comput (p1,s1,(k + m))))) . b = ((Comput (p1,s1,(k + m))) . fa) /. k1 ) by SCMFSA_2:72;
A124: ex k2 being Element of NAT st
( k2 = abs ((Comput (p2,s2,(k + m))) . a) & (Exec ((b := (fa,a)),(Comput (p2,s2,(k + m))))) . b = ((Comput (p2,s2,(k + m))) . fa) /. k2 ) by SCMFSA_2:72;
A125: a in UsedIntLoc p by A18, A118, Th9;
then A126: (Comput (p1,s1,(k + m))) . a = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . a by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . a by A10, A125, FUNCT_1:49 ;
A127: fa in UsedInt*Loc p by A18, A118, Th10;
then (Comput (p1,s1,(k + m))) . fa = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . fa by A11, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . fa by A11, A127, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A20, A21, A118, A122, A123, A124, A126; ::_thesis: verum
end;
supposeA128: x <> b ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A129: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A118, A121, SCMFSA_2:72;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A120, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A120, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A118, A121, A128, A129, SCMFSA_2:72; ::_thesis: verum
end;
end;
end;
supposeA130: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A131: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A118, SCMFSA_2:72;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A120, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A120, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A118, A130, A131, SCMFSA_2:72; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A119, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 10 ; ::_thesis: S1[b1 + 1]
then consider a, b being Int-Location, fa being FinSeq-Location  such that
A132: CurInstr (p1,(Comput (p1,s1,(k + m)))) = (fa,a) := b by SCMFSA_2:39;
A133: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p2,s2,(k + m)))) by A14, A20, A132, SCMFSA_2:73
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A19, A21, A132, SCMFSA_2:73 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A134: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x is FinSeq-Location or x is Int-Location ) by A4, A134, Th13;
supposeA135: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x = fa or x <> fa ) ;
supposeA136: x = fa ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
A137: ex k1 being Element of NAT st
( k1 = abs ((Comput (p1,s1,(k + m))) . a) & (Exec (((fa,a) := b),(Comput (p1,s1,(k + m))))) . fa = ((Comput (p1,s1,(k + m))) . fa) +* (k1,((Comput (p1,s1,(k + m))) . b)) ) by SCMFSA_2:73;
A138: ex k2 being Element of NAT st
( k2 = abs ((Comput (p2,s2,(k + m))) . a) & (Exec (((fa,a) := b),(Comput (p2,s2,(k + m))))) . fa = ((Comput (p2,s2,(k + m))) . fa) +* (k2,((Comput (p2,s2,(k + m))) . b)) ) by SCMFSA_2:73;
A139: a in UsedIntLoc p by A18, A132, Th9;
then A140: (Comput (p1,s1,(k + m))) . a = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . a by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . a by A10, A139, FUNCT_1:49 ;
A141: b in UsedIntLoc p by A18, A132, Th9;
then A142: (Comput (p1,s1,(k + m))) . b = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . b by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . b by A10, A141, FUNCT_1:49 ;
A143: fa in UsedInt*Loc p by A18, A132, Th10;
then (Comput (p1,s1,(k + m))) . fa = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . fa by A11, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . fa by A11, A143, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A20, A21, A132, A136, A137, A138, A140, A142; ::_thesis: verum
end;
supposeA144: x <> fa ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A145: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A132, A135, SCMFSA_2:73;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A134, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A134, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A132, A135, A144, A145, SCMFSA_2:73; ::_thesis: verum
end;
end;
end;
supposeA146: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A147: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A132, SCMFSA_2:73;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A134, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A134, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A132, A146, A147, SCMFSA_2:73; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A133, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 11 ; ::_thesis: S1[b1 + 1]
then consider a being Int-Location, fa being FinSeq-Location  such that
A148: CurInstr (p1,(Comput (p1,s1,(k + m)))) = a :=len fa by SCMFSA_2:40;
A149: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p2,s2,(k + m)))) by A14, A20, A148, SCMFSA_2:74
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A19, A21, A148, SCMFSA_2:74 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A150: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x is Int-Location or x is FinSeq-Location ) by A4, A150, Th13;
supposeA151: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x = a or x <> a ) ;
supposeA152: x = a ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A153: (Comput (p2,s2,(k + (m + 1)))) . x = len ((Comput (p2,s2,(k + m))) . fa) by A19, A21, A148, SCMFSA_2:74;
A154: fa in UsedInt*Loc p by A18, A148, Th12;
then (Comput (p1,s1,(k + m))) . fa = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . fa by A11, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . fa by A11, A154, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A148, A152, A153, SCMFSA_2:74; ::_thesis: verum
end;
supposeA155: x <> a ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A156: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A148, A151, SCMFSA_2:74;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A150, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A150, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A148, A151, A155, A156, SCMFSA_2:74; ::_thesis: verum
end;
end;
end;
supposeA157: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A158: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A148, SCMFSA_2:74;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A150, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A150, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A148, A157, A158, SCMFSA_2:74; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A149, FUNCT_1:96; ::_thesis: verum
end;
suppose InsCode (CurInstr (p1,(Comput (p1,s1,(k + m))))) = 12 ; ::_thesis: S1[b1 + 1]
then consider a being Int-Location, fa being FinSeq-Location  such that
A159: CurInstr (p1,(Comput (p1,s1,(k + m)))) = fa :=<0,...,0> a by SCMFSA_2:41;
A160: (Comput (p1,s1,(k + (m + 1)))) . (IC ) = succ (IC (Comput (p2,s2,(k + m)))) by A14, A20, A159, SCMFSA_2:75
.= (Comput (p2,s2,(k + (m + 1)))) . (IC ) by A19, A21, A159, SCMFSA_2:75 ;
now__::_thesis:_for_x_being_set_st_x_in_((dom_t)_\/_(UsedInt*Loc_p))_\/_(UsedIntLoc_p)_holds_
(Comput_(p1,s1,(k_+_(m_+_1))))_._x_=_(Comput_(p2,s2,(k_+_(m_+_1))))_._x
let x be set ; ::_thesis: ( x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) implies (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1 )
assume A161: x in ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x is FinSeq-Location or x is Int-Location ) by A4, A161, Th13;
supposeA162: x is FinSeq-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
percases ( x = fa or x <> fa ) ;
supposeA163: x = fa ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
A164: ex k1 being Element of NAT st
( k1 = abs ((Comput (p1,s1,(k + m))) . a) & (Exec ((fa :=<0,...,0> a),(Comput (p1,s1,(k + m))))) . fa = k1 |-> 0 ) by SCMFSA_2:75;
A165: ex k2 being Element of NAT st
( k2 = abs ((Comput (p2,s2,(k + m))) . a) & (Exec ((fa :=<0,...,0> a),(Comput (p2,s2,(k + m))))) . fa = k2 |-> 0 ) by SCMFSA_2:75;
A166: a in UsedIntLoc p by A18, A159, Th11;
then (Comput (p1,s1,(k + m))) . a = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . a by A10, A14, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . a by A10, A166, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A19, A20, A21, A159, A163, A164, A165; ::_thesis: verum
end;
supposeA167: x <> fa ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A168: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A159, A162, SCMFSA_2:75;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A161, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A161, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A159, A162, A167, A168, SCMFSA_2:75; ::_thesis: verum
end;
end;
end;
supposeA169: x is Int-Location ; ::_thesis: (Comput (p1,s1,(k + (m + 1)))) . b1 = (Comput (p2,s2,(k + (m + 1)))) . b1
then A170: (Comput (p2,s2,(k + (m + 1)))) . x = (Comput (p2,s2,(k + m))) . x by A19, A21, A159, SCMFSA_2:75;
(Comput (p1,s1,(k + m))) . x = ((Comput (p2,s2,(k + m))) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p))) . x by A14, A161, FUNCT_1:49
.= (Comput (p2,s2,(k + m))) . x by A161, FUNCT_1:49 ;
hence  (Comput (p1,s1,(k + (m + 1)))) . x = (Comput (p2,s2,(k + (m + 1)))) . x by A20, A159, A169, A170, SCMFSA_2:75; ::_thesis: verum
end;
end;
end;
hence  S1[m + 1] by A22, A160, FUNCT_1:96; ::_thesis: verum
end;
end;
end;
 for m being Element of NAT holds S1[m] from NAT_1:sch_1(A12, A13);
hence  ( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | (((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p)) ) by A9; ::_thesis: verum
end;

theorem Th15: :: SCMBSORT:15
for p1, p2 being Instruction-Sequence of SCM+FSA
for i, k being Element of NAT
for p being Program of
for s1, s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) holds
( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) )
proof
let p1, p2 be Instruction-Sequence of SCM+FSA; ::_thesis: for i, k being Element of NAT
for p being Program of
for s1, s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) holds
( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) )
let i, k be Element of NAT ; ::_thesis: for p being Program of
for s1, s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) holds
( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) )
let p be Program of ; ::_thesis: for s1, s2 being State of SCM+FSA st k <= i & p c= p1 & p c= p2 & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) holds
( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) )
let s1, s2 be State of SCM+FSA; ::_thesis: ( k <= i & p c= p1 & p c= p2 & ( for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) ) & (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) & (Comput (p1,s1,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) implies ( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) ) )
set D = (UsedInt*Loc p) \/ (UsedIntLoc p);
assume that
A1: k <= i and
A2: p c= p1 and
A3: p c= p2 and
A4: for j being Element of NAT holds
( IC (Comput (p1,s1,j)) in dom p & IC (Comput (p2,s2,j)) in dom p ) and
A5: (Comput (p1,s1,k)) . (IC ) = (Comput (p2,s2,k)) . (IC ) and
A6: (Comput (p1,s1,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,k)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) ; ::_thesis: ( (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) & (Comput (p1,s1,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) )
reconsider t = {} as PartState of SCM+FSA by FUNCT_1:104, RELAT_1:171;
set D1 = ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p);
A7: dom t c= Int-Locations \/ FinSeq-Locations by RELAT_1:38, XBOOLE_1:2;
A8: ((dom t) \/ (UsedInt*Loc p)) \/ (UsedIntLoc p) = (UsedInt*Loc p) \/ (UsedIntLoc p) by RELAT_1:38;
hence  (Comput (p1,s1,i)) . (IC ) = (Comput (p2,s2,i)) . (IC ) by A1, A2, A3, A4, A5, A6, A7, Th14; ::_thesis: (Comput (p1,s1,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p))
thus  (Comput (p1,s1,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) = (Comput (p2,s2,i)) | ((UsedInt*Loc p) \/ (UsedIntLoc p)) by A1, A2, A3, A4, A5, A6, A7, A8, Th14; ::_thesis: verum
end;

theorem Th16: :: SCMBSORT:16
for I, J being Program of
for a being Int-Location holds
( UsedIntLoc (if=0 (a,I,J)) = ({a} \/ (UsedIntLoc I)) \/ (UsedIntLoc J) & UsedIntLoc (if>0 (a,I,J)) = ({a} \/ (UsedIntLoc I)) \/ (UsedIntLoc J) )
proof
let I, J be Program of ; ::_thesis: for a being Int-Location holds
( UsedIntLoc (if=0 (a,I,J)) = ({a} \/ (UsedIntLoc I)) \/ (UsedIntLoc J) & UsedIntLoc (if>0 (a,I,J)) = ({a} \/ (UsedIntLoc I)) \/ (UsedIntLoc J) )
let a be Int-Location; ::_thesis: ( UsedIntLoc (if=0 (a,I,J)) = ({a} \/ (UsedIntLoc I)) \/ (UsedIntLoc J) & UsedIntLoc (if>0 (a,I,J)) = ({a} \/ (UsedIntLoc I)) \/ (UsedIntLoc J) )
set g1 = a =0_goto ((card J) + 3);
set g2 = Goto ((card I) + 1);
set g3 = a >0_goto ((card J) + 3);
set SS = Stop SCM+FSA;
thus  UsedIntLoc (if=0 (a,I,J)) = UsedIntLoc (((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by SCMFSA8B:def_1
.= (UsedIntLoc ((((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I)) \/ {} by SCMFSA9A:3, SF_MASTR:27
.= (UsedIntLoc (((a =0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1)))) \/ (UsedIntLoc I) by SF_MASTR:27
.= ((UsedIntLoc ((a =0_goto ((card J) + 3)) ";" J)) \/ (UsedIntLoc (Goto ((card I) + 1)))) \/ (UsedIntLoc I) by SF_MASTR:27
.= ((UsedIntLoc ((a =0_goto ((card J) + 3)) ";" J)) \/ {}) \/ (UsedIntLoc I) by SCMFSA9A:5
.= ((UsedIntLoc (a =0_goto ((card J) + 3))) \/ (UsedIntLoc J)) \/ (UsedIntLoc I) by SF_MASTR:29
.= ({a} \/ (UsedIntLoc J)) \/ (UsedIntLoc I) by SF_MASTR:16
.= ({a} \/ (UsedIntLoc I)) \/ (UsedIntLoc J) by XBOOLE_1:4 ; ::_thesis: UsedIntLoc (if>0 (a,I,J)) = ({a} \/ (UsedIntLoc I)) \/ (UsedIntLoc J)
thus  UsedIntLoc (if>0 (a,I,J)) = UsedIntLoc (((((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I) ";" (Stop SCM+FSA)) by SCMFSA8B:def_2
.= (UsedIntLoc ((((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1))) ";" I)) \/ {} by SCMFSA9A:3, SF_MASTR:27
.= (UsedIntLoc (((a >0_goto ((card J) + 3)) ";" J) ";" (Goto ((card I) + 1)))) \/ (UsedIntLoc I) by SF_MASTR:27
.= ((UsedIntLoc ((a >0_goto ((card J) + 3)) ";" J)) \/ (UsedIntLoc (Goto ((card I) + 1)))) \/ (UsedIntLoc I) by SF_MASTR:27
.= ((UsedIntLoc ((a >0_goto ((card J) + 3)) ";" J)) \/ {}) \/ (UsedIntLoc I) by SCMFSA9A:5
.= ((UsedIntLoc (a >0_goto ((card J) + 3))) \/ (UsedIntLoc J)) \/ (UsedIntLoc I) by SF_MASTR:29
.= ({a} \/ (UsedIntLoc J)) \/ (UsedIntLoc I) by SF_MASTR:16
.= ({a} \/ (UsedIntLoc I)) \/ (UsedIntLoc J) by XBOOLE_1:4 ; ::_thesis: verum
end;

theorem Th17: :: SCMBSORT:17
for I being Program of
for l being Element of NAT holds UsedIntLoc (Directed (I,l)) = UsedIntLoc I
proof
let I be Program of ; ::_thesis: for l being Element of NAT holds UsedIntLoc (Directed (I,l)) = UsedIntLoc I
let l be Element of NAT ; ::_thesis: UsedIntLoc (Directed (I,l)) = UsedIntLoc I
consider UIL being Function of the InstructionsF of SCM+FSA,(Fin Int-Locations) such that
A1: for i being Instruction of SCM+FSA holds UIL . i = UsedIntLoc i and
A2: UsedIntLoc I = Union (UIL * I) by SF_MASTR:def_2;
consider UIL2 being Function of the InstructionsF of SCM+FSA,(Fin Int-Locations) such that
A3: for i being Instruction of SCM+FSA holds UIL2 . i = UsedIntLoc i and
A4: UsedIntLoc (Directed (I,l)) = Union (UIL2 * (Directed (I,l))) by SF_MASTR:def_2;
A5: for c being Element of the InstructionsF of SCM+FSA holds UIL . c = UIL2 . c
proof
let c be Element of the InstructionsF of SCM+FSA; ::_thesis: UIL . c = UIL2 . c
reconsider d = c as Instruction of SCM+FSA ;
thus UIL . c = UsedIntLoc d by A1
.= UIL2 . c by A3 ; ::_thesis: verum
end;
A6: dom UIL = the InstructionsF of SCM+FSA by FUNCT_2:def_1;
A7: UIL . (halt SCM+FSA) = {} by A1, SF_MASTR:13;
A8: UIL . (goto l) = UsedIntLoc (goto l) by A1
.= {} by SF_MASTR:15 ;
A9: rng I c= the InstructionsF of SCM+FSA by RELAT_1:def_19;
UIL * (Directed (I,l)) = UIL * (I +~ ((halt SCM+FSA),(goto l))) by SCMFSA6A:def_1
.= UIL * (((id the InstructionsF of SCM+FSA) +* ((halt SCM+FSA),(goto l))) * I) by A9, FUNCT_7:116
.= (UIL * ((id the InstructionsF of SCM+FSA) +* ((halt SCM+FSA),(goto l)))) * I by RELAT_1:36
.= UIL * I by A6, A7, A8, FUNCT_7:108 ;
hence  UsedIntLoc (Directed (I,l)) = UsedIntLoc I by A2, A4, A5, FUNCT_2:63; ::_thesis: verum
end;

theorem Th18: :: SCMBSORT:18
for a being Int-Location
for I being Program of holds UsedIntLoc (Times (a,I)) = (UsedIntLoc I) \/ {a,(intloc 0)}
proof
let a be Int-Location; ::_thesis: for I being Program of holds UsedIntLoc (Times (a,I)) = (UsedIntLoc I) \/ {a,(intloc 0)}
let I be Program of ; ::_thesis: UsedIntLoc (Times (a,I)) = (UsedIntLoc I) \/ {a,(intloc 0)}
set g1 = Goto 2;
set SF = SubFrom (a,(intloc 0));
set if0 = if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))));
thus  UsedIntLoc (Times (a,I)) = ({a} \/ (UsedIntLoc (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))))) \/ {} by Th16, SCMFSA9A:3
.= {a} \/ (UsedIntLoc (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) by Th17
.= {a} \/ (({a} \/ (UsedIntLoc (Goto 2))) \/ (UsedIntLoc (I ";" (SubFrom (a,(intloc 0)))))) by Th16
.= {a} \/ (({a} \/ {}) \/ (UsedIntLoc (I ";" (SubFrom (a,(intloc 0)))))) by SCMFSA9A:5
.= ({a} \/ {a}) \/ (UsedIntLoc (I ";" (SubFrom (a,(intloc 0))))) by XBOOLE_1:4
.= ((UsedIntLoc I) \/ (UsedIntLoc (SubFrom (a,(intloc 0))))) \/ {a} by SF_MASTR:30
.= ((UsedIntLoc I) \/ {a,(intloc 0)}) \/ {a} by SF_MASTR:14
.= (UsedIntLoc I) \/ ({a} \/ {a,(intloc 0)}) by XBOOLE_1:4
.= (UsedIntLoc I) \/ {a,a,(intloc 0)} by ENUMSET1:2
.= (UsedIntLoc I) \/ {a,(intloc 0)} by ENUMSET1:30 ; ::_thesis: verum
end;

theorem Th19: :: SCMBSORT:19
for I being Program of
for l being Element of NAT holds UsedInt*Loc (Directed (I,l)) = UsedInt*Loc I
proof
let I be Program of ; ::_thesis: for l being Element of NAT holds UsedInt*Loc (Directed (I,l)) = UsedInt*Loc I
let l be Element of NAT ; ::_thesis: UsedInt*Loc (Directed (I,l)) = UsedInt*Loc I
consider UIL being Function of the InstructionsF of SCM+FSA,(Fin FinSeq-Locations) such that
A1: for i being Instruction of SCM+FSA holds UIL . i = UsedInt*Loc i and
A2: UsedInt*Loc I = Union (UIL * I) by SF_MASTR:def_4;
consider UIL2 being Function of the InstructionsF of SCM+FSA,(Fin FinSeq-Locations) such that
A3: for i being Instruction of SCM+FSA holds UIL2 . i = UsedInt*Loc i and
A4: UsedInt*Loc (Directed (I,l)) = Union (UIL2 * (Directed (I,l))) by SF_MASTR:def_4;
A5: for c being Element of the InstructionsF of SCM+FSA holds UIL . c = UIL2 . c
proof
let c be Element of the InstructionsF of SCM+FSA; ::_thesis: UIL . c = UIL2 . c
reconsider d = c as Instruction of SCM+FSA ;
thus UIL . c = UsedInt*Loc d by A1
.= UIL2 . c by A3 ; ::_thesis: verum
end;
A6: dom UIL = the InstructionsF of SCM+FSA by FUNCT_2:def_1;
A7: UIL . (halt SCM+FSA) = UsedInt*Loc (halt SCM+FSA) by A1
.= {} by SF_MASTR:32 ;
A8: UIL . (goto l) = UsedInt*Loc (goto l) by A1
.= {} by SF_MASTR:32 ;
A9: rng I c= the InstructionsF of SCM+FSA by RELAT_1:def_19;
UIL * (Directed (I,l)) = UIL * (I +~ ((halt SCM+FSA),(goto l))) by SCMFSA6A:def_1
.= UIL * (((id the InstructionsF of SCM+FSA) +* ((halt SCM+FSA),(goto l))) * I) by A9, FUNCT_7:116
.= (UIL * ((id the InstructionsF of SCM+FSA) +* ((halt SCM+FSA),(goto l)))) * I by RELAT_1:36
.= UIL * I by A6, A7, A8, FUNCT_7:108 ;
hence  UsedInt*Loc (Directed (I,l)) = UsedInt*Loc I by A2, A4, A5, FUNCT_2:63; ::_thesis: verum
end;

theorem Th20: :: SCMBSORT:20
for a being Int-Location
for I being Program of holds UsedInt*Loc (Times (a,I)) = UsedInt*Loc I
proof
let a be Int-Location; ::_thesis: for I being Program of holds UsedInt*Loc (Times (a,I)) = UsedInt*Loc I
let I be Program of ; ::_thesis: UsedInt*Loc (Times (a,I)) = UsedInt*Loc I
set g1 = Goto 2;
set SF = SubFrom (a,(intloc 0));
set if0 = if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))));
thus  UsedInt*Loc (Times (a,I)) = (UsedInt*Loc (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))) \/ {} by SCMFSA9A:4, SCMFSA9A:10
.= UsedInt*Loc (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))) by Th19
.= (UsedInt*Loc (Goto 2)) \/ (UsedInt*Loc (I ";" (SubFrom (a,(intloc 0))))) by SCMFSA9A:8
.= {} \/ (UsedInt*Loc (I ";" (SubFrom (a,(intloc 0))))) by SCMFSA9A:6
.= (UsedInt*Loc I) \/ (UsedInt*Loc (SubFrom (a,(intloc 0)))) by SF_MASTR:46
.= (UsedInt*Loc I) \/ {} by SF_MASTR:32
.= UsedInt*Loc I ; ::_thesis: verum
end;

theorem :: SCMBSORT:21
canceled;

theorem Th22: :: SCMBSORT:22
for a being Int-Location
for I being Program of holds card (Times (a,I)) = (card I) + 12
proof
let a be Int-Location; ::_thesis: for I being Program of holds card (Times (a,I)) = (card I) + 12
let I be Program of ; ::_thesis: card (Times (a,I)) = (card I) + 12
set g1 = Goto 2;
set SF = SubFrom (a,(intloc 0));
set if0 = if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))));
 card (Stop SCM+FSA) = 1 by COMPOS_1:4;
hence  card (Times (a,I)) = ((card (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))) + 1) + 4 by SCMFSA8B:12
.= (card (loop (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0)))))))) + (1 + 4)
.= (card (if=0 (a,(Goto 2),(I ";" (SubFrom (a,(intloc 0))))))) + 5 by SCMFSA8A:19
.= (((card (I ";" (SubFrom (a,(intloc 0))))) + (card (Goto 2))) + 4) + 5 by SCMFSA8B:11
.= (((card (I ";" (SubFrom (a,(intloc 0))))) + 1) + 4) + 5 by SCMFSA8A:15
.= ((((card I) + 2) + 1) + 4) + 5 by SCMFSA6A:34
.= (card I) + 12 ;
::_thesis: verum
end;

theorem Th23: :: SCMBSORT:23
for i1, i2, i3 being Instruction of SCM+FSA holds card ((i1 ";" i2) ";" i3) = 6
proof
let i1, i2, i3 be Instruction of SCM+FSA; ::_thesis: card ((i1 ";" i2) ";" i3) = 6
thus  card ((i1 ";" i2) ";" i3) = (card (i1 ";" i2)) + 2 by SCMFSA6A:34
.= 4 + 2 by SCMFSA6A:35
.= 6 ; ::_thesis: verum
end;

theorem :: SCMBSORT:24
canceled;

theorem :: SCMBSORT:25
canceled;

theorem Th26: :: SCMBSORT:26
for I, J being Program of
for k being Element of NAT
for i being Instruction of SCM+FSA st k < card J & i = J . k holds
(I ";" J) . ((card I) + k) = IncAddr (i,(card I))
proof
let I, J be Program of ; ::_thesis: for k being Element of NAT
for i being Instruction of SCM+FSA st k < card J & i = J . k holds
(I ";" J) . ((card I) + k) = IncAddr (i,(card I))
let k be Element of NAT ; ::_thesis: for i being Instruction of SCM+FSA st k < card J & i = J . k holds
(I ";" J) . ((card I) + k) = IncAddr (i,(card I))
let i be Instruction of SCM+FSA; ::_thesis: ( k < card J & i = J . k implies (I ";" J) . ((card I) + k) = IncAddr (i,(card I)) )
assume that
A1: k < card J and
A2: i = J . k ; ::_thesis: (I ";" J) . ((card I) + k) = IncAddr (i,(card I))
set m = (card I) + k;
A3: (card I) + k < (card I) + (card J) by A1, XREAL_1:6;
 ((card I) + k) -' (card I) = k by NAT_D:34;
hence  (I ";" J) . ((card I) + k) = IncAddr (i,(card I)) by A2, A3, NAT_1:11, SCMFSA8C:2; ::_thesis: verum
end;

theorem Th27: :: SCMBSORT:27
for I, J being Program of
for i being ins-loc-free Instruction of SCM+FSA st i <> halt SCM+FSA holds
((I ";" i) ";" J) . (card I) = i
proof
let I, J be Program of ; ::_thesis: for i being ins-loc-free Instruction of SCM+FSA st i <> halt SCM+FSA holds
((I ";" i) ";" J) . (card I) = i
let i be ins-loc-free Instruction of SCM+FSA; ::_thesis: ( i <> halt SCM+FSA implies ((I ";" i) ";" J) . (card I) = i )
assume A1: i <> halt SCM+FSA ; ::_thesis: ((I ";" i) ";" J) . (card I) = i
set x1 = card I;
A2: card (I ";" i) = (card I) + 2 by SCMFSA6A:34;
 (card I) + 0 < (card I) + 2 by XREAL_1:6;
then A3: card I in dom (I ";" i) by A2, AFINSQ_1:66;
A4: (Macro i) . 0 = i by COMPOS_1:58;
A5: card (Macro i) = 2 by COMPOS_1:56;
A6: (I ";" i) . (card I) = (I ";" (Macro i)) . ((card I) + 0) by SCMFSA6A:def_6
.= IncAddr (i,(card I)) by A4, A5, Th26
.= i by COMPOS_0:4 ;
thus ((I ";" i) ";" J) . (card I) = (Directed (I ";" i)) . (card I) by A3, SCMFSA8A:14
.= i by A1, A3, A6, SCMFSA8A:16 ; ::_thesis: verum
end;

theorem Th28: :: SCMBSORT:28
for I, J being Program of
for i being Instruction of SCM+FSA holds ((I ";" i) ";" J) . ((card I) + 1) = goto ((card I) + 2)
proof
let I, J be Program of ; ::_thesis: for i being Instruction of SCM+FSA holds ((I ";" i) ";" J) . ((card I) + 1) = goto ((card I) + 2)
let i be Instruction of SCM+FSA; ::_thesis: ((I ";" i) ";" J) . ((card I) + 1) = goto ((card I) + 2)
set x1 = card I;
A1: card (I ";" i) = (card I) + 2 by SCMFSA6A:34;
A2: card (Macro i) = 2 by COMPOS_1:56;
set x2 = (card I) + 1;
 (card I) + 1 < (card I) + 2 by XREAL_1:6;
then A3: (card I) + 1 in dom (I ";" i) by A1, AFINSQ_1:66;
 (Macro i) . 1 = halt SCM+FSA by COMPOS_1:59;
then  (I ";" (Macro i)) . ((card I) + 1) = IncAddr ((halt SCM+FSA),(card I)) by A2, Th26;
then A4: (I ";" i) . ((card I) + 1) = IncAddr ((halt SCM+FSA),(card I)) by SCMFSA6A:def_6
.= halt SCM+FSA by COMPOS_0:4 ;
thus ((I ";" i) ";" J) . ((card I) + 1) = (Directed (I ";" i)) . ((card I) + 1) by A3, SCMFSA8A:14
.= goto ((card I) + 2) by A1, A3, A4, SCMFSA8A:16 ; ::_thesis: verum
end;

theorem :: SCMBSORT:29
canceled;

theorem :: SCMBSORT:30
canceled;

theorem :: SCMBSORT:31
canceled;

theorem Th32: :: SCMBSORT:32
for p being Program of
for s being State of SCM+FSA holds (UsedInt*Loc p) \/ (UsedIntLoc p) c= dom s
proof
let p be Program of ; ::_thesis: for s being State of SCM+FSA holds (UsedInt*Loc p) \/ (UsedIntLoc p) c= dom s
let s be State of SCM+FSA; ::_thesis: (UsedInt*Loc p) \/ (UsedIntLoc p) c= dom s
 Int-Locations c= dom s by SCMFSA_2:45;
then A1: UsedIntLoc p c= dom s by XBOOLE_1:1;
 FinSeq-Locations c= dom s by SCMFSA_2:46;
then  UsedInt*Loc p c= dom s by XBOOLE_1:1;
hence  (UsedInt*Loc p) \/ (UsedIntLoc p) c= dom s by A1, XBOOLE_1:8; ::_thesis: verum
end;

theorem Th33: :: SCMBSORT:33
for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for I being Program of
for f being FinSeq-Location holds (Result ((p +* I),(Initialized s))) . f = (IExec (I,p,s)) . f
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for I being Program of
for f being FinSeq-Location holds (Result ((p +* I),(Initialized s))) . f = (IExec (I,p,s)) . f
let s be State of SCM+FSA; ::_thesis: for I being Program of
for f being FinSeq-Location holds (Result ((p +* I),(Initialized s))) . f = (IExec (I,p,s)) . f
let I be Program of ; ::_thesis: for f being FinSeq-Location holds (Result ((p +* I),(Initialized s))) . f = (IExec (I,p,s)) . f
let f be FinSeq-Location ; ::_thesis: (Result ((p +* I),(Initialized s))) . f = (IExec (I,p,s)) . f
set D = Int-Locations \/ FinSeq-Locations;
 f in FinSeq-Locations by SCMFSA_2:def_5;
then A1: f in Int-Locations \/ FinSeq-Locations by XBOOLE_0:def_3;
hence (Result ((p +* I),(Initialized s))) . f = (DataPart (Result ((p +* I),(Initialized s)))) . f by FUNCT_1:49, SCMFSA_2:100
.= (DataPart (IExec (I,p,s))) . f by SCMFSA8B:32
.= (IExec (I,p,s)) . f by A1, FUNCT_1:49, SCMFSA_2:100 ;
::_thesis: verum
end;

set a0 = intloc 0;

set a1 = intloc 1;

set a2 = intloc 2;

set a3 = intloc 3;

set a4 = intloc 4;

set a5 = intloc 5;

set a6 = intloc 6;

Lm1:  intloc 0 <> intloc 2
by SCMFSA_2:101;

Lm2:  intloc 0 <> intloc 4
by SCMFSA_2:101;

Lm3:  intloc 0 <> intloc 5
by SCMFSA_2:101;

Lm4:  intloc 0 <> intloc 6
by SCMFSA_2:101;

Lm5:  intloc 1 <> intloc 2
by SCMFSA_2:101;

Lm6:  intloc 1 <> intloc 3
by SCMFSA_2:101;

Lm7:  intloc 1 <> intloc 4
by SCMFSA_2:101;

Lm8:  intloc 1 <> intloc 5
by SCMFSA_2:101;

Lm9:  intloc 1 <> intloc 6
by SCMFSA_2:101;

Lm10:  intloc 2 <> intloc 3
by SCMFSA_2:101;

Lm11:  intloc 2 <> intloc 4
by SCMFSA_2:101;

Lm12:  intloc 2 <> intloc 5
by SCMFSA_2:101;

Lm13:  intloc 2 <> intloc 6
by SCMFSA_2:101;

Lm14:  intloc 3 <> intloc 4
by SCMFSA_2:101;

Lm15:  intloc 3 <> intloc 5
by SCMFSA_2:101;

Lm16:  intloc 3 <> intloc 6
by SCMFSA_2:101;

Lm17:  intloc 4 <> intloc 5
by SCMFSA_2:101;

Lm18:  intloc 4 <> intloc 6
by SCMFSA_2:101;

Lm19:  intloc 5 <> intloc 6
by SCMFSA_2:101;

set initializeWorkMem = (((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0));

definition
let f be FinSeq-Location ;
func bubble-sort f ->  Program of  equals :: SCMBSORT:def 1
(((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) ";" ((intloc 1) :=len f)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))));
correctness
coherence
(((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) ";" ((intloc 1) :=len f)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))))) is Program of ;
;
end;

:: deftheorem  defines bubble-sort SCMBSORT:def_1_:_
 for f being FinSeq-Location holds bubble-sort f = (((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) ";" ((intloc 1) :=len f)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))));

definition
func Bubble-Sort-Algorithm  ->  Program of  equals :: SCMBSORT:def 2
 bubble-sort (fsloc 0);
coherence
 bubble-sort (fsloc 0) is Program of ;
end;

:: deftheorem  defines Bubble-Sort-Algorithm SCMBSORT:def_2_:_
 Bubble-Sort-Algorithm = bubble-sort (fsloc 0);

set b1 = intloc (0 + 1);

set b2 = intloc (1 + 1);

set b3 = intloc (2 + 1);

set b4 = intloc (3 + 1);

set b5 = intloc (4 + 1);

set b6 = intloc (5 + 1);

set f0 = fsloc 0;

set i1 = (intloc (3 + 1)) := (intloc (2 + 1));

set i2 = SubFrom ((intloc (2 + 1)),(intloc 0));

set i3 = (intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1)));

set i4 = (intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)));

set i5 = SubFrom ((intloc (5 + 1)),(intloc (4 + 1)));

set i6 = ((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1));

set i7 = ((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1));

set SS = Stop SCM+FSA;

set ifc = if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA));

set body2 = ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)));

set T2 = Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))));

set j1 = (intloc (1 + 1)) := (intloc (0 + 1));

set j2 = SubFrom ((intloc (1 + 1)),(intloc 0));

set j3 = (intloc (2 + 1)) :=len (fsloc 0);

set Sb = (((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0));

set body1 = ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))));

set T1 = Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))));

set w2 = (intloc (1 + 1)) := (intloc 0);

set w3 = (intloc (2 + 1)) := (intloc 0);

set w4 = (intloc (3 + 1)) := (intloc 0);

set w5 = (intloc (4 + 1)) := (intloc 0);

set w6 = (intloc (5 + 1)) := (intloc 0);

set w7 = (intloc (0 + 1)) :=len (fsloc 0);

theorem Th34: :: SCMBSORT:34
for f being FinSeq-Location holds UsedIntLoc (bubble-sort f) = {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)}
proof
let f be FinSeq-Location ; ::_thesis: UsedIntLoc (bubble-sort f) = {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)}
set i1 = (intloc 4) := (intloc 3);
set i2 = SubFrom ((intloc 3),(intloc 0));
set i3 = (intloc 5) := (f,(intloc 3));
set i4 = (intloc 6) := (f,(intloc 4));
set i5 = SubFrom ((intloc 6),(intloc 5));
set i6 = (f,(intloc 3)) := (intloc 6);
set i7 = (f,(intloc 4)) := (intloc 5);
set ifc = if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA));
set Sif = UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)));
set body2 = ((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)));
A1: UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))) = ({(intloc 6)} \/ (UsedIntLoc ((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))))) \/ {} by Th16, SCMFSA9A:3
.= {(intloc 6)} \/ ((UsedIntLoc (((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6)))) \/ (UsedIntLoc ((f,(intloc 4)) := (intloc 5)))) by SF_MASTR:30
.= {(intloc 6)} \/ ((UsedIntLoc (((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6)))) \/ {(intloc 4),(intloc 5)}) by SF_MASTR:17
.= {(intloc 6)} \/ (((UsedIntLoc ((intloc 6) := (f,(intloc 4)))) \/ (UsedIntLoc ((f,(intloc 3)) := (intloc 6)))) \/ {(intloc 4),(intloc 5)}) by SF_MASTR:31
.= {(intloc 6)} \/ (((UsedIntLoc ((intloc 6) := (f,(intloc 4)))) \/ {(intloc 3),(intloc 6)}) \/ {(intloc 4),(intloc 5)}) by SF_MASTR:17
.= {(intloc 6)} \/ (({(intloc 4),(intloc 6)} \/ {(intloc 3),(intloc 6)}) \/ {(intloc 4),(intloc 5)}) by SF_MASTR:17
.= {(intloc 6)} \/ ({(intloc 4),(intloc 6),(intloc 3),(intloc 6)} \/ {(intloc 4),(intloc 5)}) by ENUMSET1:5
.= {(intloc 6)} \/ ({(intloc 6),(intloc 6),(intloc 3),(intloc 4)} \/ {(intloc 4),(intloc 5)}) by ENUMSET1:75
.= ({(intloc 6)} \/ {(intloc 6),(intloc 6),(intloc 3),(intloc 4)}) \/ {(intloc 4),(intloc 5)} by XBOOLE_1:4
.= {(intloc 6),(intloc 6),(intloc 6),(intloc 3),(intloc 4)} \/ {(intloc 4),(intloc 5)} by ENUMSET1:7
.= {(intloc 6),(intloc 3),(intloc 4)} \/ {(intloc 4),(intloc 5)} by ENUMSET1:38
.= ({(intloc 6),(intloc 3)} \/ {(intloc 4)}) \/ {(intloc 4),(intloc 5)} by ENUMSET1:3
.= {(intloc 6),(intloc 3)} \/ ({(intloc 4)} \/ {(intloc 4),(intloc 5)}) by XBOOLE_1:4
.= {(intloc 6),(intloc 3)} \/ {(intloc 4),(intloc 4),(intloc 5)} by ENUMSET1:2
.= {(intloc 4),(intloc 5)} \/ {(intloc 6),(intloc 3)} by ENUMSET1:30
.= {(intloc 4),(intloc 5),(intloc 6),(intloc 3)} by ENUMSET1:5
.= {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by ENUMSET1:64 ;
set ui12 = UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))));
A2: UsedIntLoc (((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) = (UsedIntLoc ((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5))))) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:27
.= ((UsedIntLoc (((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4))))) \/ (UsedIntLoc (SubFrom ((intloc 6),(intloc 5))))) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:30
.= ((UsedIntLoc (((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4))))) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:14
.= (((UsedIntLoc ((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3))))) \/ (UsedIntLoc ((intloc 6) := (f,(intloc 4))))) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:30
.= (((UsedIntLoc ((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3))))) \/ {(intloc 6),(intloc 4)}) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:17
.= ((((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ (UsedIntLoc ((intloc 5) := (f,(intloc 3))))) \/ {(intloc 6),(intloc 4)}) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:30
.= ((((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ {(intloc 5),(intloc 3)}) \/ {(intloc 6),(intloc 4)}) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:17
.= (((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ ({(intloc 5),(intloc 3)} \/ {(intloc 6),(intloc 4)})) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by XBOOLE_1:4
.= (((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ {(intloc 5),(intloc 3),(intloc 6),(intloc 4)}) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by ENUMSET1:5
.= (((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)}) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by ENUMSET1:75
.= (((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ ({(intloc 4),(intloc 3)} \/ {(intloc 6),(intloc 5)})) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by ENUMSET1:5
.= ((((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ {(intloc 4),(intloc 3)}) \/ {(intloc 6),(intloc 5)}) \/ {(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by XBOOLE_1:4
.= (((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ {(intloc 4),(intloc 3)}) \/ ({(intloc 6),(intloc 5)} \/ {(intloc 6),(intloc 5)})) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by XBOOLE_1:4
.= ((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ ({(intloc 4),(intloc 3)} \/ {(intloc 6),(intloc 5)})) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by XBOOLE_1:4
.= ((UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)}) \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by ENUMSET1:5
.= (UsedIntLoc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ ({(intloc 4),(intloc 3),(intloc 6),(intloc 5)} \/ (UsedIntLoc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))) by XBOOLE_1:4
.= ((UsedIntLoc ((intloc 4) := (intloc 3))) \/ (UsedIntLoc (SubFrom ((intloc 3),(intloc 0))))) \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by A1, SF_MASTR:31
.= ((UsedIntLoc ((intloc 4) := (intloc 3))) \/ {(intloc 3),(intloc 0)}) \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by SF_MASTR:14
.= ({(intloc 3),(intloc 4)} \/ {(intloc 3),(intloc 0)}) \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by SF_MASTR:14
.= {(intloc 3),(intloc 4),(intloc 3),(intloc 0)} \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by ENUMSET1:5
.= {(intloc 3),(intloc 3),(intloc 4),(intloc 0)} \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by ENUMSET1:62
.= {(intloc 3),(intloc 4),(intloc 0)} \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by ENUMSET1:31
.= {(intloc 0),(intloc 4),(intloc 3)} \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by ENUMSET1:60
.= ({(intloc 0)} \/ {(intloc 4),(intloc 3)}) \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by ENUMSET1:2
.= ({(intloc 0)} \/ {(intloc 4),(intloc 3)}) \/ ({(intloc 4),(intloc 3)} \/ {(intloc 6),(intloc 5)}) by ENUMSET1:5
.= (({(intloc 0)} \/ {(intloc 4),(intloc 3)}) \/ {(intloc 4),(intloc 3)}) \/ {(intloc 6),(intloc 5)} by XBOOLE_1:4
.= ({(intloc 0)} \/ ({(intloc 4),(intloc 3)} \/ {(intloc 4),(intloc 3)})) \/ {(intloc 6),(intloc 5)} by XBOOLE_1:4
.= {(intloc 0)} \/ ({(intloc 4),(intloc 3)} \/ {(intloc 6),(intloc 5)}) by XBOOLE_1:4
.= {(intloc 0)} \/ {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} by ENUMSET1:5 ;
set j1 = (intloc 2) := (intloc 1);
set j2 = SubFrom ((intloc 2),(intloc 0));
set j3 = (intloc 3) :=len f;
set Sfor = UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))));
set body1 = ((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))));
A3: UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))) = ({(intloc 4),(intloc 3),(intloc 6),(intloc 5)} \/ {(intloc 0)}) \/ {(intloc 2),(intloc 0)} by A2, Th18
.= {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} \/ ({(intloc 0)} \/ {(intloc 2),(intloc 0)}) by XBOOLE_1:4
.= {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} \/ {(intloc 0),(intloc 0),(intloc 2)} by ENUMSET1:2
.= {(intloc 4),(intloc 3),(intloc 6),(intloc 5)} \/ {(intloc 0),(intloc 2)} by ENUMSET1:30
.= {(intloc 4),(intloc 5),(intloc 6),(intloc 3)} \/ {(intloc 0),(intloc 2)} by ENUMSET1:64
.= ({(intloc 4),(intloc 5),(intloc 6)} \/ {(intloc 3)}) \/ {(intloc 0),(intloc 2)} by ENUMSET1:6
.= {(intloc 4),(intloc 5),(intloc 6)} \/ ({(intloc 3)} \/ {(intloc 0),(intloc 2)}) by XBOOLE_1:4
.= {(intloc 4),(intloc 5),(intloc 6)} \/ {(intloc 0),(intloc 2),(intloc 3)} by ENUMSET1:3 ;
A4: UsedIntLoc (((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) = (UsedIntLoc ((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f))) \/ (UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:27
.= ((UsedIntLoc (((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0))))) \/ (UsedIntLoc ((intloc 3) :=len f))) \/ (UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:30
.= ((UsedIntLoc (((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0))))) \/ {(intloc 3)}) \/ (UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:18
.= (((UsedIntLoc ((intloc 2) := (intloc 1))) \/ (UsedIntLoc (SubFrom ((intloc 2),(intloc 0))))) \/ {(intloc 3)}) \/ (UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:31
.= (((UsedIntLoc ((intloc 2) := (intloc 1))) \/ {(intloc 2),(intloc 0)}) \/ {(intloc 3)}) \/ (UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:14
.= (({(intloc 2),(intloc 1)} \/ {(intloc 2),(intloc 0)}) \/ {(intloc 3)}) \/ (UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:14
.= ({(intloc 2),(intloc 1)} \/ ({(intloc 0),(intloc 2)} \/ {(intloc 3)})) \/ (UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by XBOOLE_1:4
.= ({(intloc 2),(intloc 1)} \/ {(intloc 0),(intloc 2),(intloc 3)}) \/ (UsedIntLoc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by ENUMSET1:3
.= (({(intloc 2),(intloc 1)} \/ {(intloc 0),(intloc 2),(intloc 3)}) \/ {(intloc 0),(intloc 2),(intloc 3)}) \/ {(intloc 4),(intloc 5),(intloc 6)} by A3, XBOOLE_1:4
.= ({(intloc 2),(intloc 1)} \/ ({(intloc 0),(intloc 2),(intloc 3)} \/ {(intloc 0),(intloc 2),(intloc 3)})) \/ {(intloc 4),(intloc 5),(intloc 6)} by XBOOLE_1:4
.= ({(intloc 2),(intloc 1)} \/ ({(intloc 0),(intloc 2)} \/ {(intloc 3)})) \/ {(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:3
.= (({(intloc 2),(intloc 1)} \/ {(intloc 0),(intloc 2)}) \/ {(intloc 3)}) \/ {(intloc 4),(intloc 5),(intloc 6)} by XBOOLE_1:4
.= ({(intloc 2),(intloc 1),(intloc 0),(intloc 2)} \/ {(intloc 3)}) \/ {(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:5
.= ({(intloc 2),(intloc 2),(intloc 0),(intloc 1)} \/ {(intloc 3)}) \/ {(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:64
.= ({(intloc 2),(intloc 0),(intloc 1)} \/ {(intloc 3)}) \/ {(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:31
.= ({(intloc 0),(intloc 1),(intloc 2)} \/ {(intloc 3)}) \/ {(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:59
.= {(intloc 0),(intloc 1),(intloc 2),(intloc 3)} \/ {(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:6
.= {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:19 ;
set k2 = (intloc 2) := (intloc 0);
set k3 = (intloc 3) := (intloc 0);
set k4 = (intloc 4) := (intloc 0);
set k5 = (intloc 5) := (intloc 0);
A5: UsedIntLoc ((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) = (UsedIntLoc (((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0)))) \/ (UsedIntLoc ((intloc 6) := (intloc 0))) by SF_MASTR:30
.= (UsedIntLoc (((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0)))) \/ {(intloc 6),(intloc 0)} by SF_MASTR:14
.= ((UsedIntLoc ((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0)))) \/ (UsedIntLoc ((intloc 5) := (intloc 0)))) \/ {(intloc 6),(intloc 0)} by SF_MASTR:30
.= ((UsedIntLoc ((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0)))) \/ {(intloc 5),(intloc 0)}) \/ {(intloc 6),(intloc 0)} by SF_MASTR:14
.= (((UsedIntLoc (((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0)))) \/ (UsedIntLoc ((intloc 4) := (intloc 0)))) \/ {(intloc 5),(intloc 0)}) \/ {(intloc 6),(intloc 0)} by SF_MASTR:30
.= (((UsedIntLoc (((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0)))) \/ {(intloc 4),(intloc 0)}) \/ {(intloc 5),(intloc 0)}) \/ {(intloc 6),(intloc 0)} by SF_MASTR:14
.= ((((UsedIntLoc ((intloc 2) := (intloc 0))) \/ (UsedIntLoc ((intloc 3) := (intloc 0)))) \/ {(intloc 4),(intloc 0)}) \/ {(intloc 5),(intloc 0)}) \/ {(intloc 6),(intloc 0)} by SF_MASTR:31
.= ((((UsedIntLoc ((intloc 2) := (intloc 0))) \/ {(intloc 3),(intloc 0)}) \/ {(intloc 4),(intloc 0)}) \/ {(intloc 5),(intloc 0)}) \/ {(intloc 6),(intloc 0)} by SF_MASTR:14
.= ((({(intloc 2),(intloc 0)} \/ {(intloc 3),(intloc 0)}) \/ {(intloc 4),(intloc 0)}) \/ {(intloc 5),(intloc 0)}) \/ {(intloc 6),(intloc 0)} by SF_MASTR:14
.= (({(intloc 2),(intloc 0)} \/ {(intloc 3),(intloc 0)}) \/ {(intloc 4),(intloc 0)}) \/ ({(intloc 5),(intloc 0)} \/ {(intloc 6),(intloc 0)}) by XBOOLE_1:4
.= (({(intloc 2),(intloc 0)} \/ {(intloc 3),(intloc 0)}) \/ {(intloc 4),(intloc 0)}) \/ {(intloc 0),(intloc 5),(intloc 6)} by ENUMSET1:87
.= ({(intloc 0),(intloc 2),(intloc 3)} \/ {(intloc 4),(intloc 0)}) \/ {(intloc 0),(intloc 5),(intloc 6)} by ENUMSET1:87
.= ({(intloc 0),(intloc 2),(intloc 3)} \/ {(intloc 4),(intloc 0)}) \/ ({(intloc 0)} \/ {(intloc 5),(intloc 6)}) by ENUMSET1:2
.= (({(intloc 0),(intloc 2),(intloc 3)} \/ {(intloc 4),(intloc 0)}) \/ {(intloc 0)}) \/ {(intloc 5),(intloc 6)} by XBOOLE_1:4
.= ({(intloc 0),(intloc 2),(intloc 3)} \/ ({(intloc 4),(intloc 0)} \/ {(intloc 0)})) \/ {(intloc 5),(intloc 6)} by XBOOLE_1:4
.= ({(intloc 0),(intloc 2),(intloc 3)} \/ {(intloc 4),(intloc 0),(intloc 0)}) \/ {(intloc 5),(intloc 6)} by ENUMSET1:3
.= ({(intloc 0),(intloc 2),(intloc 3)} \/ ({(intloc 0),(intloc 0)} \/ {(intloc 4)})) \/ {(intloc 5),(intloc 6)} by ENUMSET1:2
.= (({(intloc 0),(intloc 2),(intloc 3)} \/ {(intloc 0),(intloc 0)}) \/ {(intloc 4)}) \/ {(intloc 5),(intloc 6)} by XBOOLE_1:4
.= ({(intloc 0),(intloc 0),(intloc 0),(intloc 2),(intloc 3)} \/ {(intloc 4)}) \/ {(intloc 5),(intloc 6)} by ENUMSET1:8
.= ({(intloc 0),(intloc 2),(intloc 3)} \/ {(intloc 4)}) \/ {(intloc 5),(intloc 6)} by ENUMSET1:38
.= {(intloc 0),(intloc 2),(intloc 3),(intloc 4)} \/ {(intloc 5),(intloc 6)} by ENUMSET1:6
.= {(intloc 0),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:14
.= {(intloc 0)} \/ {(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:11 ;
set k7 = (intloc 1) :=len f;
set Ut = UsedIntLoc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))));
thus  UsedIntLoc (bubble-sort f) = (UsedIntLoc (((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) ";" ((intloc 1) :=len f))) \/ (UsedIntLoc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by SF_MASTR:27
.= ((UsedIntLoc ((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0)))) \/ (UsedIntLoc ((intloc 1) :=len f))) \/ (UsedIntLoc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by SF_MASTR:30
.= (({(intloc 0)} \/ {(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)}) \/ {(intloc 1)}) \/ (UsedIntLoc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by A5, SF_MASTR:18
.= (({(intloc 0)} \/ {(intloc 1)}) \/ {(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)}) \/ (UsedIntLoc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by XBOOLE_1:4
.= ({(intloc 0),(intloc 1)} \/ {(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)}) \/ (UsedIntLoc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by ENUMSET1:1
.= {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} \/ (UsedIntLoc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by ENUMSET1:17
.= {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} \/ ({(intloc 1),(intloc 0)} \/ {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)}) by A4, Th18
.= ({(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} \/ {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)}) \/ {(intloc 1),(intloc 0)} by XBOOLE_1:4
.= ({(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} \/ {(intloc 0),(intloc 1)}) \/ {(intloc 0),(intloc 1)} by ENUMSET1:17
.= {(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} \/ ({(intloc 0),(intloc 1)} \/ {(intloc 0),(intloc 1)}) by XBOOLE_1:4
.= {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by ENUMSET1:17 ; ::_thesis: verum
end;

theorem Th35: :: SCMBSORT:35
for f being FinSeq-Location holds UsedInt*Loc (bubble-sort f) = {f}
proof
let f be FinSeq-Location ; ::_thesis: UsedInt*Loc (bubble-sort f) = {f}
set i1 = (intloc 4) := (intloc 3);
set i2 = SubFrom ((intloc 3),(intloc 0));
set i3 = (intloc 5) := (f,(intloc 3));
set i4 = (intloc 6) := (f,(intloc 4));
set i5 = SubFrom ((intloc 6),(intloc 5));
set i6 = (f,(intloc 3)) := (intloc 6);
set i7 = (f,(intloc 4)) := (intloc 5);
set ifc = if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA));
set Sif = UsedInt*Loc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)));
set body2 = ((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)));
A1: UsedInt*Loc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))) = (UsedInt*Loc ((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5)))) \/ {} by SCMFSA9A:4, SCMFSA9A:10
.= (UsedInt*Loc (((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6)))) \/ (UsedInt*Loc ((f,(intloc 4)) := (intloc 5))) by SF_MASTR:46
.= (UsedInt*Loc (((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6)))) \/ {f} by SF_MASTR:33
.= ((UsedInt*Loc ((intloc 6) := (f,(intloc 4)))) \/ (UsedInt*Loc ((f,(intloc 3)) := (intloc 6)))) \/ {f} by SF_MASTR:47
.= ((UsedInt*Loc ((intloc 6) := (f,(intloc 4)))) \/ {f}) \/ {f} by SF_MASTR:33
.= ({f} \/ {f}) \/ {f} by SF_MASTR:33
.= {f} ;
A2: UsedInt*Loc (((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) = (UsedInt*Loc ((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5))))) \/ (UsedInt*Loc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:43
.= ((UsedInt*Loc (((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4))))) \/ (UsedInt*Loc (SubFrom ((intloc 6),(intloc 5))))) \/ (UsedInt*Loc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:46
.= ((UsedInt*Loc (((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4))))) \/ {}) \/ (UsedInt*Loc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:32
.= ((UsedInt*Loc ((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3))))) \/ (UsedInt*Loc ((intloc 6) := (f,(intloc 4))))) \/ (UsedInt*Loc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:46
.= ((UsedInt*Loc ((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3))))) \/ {f}) \/ (UsedInt*Loc (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SF_MASTR:33
.= (UsedInt*Loc ((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3))))) \/ ({f} \/ {f}) by A1, XBOOLE_1:4
.= ((UsedInt*Loc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ (UsedInt*Loc ((intloc 5) := (f,(intloc 3))))) \/ {f} by SF_MASTR:46
.= ((UsedInt*Loc (((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0))))) \/ {f}) \/ {f} by SF_MASTR:33
.= (((UsedInt*Loc ((intloc 4) := (intloc 3))) \/ (UsedInt*Loc (SubFrom ((intloc 3),(intloc 0))))) \/ {f}) \/ {f} by SF_MASTR:47
.= (((UsedInt*Loc ((intloc 4) := (intloc 3))) \/ {}) \/ {f}) \/ {f} by SF_MASTR:32
.= (({} \/ {}) \/ {f}) \/ {f} by SF_MASTR:32
.= {f} ;
set j1 = (intloc 2) := (intloc 1);
set j2 = SubFrom ((intloc 2),(intloc 0));
set j3 = (intloc 3) :=len f;
set Sfor = UsedInt*Loc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))));
set body1 = ((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))));
A3: UsedInt*Loc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))) = {f} by A2, Th20;
A4: UsedInt*Loc (((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) = (UsedInt*Loc ((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f))) \/ (UsedInt*Loc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:43
.= ((UsedInt*Loc (((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0))))) \/ (UsedInt*Loc ((intloc 3) :=len f))) \/ (UsedInt*Loc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:46
.= ((UsedInt*Loc (((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0))))) \/ {f}) \/ (UsedInt*Loc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:34
.= (((UsedInt*Loc ((intloc 2) := (intloc 1))) \/ (UsedInt*Loc (SubFrom ((intloc 2),(intloc 0))))) \/ {f}) \/ (UsedInt*Loc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:47
.= (({} \/ (UsedInt*Loc (SubFrom ((intloc 2),(intloc 0))))) \/ {f}) \/ (UsedInt*Loc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:32
.= (({} \/ {}) \/ {f}) \/ (UsedInt*Loc (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SF_MASTR:32
.= {f} by A3 ;
set k2 = (intloc 2) := (intloc 0);
set k3 = (intloc 3) := (intloc 0);
set k4 = (intloc 4) := (intloc 0);
set k5 = (intloc 5) := (intloc 0);
A5: UsedInt*Loc ((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) = (UsedInt*Loc (((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0)))) \/ (UsedInt*Loc ((intloc 6) := (intloc 0))) by SF_MASTR:46
.= (UsedInt*Loc (((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0)))) \/ {} by SF_MASTR:32
.= (UsedInt*Loc ((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0)))) \/ (UsedInt*Loc ((intloc 5) := (intloc 0))) by SF_MASTR:46
.= (UsedInt*Loc ((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0)))) \/ {} by SF_MASTR:32
.= (UsedInt*Loc (((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0)))) \/ (UsedInt*Loc ((intloc 4) := (intloc 0))) by SF_MASTR:46
.= (UsedInt*Loc (((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0)))) \/ {} by SF_MASTR:32
.= (UsedInt*Loc ((intloc 2) := (intloc 0))) \/ (UsedInt*Loc ((intloc 3) := (intloc 0))) by SF_MASTR:47
.= (UsedInt*Loc ((intloc 2) := (intloc 0))) \/ {} by SF_MASTR:32
.= {} by SF_MASTR:32 ;
set k7 = (intloc 1) :=len f;
set Ut = UsedInt*Loc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))));
thus  UsedInt*Loc (bubble-sort f) = (UsedInt*Loc (((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) ";" ((intloc 1) :=len f))) \/ (UsedInt*Loc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by SF_MASTR:43
.= ((UsedInt*Loc ((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0)))) \/ (UsedInt*Loc ((intloc 1) :=len f))) \/ (UsedInt*Loc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by SF_MASTR:46
.= {f} \/ (UsedInt*Loc (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by A5, SF_MASTR:34
.= {f} \/ {f} by A4, Th20
.= {f} ; ::_thesis: verum
end;

theorem :: SCMBSORT:36
canceled;

theorem :: SCMBSORT:37
canceled;

theorem Th38: :: SCMBSORT:38
for f being FinSeq-Location holds card (bubble-sort f) = 63
proof
let f be FinSeq-Location ; ::_thesis: card (bubble-sort f) = 63
set i1 = (intloc 4) := (intloc 3);
set i2 = SubFrom ((intloc 3),(intloc 0));
set i3 = (intloc 5) := (f,(intloc 3));
set i4 = (intloc 6) := (f,(intloc 4));
set i5 = SubFrom ((intloc 6),(intloc 5));
set i6 = (f,(intloc 3)) := (intloc 6);
set i7 = (f,(intloc 4)) := (intloc 5);
set ifc = if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA));
set Cif = card (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)));
set body2 = ((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)));
 card (Stop SCM+FSA) = 1 by COMPOS_1:4;
then A1: card (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))) = ((card ((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5)))) + 1) + 4 by SCMFSA8B:12
.= (6 + 1) + 4 by Th23
.= 11 ;
A2: card (((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) = (card ((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5))))) + (card (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SCMFSA6A:21
.= (card (((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" (((intloc 6) := (f,(intloc 4))) ";" (SubFrom ((intloc 6),(intloc 5)))))) + (card (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SCMFSA6A:28
.= ((card ((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3))))) + (card (((intloc 6) := (f,(intloc 4))) ";" (SubFrom ((intloc 6),(intloc 5)))))) + (card (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SCMFSA6A:21
.= (6 + (card (((intloc 6) := (f,(intloc 4))) ";" (SubFrom ((intloc 6),(intloc 5)))))) + (card (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by Th23
.= (6 + 4) + (card (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))) by SCMFSA6A:35
.= 21 by A1 ;
set j1 = (intloc 2) := (intloc 1);
set j2 = SubFrom ((intloc 2),(intloc 0));
set j3 = (intloc 3) :=len f;
set body1 = ((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))));
A3: card (((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) = (card ((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f))) + (card (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by SCMFSA6A:21
.= 6 + (card (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))) by Th23
.= 6 + (21 + 12) by A2, Th22
.= 39 ;
set k2 = (intloc 2) := (intloc 0);
set k3 = (intloc 3) := (intloc 0);
set k4 = (intloc 4) := (intloc 0);
set k5 = (intloc 5) := (intloc 0);
set k6 = (intloc 6) := (intloc 0);
A4: card ((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) = (card (((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0)))) + 2 by SCMFSA6A:34
.= ((card ((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0)))) + 2) + 2 by SCMFSA6A:34
.= (card ((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0)))) + 4
.= 6 + 4 by Th23
.= 10 ;
set k7 = (intloc 1) :=len f;
set Ct = card (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))));
A5: card (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))))) = 39 + 12 by A3, Th22;
thus  card (bubble-sort f) = (card (((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) ";" ((intloc 1) :=len f))) + (card (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by SCMFSA6A:21
.= (10 + 2) + (card (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len f)) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := (f,(intloc 3)))) ";" ((intloc 6) := (f,(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := (f,(intloc 4))) ";" ((f,(intloc 3)) := (intloc 6))) ";" ((f,(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by A4, SCMFSA6A:34
.= 63 by A5 ; ::_thesis: verum
end;

theorem Th39: :: SCMBSORT:39
for P being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P holds
for f being FinSeq-Location
for k being Element of NAT st k < 63 holds
Bubble-Sort-Algorithm . k = P . k
proof
let P be Instruction-Sequence of SCM+FSA; ::_thesis: ( Bubble-Sort-Algorithm c= P implies for f being FinSeq-Location
for k being Element of NAT st k < 63 holds
Bubble-Sort-Algorithm . k = P . k )
assume A1: Bubble-Sort-Algorithm c= P ; ::_thesis: for f being FinSeq-Location
for k being Element of NAT st k < 63 holds
Bubble-Sort-Algorithm . k = P . k
let f be FinSeq-Location ; ::_thesis: for k being Element of NAT st k < 63 holds
Bubble-Sort-Algorithm . k = P . k
let k be Element of NAT ; ::_thesis: ( k < 63 implies Bubble-Sort-Algorithm . k = P . k )
assume A2: k < 63 ; ::_thesis: Bubble-Sort-Algorithm . k = P . k
 card (bubble-sort (fsloc 0)) = 63 by Th38;
then  k in dom Bubble-Sort-Algorithm by A2, AFINSQ_1:66;
hence  Bubble-Sort-Algorithm . k = P . k by A1, GRFUNC_1:2; ::_thesis: verum
end;

Lm20:  for P being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P holds
( P . 0 = (intloc 2) := (intloc 0) & P . 1 = goto 2 & P . 2 = (intloc 3) := (intloc 0) & P . 3 = goto 4 & P . 4 = (intloc 4) := (intloc 0) & P . 5 = goto 6 & P . 6 = (intloc 5) := (intloc 0) & P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
proof
set f0 = fsloc 0;
set TT = Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))));
set q = Bubble-Sort-Algorithm ;
let P be Instruction-Sequence of SCM+FSA; ::_thesis: ( Bubble-Sort-Algorithm c= P implies ( P . 0 = (intloc 2) := (intloc 0) & P . 1 = goto 2 & P . 2 = (intloc 3) := (intloc 0) & P . 3 = goto 4 & P . 4 = (intloc 4) := (intloc 0) & P . 5 = goto 6 & P . 6 = (intloc 5) := (intloc 0) & P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 ) )
assume A1: Bubble-Sort-Algorithm c= P ; ::_thesis: ( P . 0 = (intloc 2) := (intloc 0) & P . 1 = goto 2 & P . 2 = (intloc 3) := (intloc 0) & P . 3 = goto 4 & P . 4 = (intloc 4) := (intloc 0) & P . 5 = goto 6 & P . 6 = (intloc 5) := (intloc 0) & P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
set W2 = (intloc 2) := (intloc 0);
set W3 = (intloc 3) := (intloc 0);
set W4 = (intloc 4) := (intloc 0);
set W5 = (intloc 5) := (intloc 0);
set W6 = (intloc 6) := (intloc 0);
set W7 = (intloc 1) :=len (fsloc 0);
set T7 = ((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))));
set T6 = ((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))))));
set T5 = ((intloc 5) := (intloc 0)) ";" (((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))));
set T4 = ((intloc 4) := (intloc 0)) ";" (((intloc 5) := (intloc 0)) ";" (((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))))))));
set T3 = ((intloc 3) := (intloc 0)) ";" (((intloc 4) := (intloc 0)) ";" (((intloc 5) := (intloc 0)) ";" (((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))))));
set X3 = ((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0));
set X4 = (((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0));
set X5 = ((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0));
set X6 = (((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0));
A2: Bubble-Sort-Algorithm = ((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))) by SCMFSA6A:27;
then A3: Bubble-Sort-Algorithm = (((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" (((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))))))) by SCMFSA6A:27;
then A4: Bubble-Sort-Algorithm = ((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" (((intloc 5) := (intloc 0)) ";" (((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))))) by SCMFSA6A:27;
then  Bubble-Sort-Algorithm = (((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" (((intloc 4) := (intloc 0)) ";" (((intloc 5) := (intloc 0)) ";" (((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))))))))) by SCMFSA6A:27;
then  Bubble-Sort-Algorithm = ((intloc 2) := (intloc 0)) ";" (((intloc 3) := (intloc 0)) ";" (((intloc 4) := (intloc 0)) ";" (((intloc 5) := (intloc 0)) ";" (((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))))))) by SCMFSA6A:31;
then A5: Bubble-Sort-Algorithm = (Macro ((intloc 2) := (intloc 0))) ";" (((intloc 3) := (intloc 0)) ";" (((intloc 4) := (intloc 0)) ";" (((intloc 5) := (intloc 0)) ";" (((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA)))))))))))))) by SCMFSA6A:def_5;
A6: Bubble-Sort-Algorithm = ((Macro ((intloc 2) := (intloc 0))) ";" ((intloc 3) := (intloc 0))) ";" (((intloc 4) := (intloc 0)) ";" (((intloc 5) := (intloc 0)) ";" (((intloc 6) := (intloc 0)) ";" (((intloc 1) :=len (fsloc 0)) ";" (Times ((intloc 1),(((((intloc 2) := (intloc 1)) ";" (SubFrom ((intloc 2),(intloc 0)))) ";" ((intloc 3) :=len (fsloc 0))) ";" (Times ((intloc 2),(((((((intloc 4) := (intloc 3)) ";" (SubFrom ((intloc 3),(intloc 0)))) ";" ((intloc 5) := ((fsloc 0),(intloc 3)))) ";" ((intloc 6) := ((fsloc 0),(intloc 4)))) ";" (SubFrom ((intloc 6),(intloc 5)))) ";" (if>0 ((intloc 6),((((intloc 6) := ((fsloc 0),(intloc 4))) ";" (((fsloc 0),(intloc 3)) := (intloc 6))) ";" (((fsloc 0),(intloc 4)) := (intloc 5))),(Stop SCM+FSA))))))))))))) by A5, SCMFSA6A:27;
A7: dom (Macro ((intloc 2) := (intloc 0))) = {0,1} by COMPOS_1:61;
then A8: 0 in dom (Macro ((intloc 2) := (intloc 0))) by TARSKI:def_2;
A9: 1 in dom (Macro ((intloc 2) := (intloc 0))) by A7, TARSKI:def_2;
thus P . 0 = Bubble-Sort-Algorithm . 0 by A1, Th39
.= (Directed (Macro ((intloc 2) := (intloc 0)))) . 0 by A8, A5, SCMFSA8A:14
.= (intloc 2) := (intloc 0) by SCMFSA7B:1 ; ::_thesis: ( P . 1 = goto 2 & P . 2 = (intloc 3) := (intloc 0) & P . 3 = goto 4 & P . 4 = (intloc 4) := (intloc 0) & P . 5 = goto 6 & P . 6 = (intloc 5) := (intloc 0) & P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
thus P . 1 = Bubble-Sort-Algorithm . 1 by A1, Th39
.= (Directed (Macro ((intloc 2) := (intloc 0)))) . 1 by A9, A5, SCMFSA8A:14
.= goto 2 by SCMFSA7B:2 ; ::_thesis: ( P . 2 = (intloc 3) := (intloc 0) & P . 3 = goto 4 & P . 4 = (intloc 4) := (intloc 0) & P . 5 = goto 6 & P . 6 = (intloc 5) := (intloc 0) & P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
A10: card (Macro ((intloc 2) := (intloc 0))) = 2 by COMPOS_1:56;
thus P . 2 = Bubble-Sort-Algorithm . 2 by A1, Th39
.= (intloc 3) := (intloc 0) by A6, A10, Th27 ; ::_thesis: ( P . 3 = goto 4 & P . 4 = (intloc 4) := (intloc 0) & P . 5 = goto 6 & P . 6 = (intloc 5) := (intloc 0) & P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
thus P . 3 = Bubble-Sort-Algorithm . (2 + 1) by A1, Th39
.= goto (2 + 2) by A6, A10, Th28
.= goto 4 ; ::_thesis: ( P . 4 = (intloc 4) := (intloc 0) & P . 5 = goto 6 & P . 6 = (intloc 5) := (intloc 0) & P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
A11: card (((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) = 4 by SCMFSA6A:35;
thus P . 4 = Bubble-Sort-Algorithm . 4 by A1, Th39
.= (intloc 4) := (intloc 0) by A4, A11, Th27 ; ::_thesis: ( P . 5 = goto 6 & P . 6 = (intloc 5) := (intloc 0) & P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
thus P . 5 = Bubble-Sort-Algorithm . (4 + 1) by A1, Th39
.= goto (4 + 2) by A4, A11, Th28
.= goto 6 ; ::_thesis: ( P . 6 = (intloc 5) := (intloc 0) & P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
A12: card ((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) = 6 by Th23;
thus P . 6 = Bubble-Sort-Algorithm . 6 by A1, Th39
.= (intloc 5) := (intloc 0) by A3, A12, Th27 ; ::_thesis: ( P . 7 = goto 8 & P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
thus P . 7 = Bubble-Sort-Algorithm . (6 + 1) by A1, Th39
.= goto (6 + 2) by A3, A12, Th28
.= goto 8 ; ::_thesis: ( P . 8 = (intloc 6) := (intloc 0) & P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
A13: card (((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) = 6 + 2 by A12, SCMFSA6A:34;
thus P . 8 = Bubble-Sort-Algorithm . 8 by A1, Th39
.= (intloc 6) := (intloc 0) by A2, A13, Th27 ; ::_thesis: ( P . 9 = goto 10 & P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
thus P . 9 = Bubble-Sort-Algorithm . (8 + 1) by A1, Th39
.= goto (8 + 2) by A2, A13, Th28
.= goto 10 ; ::_thesis: ( P . 10 = (intloc 1) :=len (fsloc 0) & P . 11 = goto 12 )
A14: card ((((((intloc 2) := (intloc 0)) ";" ((intloc 3) := (intloc 0))) ";" ((intloc 4) := (intloc 0))) ";" ((intloc 5) := (intloc 0))) ";" ((intloc 6) := (intloc 0))) = 8 + 2 by A13, SCMFSA6A:34;
thus P . 10 = Bubble-Sort-Algorithm . 10 by A1, Th39
.= (intloc 1) :=len (fsloc 0) by A14, Th27 ; ::_thesis: P . 11 = goto 12
thus P . 11 = Bubble-Sort-Algorithm . (10 + 1) by A1, Th39
.= goto (10 + 2) by A14, Th28
.= goto 12 ; ::_thesis: verum
end;

Lm21:  for s being 0 -started State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P holds
( (Comput (P,s,1)) . (IC ) = 1 & (Comput (P,s,1)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,1)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,2)) . (IC ) = 2 & (Comput (P,s,2)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
proof
let s be 0 -started State of SCM+FSA; ::_thesis: for P being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P holds
( (Comput (P,s,1)) . (IC ) = 1 & (Comput (P,s,1)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,1)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,2)) . (IC ) = 2 & (Comput (P,s,2)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
let P be Instruction-Sequence of SCM+FSA; ::_thesis: ( Bubble-Sort-Algorithm c= P implies ( (Comput (P,s,1)) . (IC ) = 1 & (Comput (P,s,1)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,1)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,2)) . (IC ) = 2 & (Comput (P,s,2)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) ) )
assume A1: Bubble-Sort-Algorithm c= P ; ::_thesis: ( (Comput (P,s,1)) . (IC ) = 1 & (Comput (P,s,1)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,1)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,2)) . (IC ) = 2 & (Comput (P,s,2)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A2: IC (Comput (P,s,0)) = 0 by MEMSTR_0:def_11;
then A3: Comput (P,s,(0 + 1)) = Exec ((P . 0),(Comput (P,s,0))) by EXTPRO_1:6
.= Exec (((intloc 2) := (intloc 0)),(Comput (P,s,0))) by A1, Lm20 ;
hence (Comput (P,s,1)) . (IC ) = succ (IC (Comput (P,s,0))) by SCMFSA_2:63
.= 1 by A2 ;
::_thesis: ( (Comput (P,s,1)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,1)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,2)) . (IC ) = 2 & (Comput (P,s,2)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
then A4: IC (Comput (P,s,1)) = 1 ;
A5: (Comput (P,s,1)) . (intloc 2) = s . (intloc 0) by A3, SCMFSA_2:63;
thus A6: (Comput (P,s,1)) . (intloc 0) = s . (intloc 0) by A3, Lm1, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,1)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,2)) . (IC ) = 2 & (Comput (P,s,2)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A7: (Comput (P,s,1)) . (fsloc 0) = s . (fsloc 0) by A3, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,2)) . (IC ) = 2 & (Comput (P,s,2)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A8: Comput (P,s,(1 + 1)) = Exec ((P . 1),(Comput (P,s,1))) by A4, EXTPRO_1:6
.= Exec ((goto 2),(Comput (P,s,1))) by A1, Lm20 ;
hence A9: (Comput (P,s,2)) . (IC ) = 2 by SCMFSA_2:69; ::_thesis: ( (Comput (P,s,2)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A10: IC (Comput (P,s,2)) = 2 by A8, SCMFSA_2:69;
thus A11: (Comput (P,s,2)) . (intloc 0) = s . (intloc 0) by A6, A8, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A12: (Comput (P,s,2)) . (fsloc 0) = s . (fsloc 0) by A7, A8, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,3)) . (IC ) = 3 & (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A13: (Comput (P,s,2)) . (intloc 2) = s . (intloc 0) by A5, A8, SCMFSA_2:69;
A14: Comput (P,s,(2 + 1)) = Exec ((P . 2),(Comput (P,s,2))) by A10, EXTPRO_1:6
.= Exec (((intloc 3) := (intloc 0)),(Comput (P,s,2))) by A1, Lm20 ;
hence (Comput (P,s,3)) . (IC ) = succ (IC (Comput (P,s,2))) by SCMFSA_2:63
.= 3 by A9 ;
::_thesis: ( (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
then A15: IC (Comput (P,s,3)) = 3 ;
A16: (Comput (P,s,3)) . (intloc 3) = s . (intloc 0) by A11, A14, SCMFSA_2:63;
thus A17: (Comput (P,s,3)) . (intloc 0) = s . (intloc 0) by A11, A14, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A18: (Comput (P,s,3)) . (fsloc 0) = s . (fsloc 0) by A12, A14, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,4)) . (IC ) = 4 & (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A19: (Comput (P,s,3)) . (intloc 2) = s . (intloc 0) by A13, A14, Lm10, SCMFSA_2:63;
A20: Comput (P,s,(3 + 1)) = Exec ((P . 3),(Comput (P,s,3))) by A15, EXTPRO_1:6
.= Exec ((goto 4),(Comput (P,s,3))) by A1, Lm20 ;
hence A21: (Comput (P,s,4)) . (IC ) = 4 by SCMFSA_2:69; ::_thesis: ( (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A22: IC (Comput (P,s,4)) = 4 by A20, SCMFSA_2:69;
thus A23: (Comput (P,s,4)) . (intloc 0) = s . (intloc 0) by A17, A20, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A24: (Comput (P,s,4)) . (fsloc 0) = s . (fsloc 0) by A18, A20, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,5)) . (IC ) = 5 & (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A25: (Comput (P,s,4)) . (intloc 2) = s . (intloc 0) by A19, A20, SCMFSA_2:69;
A26: (Comput (P,s,4)) . (intloc 3) = s . (intloc 0) by A16, A20, SCMFSA_2:69;
A27: Comput (P,s,(4 + 1)) = Exec ((P . 4),(Comput (P,s,4))) by A22, EXTPRO_1:6
.= Exec (((intloc 4) := (intloc 0)),(Comput (P,s,4))) by A1, Lm20 ;
hence (Comput (P,s,5)) . (IC ) = succ (IC (Comput (P,s,4))) by SCMFSA_2:63
.= 5 by A21 ;
::_thesis: ( (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
then A28: IC (Comput (P,s,5)) = 5 ;
A29: (Comput (P,s,5)) . (intloc 4) = s . (intloc 0) by A23, A27, SCMFSA_2:63;
thus A30: (Comput (P,s,5)) . (intloc 0) = s . (intloc 0) by A23, A27, Lm2, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A31: (Comput (P,s,5)) . (fsloc 0) = s . (fsloc 0) by A24, A27, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,6)) . (IC ) = 6 & (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A32: (Comput (P,s,5)) . (intloc 2) = s . (intloc 0) by A25, A27, Lm11, SCMFSA_2:63;
A33: (Comput (P,s,5)) . (intloc 3) = s . (intloc 0) by A26, A27, Lm14, SCMFSA_2:63;
A34: Comput (P,s,(5 + 1)) = Exec ((P . 5),(Comput (P,s,5))) by A28, EXTPRO_1:6
.= Exec ((goto 6),(Comput (P,s,5))) by A1, Lm20 ;
hence A35: (Comput (P,s,6)) . (IC ) = 6 by SCMFSA_2:69; ::_thesis: ( (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A36: IC (Comput (P,s,6)) = 6 by A34, SCMFSA_2:69;
thus A37: (Comput (P,s,6)) . (intloc 0) = s . (intloc 0) by A30, A34, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A38: (Comput (P,s,6)) . (fsloc 0) = s . (fsloc 0) by A31, A34, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,7)) . (IC ) = 7 & (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A39: (Comput (P,s,6)) . (intloc 2) = s . (intloc 0) by A32, A34, SCMFSA_2:69;
A40: (Comput (P,s,6)) . (intloc 3) = s . (intloc 0) by A33, A34, SCMFSA_2:69;
A41: (Comput (P,s,6)) . (intloc 4) = s . (intloc 0) by A29, A34, SCMFSA_2:69;
A42: Comput (P,s,(6 + 1)) = Exec ((P . 6),(Comput (P,s,6))) by A36, EXTPRO_1:6
.= Exec (((intloc 5) := (intloc 0)),(Comput (P,s,6))) by A1, Lm20 ;
hence (Comput (P,s,7)) . (IC ) = succ (IC (Comput (P,s,6))) by SCMFSA_2:63
.= 7 by A35 ;
::_thesis: ( (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
then A43: IC (Comput (P,s,7)) = 7 ;
A44: (Comput (P,s,7)) . (intloc 5) = s . (intloc 0) by A37, A42, SCMFSA_2:63;
thus A45: (Comput (P,s,7)) . (intloc 0) = s . (intloc 0) by A37, A42, Lm3, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A46: (Comput (P,s,7)) . (fsloc 0) = s . (fsloc 0) by A38, A42, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,8)) . (IC ) = 8 & (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A47: (Comput (P,s,7)) . (intloc 2) = s . (intloc 0) by A39, A42, Lm12, SCMFSA_2:63;
A48: (Comput (P,s,7)) . (intloc 3) = s . (intloc 0) by A40, A42, Lm15, SCMFSA_2:63;
A49: (Comput (P,s,7)) . (intloc 4) = s . (intloc 0) by A41, A42, Lm17, SCMFSA_2:63;
A50: Comput (P,s,(7 + 1)) = Exec ((P . 7),(Comput (P,s,7))) by A43, EXTPRO_1:6
.= Exec ((goto 8),(Comput (P,s,7))) by A1, Lm20 ;
hence A51: (Comput (P,s,8)) . (IC ) = 8 by SCMFSA_2:69; ::_thesis: ( (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A52: IC (Comput (P,s,8)) = 8 by A50, SCMFSA_2:69;
thus A53: (Comput (P,s,8)) . (intloc 0) = s . (intloc 0) by A45, A50, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A54: (Comput (P,s,8)) . (fsloc 0) = s . (fsloc 0) by A46, A50, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,9)) . (IC ) = 9 & (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A55: (Comput (P,s,8)) . (intloc 2) = s . (intloc 0) by A47, A50, SCMFSA_2:69;
A56: (Comput (P,s,8)) . (intloc 3) = s . (intloc 0) by A48, A50, SCMFSA_2:69;
A57: (Comput (P,s,8)) . (intloc 4) = s . (intloc 0) by A49, A50, SCMFSA_2:69;
A58: (Comput (P,s,8)) . (intloc 5) = s . (intloc 0) by A44, A50, SCMFSA_2:69;
A59: Comput (P,s,(8 + 1)) = Exec ((P . 8),(Comput (P,s,8))) by A52, EXTPRO_1:6
.= Exec (((intloc 6) := (intloc 0)),(Comput (P,s,8))) by A1, Lm20 ;
hence (Comput (P,s,9)) . (IC ) = succ (IC (Comput (P,s,8))) by SCMFSA_2:63
.= 9 by A51 ;
::_thesis: ( (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
then A60: IC (Comput (P,s,9)) = 9 ;
A61: (Comput (P,s,9)) . (intloc 6) = s . (intloc 0) by A53, A59, SCMFSA_2:63;
thus A62: (Comput (P,s,9)) . (intloc 0) = s . (intloc 0) by A53, A59, Lm4, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A63: (Comput (P,s,9)) . (fsloc 0) = s . (fsloc 0) by A54, A59, SCMFSA_2:63; ::_thesis: ( (Comput (P,s,10)) . (IC ) = 10 & (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A64: (Comput (P,s,9)) . (intloc 2) = s . (intloc 0) by A55, A59, Lm13, SCMFSA_2:63;
A65: (Comput (P,s,9)) . (intloc 3) = s . (intloc 0) by A56, A59, Lm16, SCMFSA_2:63;
A66: (Comput (P,s,9)) . (intloc 4) = s . (intloc 0) by A57, A59, Lm18, SCMFSA_2:63;
A67: (Comput (P,s,9)) . (intloc 5) = s . (intloc 0) by A58, A59, Lm19, SCMFSA_2:63;
A68: Comput (P,s,(9 + 1)) = Exec ((P . 9),(Comput (P,s,9))) by A60, EXTPRO_1:6
.= Exec ((goto 10),(Comput (P,s,9))) by A1, Lm20 ;
hence A69: (Comput (P,s,10)) . (IC ) = 10 by SCMFSA_2:69; ::_thesis: ( (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A70: IC (Comput (P,s,10)) = 10 by A68, SCMFSA_2:69;
thus A71: (Comput (P,s,10)) . (intloc 0) = s . (intloc 0) by A62, A68, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus A72: (Comput (P,s,10)) . (fsloc 0) = s . (fsloc 0) by A63, A68, SCMFSA_2:69; ::_thesis: ( (Comput (P,s,11)) . (IC ) = 11 & (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
A73: (Comput (P,s,10)) . (intloc 2) = s . (intloc 0) by A64, A68, SCMFSA_2:69;
A74: (Comput (P,s,10)) . (intloc 3) = s . (intloc 0) by A65, A68, SCMFSA_2:69;
A75: (Comput (P,s,10)) . (intloc 4) = s . (intloc 0) by A66, A68, SCMFSA_2:69;
A76: (Comput (P,s,10)) . (intloc 5) = s . (intloc 0) by A67, A68, SCMFSA_2:69;
A77: (Comput (P,s,10)) . (intloc 6) = s . (intloc 0) by A61, A68, SCMFSA_2:69;
A78: Comput (P,s,(10 + 1)) = Exec ((P . 10),(Comput (P,s,10))) by A70, EXTPRO_1:6
.= Exec (((intloc 1) :=len (fsloc 0)),(Comput (P,s,10))) by A1, Lm20 ;
hence (Comput (P,s,11)) . (IC ) = succ (IC (Comput (P,s,10))) by SCMFSA_2:74
.= 11 by A69 ;
::_thesis: ( (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) & (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus  (Comput (P,s,11)) . (intloc 0) = s . (intloc 0) by A71, A78, SCMFSA_2:74; ::_thesis: ( (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) & (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus  (Comput (P,s,11)) . (fsloc 0) = s . (fsloc 0) by A72, A78, SCMFSA_2:74; ::_thesis: ( (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) & (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus  (Comput (P,s,11)) . (intloc 1) = len (s . (fsloc 0)) by A72, A78, SCMFSA_2:74; ::_thesis: ( (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus  (Comput (P,s,11)) . (intloc 2) = s . (intloc 0) by A73, A78, Lm5, SCMFSA_2:74; ::_thesis: ( (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus  (Comput (P,s,11)) . (intloc 3) = s . (intloc 0) by A74, A78, Lm6, SCMFSA_2:74; ::_thesis: ( (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus  (Comput (P,s,11)) . (intloc 4) = s . (intloc 0) by A75, A78, Lm7, SCMFSA_2:74; ::_thesis: ( (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) & (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) )
thus  (Comput (P,s,11)) . (intloc 5) = s . (intloc 0) by A76, A78, Lm8, SCMFSA_2:74; ::_thesis: (Comput (P,s,11)) . (intloc 6) = s . (intloc 0)
thus  (Comput (P,s,11)) . (intloc 6) = s . (intloc 0) by A77, A78, Lm9, SCMFSA_2:74; ::_thesis: verum
end;

Lm22:  not ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))) destroys intloc (1 + 1)
proof
A1: not (intloc (3 + 1)) := (intloc (2 + 1)) destroys intloc (1 + 1) by SCMFSA7B:6, SCMFSA_2:101;
A2: not SubFrom ((intloc (2 + 1)),(intloc 0)) destroys intloc (1 + 1) by SCMFSA7B:8, SCMFSA_2:101;
A3: not (intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))) destroys intloc (1 + 1) by SCMFSA7B:14, SCMFSA_2:101;
A4: not (intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))) destroys intloc (1 + 1) by SCMFSA7B:14, SCMFSA_2:101;
A5: not ((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)) destroys intloc (1 + 1) by SCMFSA7B:15;
A6: not ((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)) destroys intloc (1 + 1) by SCMFSA7B:15;
A7: not Stop SCM+FSA destroys intloc (1 + 1) by SCMFSA8C:56;
 not (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1))) destroys intloc (1 + 1) by A4, A5, A6, SCMFSA8C:54, SCMFSA8C:55;
then A8: not if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)) destroys intloc (1 + 1) by A7, SCMFSA8C:88;
 not (((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1)))) destroys intloc (1 + 1) by A1, A2, A3, SCMFSA8C:54, SCMFSA8C:55;
then  not ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) destroys intloc (1 + 1) by Lm13, SCMFSA7B:14, SCMFSA8C:54;
then  not (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1)))) destroys intloc (1 + 1) by Lm13, SCMFSA7B:8, SCMFSA8C:54;
hence  not ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))) destroys intloc (1 + 1) by A8, SCMFSA8C:52; ::_thesis: verum
end;

Lm23:  ( Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) is good & Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) is InitHalting )
proof
thus  Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) is good ; ::_thesis: Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) is InitHalting
let s be State of SCM+FSA; :: according to SCM_HALT:def_2 ::_thesis: ( not Initialize K570((intloc 0),1) c= s or for b1 being set holds
( not Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) c= b1 or b1 halts_on s ) )
assume A1: Initialize ((intloc 0) .--> 1) c= s ; ::_thesis: for b1 being set holds
( not Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCM+FSA; ::_thesis: ( not Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) c= P or P halts_on s )
assume A2: Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) c= P ; ::_thesis: P halts_on s
A3: P +* (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))) = P by A2, FUNCT_4:98;
A4: dom ((intloc 0) .--> 1) = {(intloc 0)} by FUNCOP_1:13;
A5: dom ((intloc 0) .--> 1) misses {(IC )} by A4, SCMFSA_2:56, ZFMISC_1:11;
 Start-At (0,SCM+FSA) c= Initialize ((intloc 0) .--> 1) by FUNCT_4:25;
then A6: s = Initialize s by A1, FUNCT_4:98, XBOOLE_1:1;
A7: intloc 0 in dom ((intloc 0) .--> 1) by A4, TARSKI:def_1;
 dom ((intloc 0) .--> 1) misses dom (Start-At (0,SCM+FSA)) by A5, FUNCOP_1:13;
then  (intloc 0) .--> 1 c= Initialize ((intloc 0) .--> 1) by FUNCT_4:32;
then  (intloc 0) .--> 1 c= s by A1, XBOOLE_1:1;
then A8: s . (intloc 0) = ((intloc 0) .--> 1) . (intloc 0) by A7, GRFUNC_1:2
.= 1 by FUNCOP_1:72 ;
 Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) is_halting_on s,P by Lm22, A8, SCM_HALT:62;
hence  P halts_on s by A6, A3, SCMFSA7B:def_7; ::_thesis: verum
end;

Lm24:  not ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))) destroys intloc (0 + 1)
proof
A1: not (intloc (3 + 1)) := (intloc (2 + 1)) destroys intloc (0 + 1) by SCMFSA7B:6, SCMFSA_2:101;
A2: not SubFrom ((intloc (2 + 1)),(intloc 0)) destroys intloc (0 + 1) by SCMFSA7B:8, SCMFSA_2:101;
A3: not (intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))) destroys intloc (0 + 1) by SCMFSA7B:14, SCMFSA_2:101;
A4: not (intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))) destroys intloc (0 + 1) by SCMFSA7B:14, SCMFSA_2:101;
A5: not ((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)) destroys intloc (0 + 1) by SCMFSA7B:15;
A6: not ((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)) destroys intloc (0 + 1) by SCMFSA7B:15;
A7: not Stop SCM+FSA destroys intloc (0 + 1) by SCMFSA8C:56;
 not (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1))) destroys intloc (0 + 1) by A4, A5, A6, SCMFSA8C:54, SCMFSA8C:55;
then A8: not if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)) destroys intloc (0 + 1) by A7, SCMFSA8C:88;
 not (((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1)))) destroys intloc (0 + 1) by A1, A2, A3, SCMFSA8C:54, SCMFSA8C:55;
then  not ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) destroys intloc (0 + 1) by Lm9, SCMFSA7B:14, SCMFSA8C:54;
then  not (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1)))) destroys intloc (0 + 1) by Lm9, SCMFSA7B:8, SCMFSA8C:54;
hence  not ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))) destroys intloc (0 + 1) by A8, SCMFSA8C:52; ::_thesis: verum
end;

Lm25:  not ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))) destroys intloc (0 + 1)
proof
A1: not (intloc (1 + 1)) := (intloc (0 + 1)) destroys intloc (0 + 1) by SCMFSA7B:6, SCMFSA_2:101;
A2: not SubFrom ((intloc (1 + 1)),(intloc 0)) destroys intloc (0 + 1) by SCMFSA7B:8, SCMFSA_2:101;
A3: not (intloc (2 + 1)) :=len (fsloc 0) destroys intloc (0 + 1) by SCMFSA7B:16, SCMFSA_2:101;
A4: not Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) destroys intloc (0 + 1) by Lm24, Th1, SCMFSA_2:101;
 not (((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0)) destroys intloc (0 + 1) by A1, A2, A3, SCMFSA8C:54, SCMFSA8C:55;
hence  not ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))) destroys intloc (0 + 1) by A4, SCMFSA8C:52; ::_thesis: verum
end;

Lm26:  ( Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))))) is good & Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))))) is InitHalting )
proof
thus  Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))))) is good ; ::_thesis: Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))))) is InitHalting
let s be State of SCM+FSA; :: according to SCM_HALT:def_2 ::_thesis: ( not Initialize K570((intloc 0),1) c= s or for b1 being set holds
( not Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))))) c= b1 or b1 halts_on s ) )
assume A1: Initialize ((intloc 0) .--> 1) c= s ; ::_thesis: for b1 being set holds
( not Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))))) c= b1 or b1 halts_on s )
let P be Instruction-Sequence of SCM+FSA; ::_thesis: ( not Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))))) c= P or P halts_on s )
assume A2: Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))))) c= P ; ::_thesis: P halts_on s
A3: P +* (Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))) = P by A2, FUNCT_4:98;
A4: dom ((intloc 0) .--> 1) = {(intloc 0)} by FUNCOP_1:13;
A5: dom ((intloc 0) .--> 1) misses {(IC )} by A4, SCMFSA_2:56, ZFMISC_1:11;
 Start-At (0,SCM+FSA) c= Initialize ((intloc 0) .--> 1) by FUNCT_4:25;
then A6: s = Initialize s by A1, FUNCT_4:98, XBOOLE_1:1;
A7: intloc 0 in dom ((intloc 0) .--> 1) by A4, TARSKI:def_1;
 dom ((intloc 0) .--> 1) misses dom (Start-At (0,SCM+FSA)) by A5, FUNCOP_1:13;
then  (intloc 0) .--> 1 c= Initialize ((intloc 0) .--> 1) by FUNCT_4:32;
then  (intloc 0) .--> 1 c= s by A1, XBOOLE_1:1;
then A8: s . (intloc 0) = ((intloc 0) .--> 1) . (intloc 0) by A7, GRFUNC_1:2
.= 1 by FUNCOP_1:72 ;
reconsider TT = Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))) as InitHalting good Program of by Lm23;
 ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))) = ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" TT ;
then  Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))))) is_halting_on s,P by Lm25, A8, SCM_HALT:62;
hence  P halts_on s by A6, A3, SCMFSA7B:def_7; ::_thesis: verum
end;

theorem :: SCMBSORT:40
( bubble-sort (fsloc 0) is keepInt0_1 & bubble-sort (fsloc 0) is InitHalting ) by Lm26;

Lm27:  for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA holds
( ( s . (intloc (5 + 1)) > 0 implies (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = ((s . (fsloc 0)) +* ((abs (s . (intloc (2 + 1)))),((s . (fsloc 0)) /. (abs (s . (intloc (3 + 1))))))) +* ((abs (s . (intloc (3 + 1)))),(s . (intloc (4 + 1)))) ) & ( s . (intloc (5 + 1)) <= 0 implies (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = s . (fsloc 0) ) )
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA holds
( ( s . (intloc (5 + 1)) > 0 implies (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = ((s . (fsloc 0)) +* ((abs (s . (intloc (2 + 1)))),((s . (fsloc 0)) /. (abs (s . (intloc (3 + 1))))))) +* ((abs (s . (intloc (3 + 1)))),(s . (intloc (4 + 1)))) ) & ( s . (intloc (5 + 1)) <= 0 implies (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = s . (fsloc 0) ) )
let s be State of SCM+FSA; ::_thesis: ( ( s . (intloc (5 + 1)) > 0 implies (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = ((s . (fsloc 0)) +* ((abs (s . (intloc (2 + 1)))),((s . (fsloc 0)) /. (abs (s . (intloc (3 + 1))))))) +* ((abs (s . (intloc (3 + 1)))),(s . (intloc (4 + 1)))) ) & ( s . (intloc (5 + 1)) <= 0 implies (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = s . (fsloc 0) ) )
set s0 = Initialized s;
set s1 = Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s));
set s2 = IExec ((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))),p,s);
A1: (Initialized s) . (fsloc 0) = s . (fsloc 0) by SCMFSA_M:37;
 (Initialized s) . (intloc (3 + 1)) = s . (intloc (3 + 1)) by SCMFSA_M:37;
then A2: (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))) . (intloc (5 + 1)) = (s . (fsloc 0)) /. (abs (s . (intloc (3 + 1)))) by A1, Th2;
A3: (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))) . (fsloc 0) = s . (fsloc 0) by A1, SCMFSA_2:72;
A4: (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))) . (intloc (2 + 1)) = s . (intloc (2 + 1)) by Th4;
A5: (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))) . (intloc (3 + 1)) = s . (intloc (3 + 1)) by Th4;
A6: (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))) . (intloc (4 + 1)) = s . (intloc (4 + 1)) by Th4;
A7: (IExec ((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))),p,s)) . (fsloc 0) = (Exec ((((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1))),(Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))))) . (fsloc 0) by SCMFSA6C:9
.= (s . (fsloc 0)) +* ((abs (s . (intloc (2 + 1)))),((s . (fsloc 0)) /. (abs (s . (intloc (3 + 1)))))) by A2, A3, A4, Th3 ;
A8: (IExec ((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))),p,s)) . (intloc (3 + 1)) = (Exec ((((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1))),(Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))))) . (intloc (3 + 1)) by SCMFSA6C:8
.= s . (intloc (3 + 1)) by A5, SCMFSA_2:73 ;
A9: (IExec ((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))),p,s)) . (intloc (4 + 1)) = (Exec ((((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1))),(Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))))) . (intloc (4 + 1)) by SCMFSA6C:8
.= s . (intloc (4 + 1)) by A6, SCMFSA_2:73 ;
set I = (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)));
set J = Stop SCM+FSA;
hereby  ::_thesis: ( s . (intloc (5 + 1)) <= 0 implies (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = s . (fsloc 0) )
assume  s . (intloc (5 + 1)) > 0 ; ::_thesis: (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = ((s . (fsloc 0)) +* ((abs (s . (intloc (2 + 1)))),((s . (fsloc 0)) /. (abs (s . (intloc (3 + 1))))))) +* ((abs (s . (intloc (3 + 1)))),(s . (intloc (4 + 1))))
hence (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = (IExec (((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),p,s)) . (fsloc 0) by SCM_HALT:44
.= (Exec ((((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1))),(IExec ((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))),p,s)))) . (fsloc 0) by SCMFSA6C:7
.= ((s . (fsloc 0)) +* ((abs (s . (intloc (2 + 1)))),((s . (fsloc 0)) /. (abs (s . (intloc (3 + 1))))))) +* ((abs (s . (intloc (3 + 1)))),(s . (intloc (4 + 1)))) by A7, A8, A9, Th3 ;
::_thesis: verum
end;
assume  s . (intloc (5 + 1)) <= 0 ; ::_thesis: (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = s . (fsloc 0)
hence (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (fsloc 0) = (IExec ((Stop SCM+FSA),p,s)) . (fsloc 0) by SCM_HALT:44
.= s . (fsloc 0) by Th6 ;
::_thesis: verum
end;

Lm28:  for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA holds (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (intloc (2 + 1)) = s . (intloc (2 + 1))
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA holds (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (intloc (2 + 1)) = s . (intloc (2 + 1))
let s be State of SCM+FSA; ::_thesis: (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (intloc (2 + 1)) = s . (intloc (2 + 1))
set s1 = Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s));
set s2 = IExec ((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))),p,s);
A1: (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))) . (intloc (2 + 1)) = s . (intloc (2 + 1)) by Th4;
A2: (IExec ((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))),p,s)) . (intloc (2 + 1)) = (Exec ((((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1))),(Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(Initialized s))))) . (intloc (2 + 1)) by SCMFSA6C:8
.= s . (intloc (2 + 1)) by A1, SCMFSA_2:73 ;
percases ( s . (intloc (5 + 1)) > 0 or s . (intloc (5 + 1)) <= 0 ) ;
suppose s . (intloc (5 + 1)) > 0 ; ::_thesis: (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (intloc (2 + 1)) = s . (intloc (2 + 1))
hence (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (intloc (2 + 1)) = (IExec (((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),p,s)) . (intloc (2 + 1)) by SCM_HALT:44
.= (Exec ((((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1))),(IExec ((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))),p,s)))) . (intloc (2 + 1)) by SCMFSA6C:6
.= s . (intloc (2 + 1)) by A2, SCMFSA_2:73 ;
::_thesis: verum
end;
suppose s . (intloc (5 + 1)) <= 0 ; ::_thesis: (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (intloc (2 + 1)) = s . (intloc (2 + 1))
hence (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,s)) . (intloc (2 + 1)) = (IExec ((Stop SCM+FSA),p,s)) . (intloc (2 + 1)) by SCM_HALT:44
.= s . (intloc (2 + 1)) by Th6 ;
::_thesis: verum
end;
end;
end;

Lm29:  for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA st s . (intloc (2 + 1)) <= len (s . (fsloc 0)) & s . (intloc (2 + 1)) >= 2 holds
( (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (intloc (2 + 1)) = (s . (intloc (2 + 1))) - 1 & s . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA st s . (intloc (2 + 1)) <= len (s . (fsloc 0)) & s . (intloc (2 + 1)) >= 2 holds
( (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (intloc (2 + 1)) = (s . (intloc (2 + 1))) - 1 & s . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
let s be State of SCM+FSA; ::_thesis: ( s . (intloc (2 + 1)) <= len (s . (fsloc 0)) & s . (intloc (2 + 1)) >= 2 implies ( (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (intloc (2 + 1)) = (s . (intloc (2 + 1))) - 1 & s . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) ) )
assume that
A1: s . (intloc (2 + 1)) <= len (s . (fsloc 0)) and
A2: s . (intloc (2 + 1)) >= 2 ; ::_thesis: ( (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (intloc (2 + 1)) = (s . (intloc (2 + 1))) - 1 & s . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
A3: (s . (intloc (2 + 1))) - 1 >= 2 - 1 by A2, XREAL_1:9;
then A4: abs ((s . (intloc (2 + 1))) - 1) = (s . (intloc (2 + 1))) - 1 by ABSVALUE:def_1;
A5: (s . (intloc (2 + 1))) - 1 <= len (s . (fsloc 0)) by A1, XREAL_1:146, XXREAL_0:2;
A6: s . (intloc (2 + 1)) >= 1 by A2, XXREAL_0:2;
A7: abs (s . (intloc (2 + 1))) = s . (intloc (2 + 1)) by A2, ABSVALUE:def_1;
reconsider k1 = (s . (intloc (2 + 1))) - 1 as Element of NAT by A3, INT_1:3;
reconsider k2 = s . (intloc (2 + 1)) as Element of NAT by A2, INT_1:3;
A8: k1 in dom (s . (fsloc 0)) by A3, A5, FINSEQ_3:25;
reconsider n1 = (s . (fsloc 0)) . k1 as Integer ;
A9: k2 in dom (s . (fsloc 0)) by A1, A6, FINSEQ_3:25;
reconsider n2 = (s . (fsloc 0)) . k2 as Integer ;
set s0 = Initialized s;
set s1 = Exec (((intloc (3 + 1)) := (intloc (2 + 1))),(Initialized s));
set s2 = IExec ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))),p,s);
set s3 = IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s);
set s4 = IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s);
set s5 = IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s);
set s6 = IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s);
A10: (Exec (((intloc (3 + 1)) := (intloc (2 + 1))),(Initialized s))) . (intloc (3 + 1)) = (Initialized s) . (intloc (2 + 1)) by SCMFSA_2:63
.= s . (intloc (2 + 1)) by SCMFSA_M:37 ;
A11: (Exec (((intloc (3 + 1)) := (intloc (2 + 1))),(Initialized s))) . (fsloc 0) = (Initialized s) . (fsloc 0) by SCMFSA_2:63
.= s . (fsloc 0) by SCMFSA_M:37 ;
A12: (Exec (((intloc (3 + 1)) := (intloc (2 + 1))),(Initialized s))) . (intloc (2 + 1)) = s . (intloc (2 + 1)) by Th5;
A13: (Exec (((intloc (3 + 1)) := (intloc (2 + 1))),(Initialized s))) . (intloc 0) = (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA_M:9 ;
A14: (IExec ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))),p,s)) . (fsloc 0) = (Exec ((SubFrom ((intloc (2 + 1)),(intloc 0))),(Exec (((intloc (3 + 1)) := (intloc (2 + 1))),(Initialized s))))) . (fsloc 0) by SCMFSA6C:9
.= s . (fsloc 0) by A11, SCMFSA_2:65 ;
A15: (IExec ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))),p,s)) . (intloc (2 + 1)) = (Exec ((SubFrom ((intloc (2 + 1)),(intloc 0))),(Exec (((intloc (3 + 1)) := (intloc (2 + 1))),(Initialized s))))) . (intloc (2 + 1)) by SCMFSA6C:8
.= (s . (intloc (2 + 1))) - 1 by A12, A13, SCMFSA_2:65 ;
A16: (IExec ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))),p,s)) . (intloc (3 + 1)) = (Exec ((SubFrom ((intloc (2 + 1)),(intloc 0))),(Exec (((intloc (3 + 1)) := (intloc (2 + 1))),(Initialized s))))) . (intloc (3 + 1)) by SCMFSA6C:8
.= s . (intloc (2 + 1)) by A10, Lm14, SCMFSA_2:65 ;
A17: (IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s)) . (fsloc 0) = (Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1)))),(IExec ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))),p,s)))) . (fsloc 0) by SCMFSA6C:7
.= s . (fsloc 0) by A14, SCMFSA_2:72 ;
A18: (s . (fsloc 0)) /. k1 = n1 by A3, A5, FINSEQ_4:15;
A19: (IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s)) . (intloc (4 + 1)) = (Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1)))),(IExec ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))),p,s)))) . (intloc (4 + 1)) by SCMFSA6C:6
.= n1 by A4, A14, A15, A18, Th2 ;
A20: (IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s)) . (intloc (3 + 1)) = (Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1)))),(IExec ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))),p,s)))) . (intloc (3 + 1)) by SCMFSA6C:6
.= s . (intloc (2 + 1)) by A16, Lm17, SCMFSA_2:72 ;
A21: (IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s)) . (intloc (2 + 1)) = (Exec (((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1)))),(IExec ((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))),p,s)))) . (intloc (2 + 1)) by SCMFSA6C:6
.= (s . (intloc (2 + 1))) - 1 by A15, Lm15, SCMFSA_2:72 ;
A22: (IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)) . (fsloc 0) = (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s)))) . (fsloc 0) by SCMFSA6C:7
.= s . (fsloc 0) by A17, SCMFSA_2:72 ;
A23: (s . (fsloc 0)) /. k2 = n2 by A1, A6, FINSEQ_4:15;
A24: (IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)) . (intloc (5 + 1)) = (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s)))) . (intloc (5 + 1)) by SCMFSA6C:6
.= n2 by A7, A17, A20, A23, Th2 ;
A25: (IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)) . (intloc (2 + 1)) = (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s)))) . (intloc (2 + 1)) by SCMFSA6C:6
.= (s . (intloc (2 + 1))) - 1 by A21, Lm16, SCMFSA_2:72 ;
A26: (IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)) . (intloc (3 + 1)) = (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s)))) . (intloc (3 + 1)) by SCMFSA6C:6
.= s . (intloc (2 + 1)) by A20, Lm18, SCMFSA_2:72 ;
A27: (IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)) . (intloc (4 + 1)) = (Exec (((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))),(IExec (((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))),p,s)))) . (intloc (4 + 1)) by SCMFSA6C:6
.= (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) by A19, Lm19, SCMFSA_2:72 ;
A28: (IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)) . (fsloc 0) = (Exec ((SubFrom ((intloc (5 + 1)),(intloc (4 + 1)))),(IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)))) . (fsloc 0) by SCMFSA6C:7
.= s . (fsloc 0) by A22, SCMFSA_2:65 ;
A29: (IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)) . (intloc (2 + 1)) = (Exec ((SubFrom ((intloc (5 + 1)),(intloc (4 + 1)))),(IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)))) . (intloc (2 + 1)) by SCMFSA6C:6
.= (s . (intloc (2 + 1))) - 1 by A25, Lm16, SCMFSA_2:65 ;
A30: (IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)) . (intloc (3 + 1)) = (Exec ((SubFrom ((intloc (5 + 1)),(intloc (4 + 1)))),(IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)))) . (intloc (3 + 1)) by SCMFSA6C:6
.= s . (intloc (2 + 1)) by A26, Lm18, SCMFSA_2:65 ;
A31: (IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)) . (intloc (4 + 1)) = (Exec ((SubFrom ((intloc (5 + 1)),(intloc (4 + 1)))),(IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)))) . (intloc (4 + 1)) by SCMFSA6C:6
.= n1 by A27, Lm19, SCMFSA_2:65 ;
A32: (IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)) . (intloc (5 + 1)) = (Exec ((SubFrom ((intloc (5 + 1)),(intloc (4 + 1)))),(IExec ((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))),p,s)))) . (intloc (5 + 1)) by SCMFSA6C:6
.= n2 - n1 by A24, A27, SCMFSA_2:65 ;
A33: (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) = (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,(IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)))) . (fsloc 0) by SCMFSA6C:2;
thus (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (intloc (2 + 1)) = (IExec ((if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))),p,(IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)))) . (intloc (2 + 1)) by SCMFSA6C:1
.= (s . (intloc (2 + 1))) - 1 by A29, Lm28 ; ::_thesis: ( s . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
percases ( (IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)) . (intloc (5 + 1)) > 0 or (IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)) . (intloc (5 + 1)) <= 0 ) ;
supposeA34: (IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)) . (intloc (5 + 1)) > 0 ; ::_thesis: ( s . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
then A35: (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) = ((s . (fsloc 0)) +* (k1,n2)) +* (k2,n1) by A4, A7, A23, A28, A29, A30, A31, A33, Lm27;
A36: dom ((s . (fsloc 0)) +* (k1,n2)) = dom (s . (fsloc 0)) by FUNCT_7:30;
then A37: dom ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) = dom (s . (fsloc 0)) by A35, FUNCT_7:30;
A38: k2 in dom ((s . (fsloc 0)) +* (k1,n2)) by A1, A6, A36, FINSEQ_3:25;
A39: ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k2 = (s . (fsloc 0)) . k1 by A9, A35, A36, FUNCT_7:31;
A40: now__::_thesis:_((IExec_((((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))),p,s))_._(fsloc_0))_._k1_=_(s_._(fsloc_0))_._k2
percases ( k1 = k2 or k1 <> k2 ) ;
suppose k1 = k2 ; ::_thesis: ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k1 = (s . (fsloc 0)) . k2
hence  ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k1 = (s . (fsloc 0)) . k2 ; ::_thesis: verum
end;
suppose k1 <> k2 ; ::_thesis: ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k1 = (s . (fsloc 0)) . k2
hence ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k1 = ((s . (fsloc 0)) +* (k1,n2)) . k1 by A35, FUNCT_7:32
.= (s . (fsloc 0)) . k2 by A8, FUNCT_7:31 ;
::_thesis: verum
end;
end;
end;
A41: now__::_thesis:_for_k_being_set_st_k_<>_k1_&_k_<>_k2_&_k_in_dom_(s_._(fsloc_0))_holds_
((IExec_((((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))),p,s))_._(fsloc_0))_._k_=_(s_._(fsloc_0))_._k
let k be set ; ::_thesis: ( k <> k1 & k <> k2 & k in dom (s . (fsloc 0)) implies ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k = (s . (fsloc 0)) . k )
assume that
A42: k <> k1 and
A43: k <> k2 and
 k in dom (s . (fsloc 0)) ; ::_thesis: ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k = (s . (fsloc 0)) . k
thus ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k = ((s . (fsloc 0)) +* (k1,n2)) . k by A35, A43, FUNCT_7:32
.= (s . (fsloc 0)) . k by A42, FUNCT_7:32 ; ::_thesis: verum
end;
hence  s . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) are_fiberwise_equipotent by A8, A9, A37, A39, A40, RFINSEQ:28; ::_thesis: ( ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
thus  ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) by A40; ::_thesis: ( ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
thus  ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) by A35, A38, FUNCT_7:31; ::_thesis: ( ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
thus  ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) by A40; ::_thesis: ( ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
thus  for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k by A41; ::_thesis: ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 )
A44: (n2 - n1) + n1 > 0 + n1 by A32, A34, XREAL_1:6;
take  n2 ; ::_thesis: ex x2 being Integer st
( n2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & n2 >= x2 )
take  n1 ; ::_thesis: ( n2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & n1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & n2 >= n1 )
thus  ( n2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & n1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & n2 >= n1 ) by A9, A35, A36, A40, A44, FUNCT_7:31; ::_thesis: verum
end;
supposeA45: (IExec (((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))),p,s)) . (intloc (5 + 1)) <= 0 ; ::_thesis: ( s . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
hence  s . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0) are_fiberwise_equipotent by A28, A33, Lm27; ::_thesis: ( ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
thus  ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) by A28, A33, A45, Lm27; ::_thesis: ( ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) & ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
thus  ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) ) by A28, A33, A45, Lm27; ::_thesis: ( ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) & ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
thus  ( (s . (fsloc 0)) . (s . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) or (s . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) ) by A28, A33, A45, Lm27; ::_thesis: ( ( for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k ) & ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 ) )
thus  for k being set st k <> (s . (intloc (2 + 1))) - 1 & k <> s . (intloc (2 + 1)) & k in dom (s . (fsloc 0)) holds
(s . (fsloc 0)) . k = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . k by A28, A33, A45, Lm27; ::_thesis: ex x1, x2 being Integer st
( x1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & x1 >= x2 )
A46: (n2 - n1) + n1 <= 0 + n1 by A32, A45, XREAL_1:6;
take  n1 ; ::_thesis: ex x2 being Integer st
( n1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & x2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & n1 >= x2 )
take  n2 ; ::_thesis: ( n1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & n2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & n1 >= n2 )
thus  ( n1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . ((s . (intloc (2 + 1))) - 1) & n2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),p,s)) . (fsloc 0)) . (s . (intloc (2 + 1))) & n1 >= n2 ) by A28, A33, A45, A46, Lm27; ::_thesis: verum
end;
end;
end;

Lm30:  for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA st s . (intloc (1 + 1)) >= 0 & s . (intloc (1 + 1)) < s . (intloc (2 + 1)) & s . (intloc (2 + 1)) <= len (s . (fsloc 0)) holds
ex k being Element of NAT st
( k <= s . (intloc (2 + 1)) & k >= (s . (intloc (2 + 1))) - (s . (intloc (1 + 1))) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),p,s)) . (fsloc 0)) . k = (s . (fsloc 0)) . (s . (intloc (2 + 1))) )
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA st s . (intloc (1 + 1)) >= 0 & s . (intloc (1 + 1)) < s . (intloc (2 + 1)) & s . (intloc (2 + 1)) <= len (s . (fsloc 0)) holds
ex k being Element of NAT st
( k <= s . (intloc (2 + 1)) & k >= (s . (intloc (2 + 1))) - (s . (intloc (1 + 1))) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),p,s)) . (fsloc 0)) . k = (s . (fsloc 0)) . (s . (intloc (2 + 1))) )
let s be State of SCM+FSA; ::_thesis: ( s . (intloc (1 + 1)) >= 0 & s . (intloc (1 + 1)) < s . (intloc (2 + 1)) & s . (intloc (2 + 1)) <= len (s . (fsloc 0)) implies ex k being Element of NAT st
( k <= s . (intloc (2 + 1)) & k >= (s . (intloc (2 + 1))) - (s . (intloc (1 + 1))) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),p,s)) . (fsloc 0)) . k = (s . (fsloc 0)) . (s . (intloc (2 + 1))) ) )
assume that
A1: s . (intloc (1 + 1)) >= 0 and
A2: s . (intloc (1 + 1)) < s . (intloc (2 + 1)) and
A3: s . (intloc (2 + 1)) <= len (s . (fsloc 0)) ; ::_thesis: ex k being Element of NAT st
( k <= s . (intloc (2 + 1)) & k >= (s . (intloc (2 + 1))) - (s . (intloc (1 + 1))) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),p,s)) . (fsloc 0)) . k = (s . (fsloc 0)) . (s . (intloc (2 + 1))) )
defpred S1[ Nat] means  for t being State of SCM+FSA
for q being Instruction-Sequence of SCM+FSA st t . (intloc (1 + 1)) = $1 & t . (intloc (1 + 1)) < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) holds
( ( for m being Element of NAT st m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - $1 & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) ) );
A4: S1[ 0 ]
proof
let t be State of SCM+FSA; ::_thesis: for q being Instruction-Sequence of SCM+FSA st t . (intloc (1 + 1)) = 0 & t . (intloc (1 + 1)) < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) holds
( ( for m being Element of NAT st m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - 0 & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) ) )
let q be Instruction-Sequence of SCM+FSA; ::_thesis: ( t . (intloc (1 + 1)) = 0 & t . (intloc (1 + 1)) < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) implies ( ( for m being Element of NAT st m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - 0 & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) ) ) )
assume that
A5: t . (intloc (1 + 1)) = 0 and
A6: t . (intloc (1 + 1)) < t . (intloc (2 + 1)) and
 t . (intloc (2 + 1)) <= len (t . (fsloc 0)) ; ::_thesis: ( ( for m being Element of NAT st m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - 0 & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) ) )
set If0 = (IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0);
thus  for m being Element of NAT st m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m by A5, SCM_HALT:67; ::_thesis: ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - 0 & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
reconsider n = t . (intloc (2 + 1)) as Element of NAT by A5, A6, INT_1:3;
take  n ; ::_thesis: ( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - 0 & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
thus  n <= t . (intloc (2 + 1)) ; ::_thesis: ( n >= (t . (intloc (2 + 1))) - 0 & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
thus  n >= (t . (intloc (2 + 1))) - 0 ; ::_thesis: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1)))
thus  ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) by A5, SCM_HALT:67; ::_thesis: verum
end;
set sb2 = SubFrom ((intloc (1 + 1)),(intloc 0));
A7: 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 A8: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_t_being_State_of_SCM+FSA
for_q_being_Instruction-Sequence_of_SCM+FSA_st_t_._(intloc_(1_+_1))_=_k_+_1_&_t_._(intloc_(1_+_1))_<_t_._(intloc_(2_+_1))_&_t_._(intloc_(2_+_1))_<=_len_(t_._(fsloc_0))_holds_
(_(_for_m_being_Element_of_NAT_st_m_>_t_._(intloc_(2_+_1))_&_m_<=_len_(t_._(fsloc_0))_holds_
(t_._(fsloc_0))_._m_=_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._m_)_&_ex_n_being_Element_of_NAT_st_
(_n_<=_t_._(intloc_(2_+_1))_&_n_>=_(t_._(intloc_(2_+_1)))_-_(k_+_1)_&_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._n_=_(t_._(fsloc_0))_._(t_._(intloc_(2_+_1)))_)_)
let t be State of SCM+FSA; ::_thesis: for q being Instruction-Sequence of SCM+FSA st t . (intloc (1 + 1)) = k + 1 & t . (intloc (1 + 1)) < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) holds
( ( for m being Element of NAT st m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) ) )
let q be Instruction-Sequence of SCM+FSA; ::_thesis: ( t . (intloc (1 + 1)) = k + 1 & t . (intloc (1 + 1)) < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) implies ( ( for m being Element of NAT st m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) ) ) )
assume that
A9: t . (intloc (1 + 1)) = k + 1 and
A10: t . (intloc (1 + 1)) < t . (intloc (2 + 1)) and
A11: t . (intloc (2 + 1)) <= len (t . (fsloc 0)) ; ::_thesis: ( ( for m being Element of NAT st m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) ) )
set t1 = IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t);
set IB = IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t);
set t2 = IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)));
A12: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (1 + 1)) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)))) . (intloc (1 + 1)) by SCM_HALT:23
.= ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (intloc (1 + 1))) - ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (intloc 0)) by SCMFSA_2:65
.= ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (intloc (1 + 1))) - 1 by SCM_HALT:9
.= ((Initialized t) . (intloc (1 + 1))) - 1 by Lm22, SCM_HALT:53
.= (t . (intloc (1 + 1))) - 1 by SCMFSA_M:37 ;
A13: 2 <= k + 2 by NAT_1:11;
 (k + 1) + 1 <= t . (intloc (2 + 1)) by A9, A10, INT_1:7;
then A14: 2 <= t . (intloc (2 + 1)) by A13, XXREAL_0:2;
A15: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)))) . (intloc (2 + 1)) by SCM_HALT:23
.= (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (intloc (2 + 1)) by Lm10, SCMFSA_2:65
.= (t . (intloc (2 + 1))) - 1 by A11, A14, Lm29 ;
A16: (t . (intloc (1 + 1))) - 1 < (t . (intloc (2 + 1))) - 1 by A10, XREAL_1:9;
A17: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (1 + 1)) < (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) by A10, A12, A15, XREAL_1:9;
A18: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)))) . (fsloc 0) by SCM_HALT:24
.= (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0) by SCMFSA_2:65 ;
A19: t . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0) are_fiberwise_equipotent by A11, A14, Lm29;
then A20: len (t . (fsloc 0)) = len ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A18, RFINSEQ:3;
then A21: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) <= len ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A11, A15, XREAL_1:146, XXREAL_0:2;
A22: (IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) = (IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0) by A9, Lm22, SCM_HALT:69;
A23: (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)))) . (fsloc 0) by SCMFSA_2:65
.= (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0) by SCM_HALT:24 ;
thus  for m being Element of NAT st m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ::_thesis: ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
proof
let m be Element of NAT ; ::_thesis: ( m > t . (intloc (2 + 1)) & m <= len (t . (fsloc 0)) implies (t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m )
assume that
A24: m > t . (intloc (2 + 1)) and
A25: m <= len (t . (fsloc 0)) ; ::_thesis: (t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m
A26: t . (intloc (2 + 1)) > (t . (intloc (2 + 1))) - 1 by XREAL_1:146;
A27: m > (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) by A15, A24, XREAL_1:146, XXREAL_0:2;
A28: m <= len ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A18, A19, A25, RFINSEQ:3;
 m >= 2 by A14, A24, XXREAL_0:2;
then  m >= 1 by XXREAL_0:2;
then  m in dom (t . (fsloc 0)) by A25, FINSEQ_3:25;
hence (t . (fsloc 0)) . m = ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) . m by A11, A14, A23, A24, A26, Lm29
.= ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m by A8, A9, A12, A17, A21, A22, A27, A28 ;
::_thesis: verum
end;
hereby  ::_thesis: verum
reconsider n = t . (intloc (2 + 1)) as Element of NAT by A9, A10, INT_1:3;
percases ( (t . (fsloc 0)) . (t . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . (t . (intloc (2 + 1))) or (t . (fsloc 0)) . (t . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) ) by A11, A14, Lm29;
supposeA29: (t . (fsloc 0)) . (t . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . (t . (intloc (2 + 1))) ; ::_thesis: ex n being Element of NAT st
( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
take n = n; ::_thesis: ( n <= t . (intloc (2 + 1)) & n >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
thus  n <= t . (intloc (2 + 1)) ; ::_thesis: ( n >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
 n <= n + (k + 1) by NAT_1:11;
hence  n >= (t . (intloc (2 + 1))) - (k + 1) by XREAL_1:20; ::_thesis: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1)))
thus  ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . n = (t . (fsloc 0)) . (t . (intloc (2 + 1))) by A8, A9, A11, A12, A15, A16, A20, A21, A22, A23, A29, XREAL_1:146; ::_thesis: verum
end;
supposeA30: (t . (fsloc 0)) . (t . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) ; ::_thesis: ex m being Element of NAT st
( m <= t . (intloc (2 + 1)) & m >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
consider m being Element of NAT  such that
A31: m <= (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) and
A32: m >= ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) - k and
A33: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0)) . m = ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) . ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) by A8, A9, A12, A17, A21;
take m = m; ::_thesis: ( m <= t . (intloc (2 + 1)) & m >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
thus  m <= t . (intloc (2 + 1)) by A15, A31, XREAL_1:146, XXREAL_0:2; ::_thesis: ( m >= (t . (intloc (2 + 1))) - (k + 1) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m = (t . (fsloc 0)) . (t . (intloc (2 + 1))) )
thus  m >= (t . (intloc (2 + 1))) - (k + 1) by A15, A32; ::_thesis: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m = (t . (fsloc 0)) . (t . (intloc (2 + 1)))
thus  ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m = (t . (fsloc 0)) . (t . (intloc (2 + 1))) by A9, A15, A23, A30, A33, Lm22, SCM_HALT:69; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A34: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A4, A7);
reconsider i = s . (intloc (1 + 1)) as Element of NAT by A1, INT_1:3;
 S1[i] by A34;
hence  ex k being Element of NAT st
( k <= s . (intloc (2 + 1)) & k >= (s . (intloc (2 + 1))) - (s . (intloc (1 + 1))) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),p,s)) . (fsloc 0)) . k = (s . (fsloc 0)) . (s . (intloc (2 + 1))) ) by A2, A3; ::_thesis: verum
end;

Lm31:  for k being Element of NAT
for t being State of SCM+FSA
for q being Instruction-Sequence of SCM+FSA st k = t . (intloc (1 + 1)) & k < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) holds
( t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - k & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - k & m <= t . (intloc (2 + 1)) holds
ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - k) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - k & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - k & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
proof
defpred S1[ Nat] means  for q being Instruction-Sequence of SCM+FSA
for t being State of SCM+FSA st $1 = t . (intloc (1 + 1)) & $1 < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) holds
( t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - $1 & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - $1 & m <= t . (intloc (2 + 1)) holds
ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - $1) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - $1 & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - $1 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) );
now__::_thesis:_for_t_being_State_of_SCM+FSA
for_q_being_Instruction-Sequence_of_SCM+FSA_st_0_=_t_._(intloc_(1_+_1))_&_0_<_t_._(intloc_(2_+_1))_&_t_._(intloc_(2_+_1))_<=_len_(t_._(fsloc_0))_holds_
(_t_._(fsloc_0),(IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0)_are_fiberwise_equipotent_&_(_for_m_being_Element_of_NAT_st_(_(_m_<_(t_._(intloc_(2_+_1)))_-_0_&_m_>=_1_)_or_(_m_>_t_._(intloc_(2_+_1))_&_m_in_dom_(t_._(fsloc_0))_)_)_holds_
(t_._(fsloc_0))_._m_=_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._m_)_&_(_for_m_being_Element_of_NAT_st_m_>=_(t_._(intloc_(2_+_1)))_-_0_&_m_<=_t_._(intloc_(2_+_1))_holds_
ex_x1,_x2_being_Integer_st_
(_x1_=_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._((t_._(intloc_(2_+_1)))_-_0)_&_x2_=_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._m_&_x1_>=_x2_)_)_&_(_for_i_being_Element_of_NAT_st_i_>=_(t_._(intloc_(2_+_1)))_-_0_&_i_<=_t_._(intloc_(2_+_1))_holds_
ex_n_being_Element_of_NAT_st_
(_n_>=_(t_._(intloc_(2_+_1)))_-_0_&_n_<=_t_._(intloc_(2_+_1))_&_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._i_=_(t_._(fsloc_0))_._n_)_)_)
let t be State of SCM+FSA; ::_thesis: for q being Instruction-Sequence of SCM+FSA st 0 = t . (intloc (1 + 1)) & 0 < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) holds
( t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - 0 & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - 0 & m <= t . (intloc (2 + 1)) holds
ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - 0 & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
let q be Instruction-Sequence of SCM+FSA; ::_thesis: ( 0 = t . (intloc (1 + 1)) & 0 < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) implies ( t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - 0 & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - 0 & m <= t . (intloc (2 + 1)) holds
ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - 0 & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) ) )
assume that
A1: 0 = t . (intloc (1 + 1)) and
 0 < t . (intloc (2 + 1)) and
 t . (intloc (2 + 1)) <= len (t . (fsloc 0)) ; ::_thesis: ( t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - 0 & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - 0 & m <= t . (intloc (2 + 1)) holds
ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - 0 & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
set If0 = (IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0);
thus  t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent by A1, SCM_HALT:67; ::_thesis: ( ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - 0 & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - 0 & m <= t . (intloc (2 + 1)) holds
ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - 0 & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
thus  for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - 0 & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m by A1, SCM_HALT:67; ::_thesis: ( ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - 0 & m <= t . (intloc (2 + 1)) holds
ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - 0 & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
hereby  ::_thesis: for i being Element of NAT st i >= (t . (intloc (2 + 1))) - 0 & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
let m be Element of NAT ; ::_thesis: ( m >= (t . (intloc (2 + 1))) - 0 & m <= t . (intloc (2 + 1)) implies ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 ) )
assume that
A2: m >= (t . (intloc (2 + 1))) - 0 and
A3: m <= t . (intloc (2 + 1)) ; ::_thesis: ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 )
A4: m = t . (intloc (2 + 1)) by A2, A3, XXREAL_0:1;
reconsider n1 = (t . (fsloc 0)) . m as Integer ;
take x1 = n1; ::_thesis: ex x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 )
take x2 = n1; ::_thesis: ( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 )
thus  x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 0) by A1, A4, SCM_HALT:67; ::_thesis: ( x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 )
thus  x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m by A1, SCM_HALT:67; ::_thesis: x1 >= x2
thus  x1 >= x2 ; ::_thesis: verum
end;
let i be Element of NAT ; ::_thesis: ( i >= (t . (intloc (2 + 1))) - 0 & i <= t . (intloc (2 + 1)) implies ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) )
assume that
A5: i >= (t . (intloc (2 + 1))) - 0 and
A6: i <= t . (intloc (2 + 1)) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
take n = i; ::_thesis: ( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  ( n >= (t . (intloc (2 + 1))) - 0 & n <= t . (intloc (2 + 1)) ) by A5, A6; ::_thesis: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n
thus  ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n by A1, SCM_HALT:67; ::_thesis: verum
end;
then A7: S1[ 0 ] ;
set sb2 = SubFrom ((intloc (1 + 1)),(intloc 0));
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_State_of_SCM+FSA
for_q_being_Instruction-Sequence_of_SCM+FSA_st_k_+_1_=_t_._(intloc_(1_+_1))_&_k_+_1_<_t_._(intloc_(2_+_1))_&_t_._(intloc_(2_+_1))_<=_len_(t_._(fsloc_0))_holds_
(_t_._(fsloc_0),(IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0)_are_fiberwise_equipotent_&_(_for_m_being_Element_of_NAT_st_(_(_m_<_(t_._(intloc_(2_+_1)))_-_(k_+_1)_&_m_>=_1_)_or_(_m_>_t_._(intloc_(2_+_1))_&_m_in_dom_(t_._(fsloc_0))_)_)_holds_
(t_._(fsloc_0))_._m_=_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._m_)_&_(_for_m_being_Element_of_NAT_st_m_>=_(t_._(intloc_(2_+_1)))_-_(k_+_1)_&_m_<=_t_._(intloc_(2_+_1))_holds_
ex_y1,_n2_being_Integer_st_
(_y1_=_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._((t_._(intloc_(2_+_1)))_-_(k_+_1))_&_n2_=_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._m_&_y1_>=_n2_)_)_&_(_for_i_being_Element_of_NAT_st_i_>=_(t_._(intloc_(2_+_1)))_-_(k_+_1)_&_i_<=_t_._(intloc_(2_+_1))_holds_
ex_n_being_Element_of_NAT_st_
(_n_>=_(t_._(intloc_(2_+_1)))_-_(k_+_1)_&_n_<=_t_._(intloc_(2_+_1))_&_((IExec_((Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA)))))),q,t))_._(fsloc_0))_._i_=_(t_._(fsloc_0))_._n_)_)_)
let t be State of SCM+FSA; ::_thesis: for q being Instruction-Sequence of SCM+FSA st k + 1 = t . (intloc (1 + 1)) & k + 1 < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) holds
( t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - (k + 1) & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - (k + 1) & m <= t . (intloc (2 + 1)) holds
ex y1, n2 being Integer st
( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= n2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - (k + 1) & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
let q be Instruction-Sequence of SCM+FSA; ::_thesis: ( k + 1 = t . (intloc (1 + 1)) & k + 1 < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) implies ( t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - (k + 1) & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - (k + 1) & m <= t . (intloc (2 + 1)) holds
ex y1, n2 being Integer st
( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= n2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - (k + 1) & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) ) )
set t1 = IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t);
set IB = IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t);
set t2 = IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)));
assume that
A10: k + 1 = t . (intloc (1 + 1)) and
A11: k + 1 < t . (intloc (2 + 1)) and
A12: t . (intloc (2 + 1)) <= len (t . (fsloc 0)) ; ::_thesis: ( t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - (k + 1) & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - (k + 1) & m <= t . (intloc (2 + 1)) holds
ex y1, n2 being Integer st
( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= n2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - (k + 1) & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
A13: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (1 + 1)) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)))) . (intloc (1 + 1)) by SCM_HALT:23
.= ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (intloc (1 + 1))) - ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (intloc 0)) by SCMFSA_2:65
.= ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (intloc (1 + 1))) - 1 by SCM_HALT:9
.= ((Initialized t) . (intloc (1 + 1))) - 1 by Lm22, SCM_HALT:53
.= (k + 1) - 1 by A10, SCMFSA_M:37
.= k ;
A14: 2 <= k + 2 by NAT_1:11;
 (k + 1) + 1 <= t . (intloc (2 + 1)) by A11, INT_1:7;
then A15: 2 <= t . (intloc (2 + 1)) by A14, XXREAL_0:2;
A16: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)))) . (intloc (2 + 1)) by SCM_HALT:23
.= (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (intloc (2 + 1)) by Lm10, SCMFSA_2:65
.= (t . (intloc (2 + 1))) - 1 by A12, A15, Lm29 ;
A17: (k + 1) - 1 < (t . (intloc (2 + 1))) - 1 by A11, XREAL_1:9;
A18: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)))) . (fsloc 0) by SCM_HALT:24
.= (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0) by SCMFSA_2:65 ;
A19: t . (fsloc 0),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0) are_fiberwise_equipotent by A12, A15, Lm29;
then A20: len (t . (fsloc 0)) = len ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A18, RFINSEQ:3;
A21: t . (intloc (2 + 1)) <= len ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A12, A18, A19, RFINSEQ:3;
A22: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) <= len ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A12, A16, A20, XREAL_1:146, XXREAL_0:2;
A23: t . (intloc (2 + 1)) = ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) + 1 by A16;
A24: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0) are_fiberwise_equipotent by A9, A13, A16, A17, A22;
A25: (IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) = (IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0) by A10, Lm22, SCM_HALT:69;
 (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent by A10, A24, Lm22, SCM_HALT:69;
hence  t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent by A18, A19, CLASSES1:76; ::_thesis: ( ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - (k + 1) & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - (k + 1) & m <= t . (intloc (2 + 1)) holds
ex y1, n2 being Integer st
( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= n2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - (k + 1) & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
A26: (t . (intloc (2 + 1))) - (k + 1) = ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) - k by A16;
consider n1, n2 being Integer such that
A27: n1 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) and
A28: n2 = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . (t . (intloc (2 + 1))) and
A29: n1 >= n2 by A12, A15, Lm29;
A30: (IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)))) . (fsloc 0) by SCMFSA_2:65
.= (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0) by SCM_HALT:24 ;
A31: t . (intloc (2 + 1)) is Element of NAT by A11, INT_1:3;
A32: t . (intloc (2 + 1)) >= 1 by A15, XXREAL_0:2;
then A33: t . (intloc (2 + 1)) in dom ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A12, A20, A31, FINSEQ_3:25;
hereby  ::_thesis: ( ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - (k + 1) & m <= t . (intloc (2 + 1)) holds
ex y1, n2 being Integer st
( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= n2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - (k + 1) & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
let m be Element of NAT ; ::_thesis: ( ( ( m < (t . (intloc (2 + 1))) - (k + 1) & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) implies (t . (fsloc 0)) . b1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . b1 )
assume A34: ( ( m < (t . (intloc (2 + 1))) - (k + 1) & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) ; ::_thesis: (t . (fsloc 0)) . b1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . b1
percases ( ( m < (t . (intloc (2 + 1))) - (k + 1) & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) by A34;
supposeA35: ( m < (t . (intloc (2 + 1))) - (k + 1) & m >= 1 ) ; ::_thesis: (t . (fsloc 0)) . b1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . b1
A36: ((t . (intloc (2 + 1))) - (k + 1)) + (k + 1) = t . (intloc (2 + 1)) ;
A37: m + (k + 1) < ((t . (intloc (2 + 1))) - (k + 1)) + (k + 1) by A35, XREAL_1:6;
A38: m + (k + 1) < t . (intloc (2 + 1)) by A35, A36, XREAL_1:6;
 m <= m + (k + 1) by NAT_1:11;
then  m <= t . (intloc (2 + 1)) by A37, XXREAL_0:2;
then  m <= len ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A12, A20, XXREAL_0:2;
then A39: m in dom (t . (fsloc 0)) by A20, A35, FINSEQ_3:25;
A40: m <> t . (intloc (2 + 1)) by A35, A36, XREAL_1:29;
 m <> (t . (intloc (2 + 1))) - 1
proof
assume A41: m = (t . (intloc (2 + 1))) - 1 ; ::_thesis: contradiction
 m + (k + 1) = (m + 1) + k ;
hence  contradiction by A38, A41, NAT_1:11; ::_thesis: verum
end;
hence (t . (fsloc 0)) . m = ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) . m by A12, A15, A30, A39, A40, Lm29
.= ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m by A9, A13, A17, A22, A25, A26, A35 ;
::_thesis: verum
end;
supposeA42: ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ; ::_thesis: (t . (fsloc 0)) . b1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . b1
then A43: m in dom ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A18, A19, RFINSEQ:3;
A44: t . (intloc (2 + 1)) > (t . (intloc (2 + 1))) - 1 by XREAL_1:146;
A45: m > (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) by A16, A42, XREAL_1:146, XXREAL_0:2;
thus (t . (fsloc 0)) . m = ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) . m by A12, A15, A30, A42, A44, Lm29
.= ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m by A9, A13, A16, A17, A22, A25, A43, A45 ; ::_thesis: verum
end;
end;
end;
hereby  ::_thesis: for i being Element of NAT st i >= (t . (intloc (2 + 1))) - (k + 1) & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
let m be Element of NAT ; ::_thesis: ( m >= (t . (intloc (2 + 1))) - (k + 1) & m <= t . (intloc (2 + 1)) implies ex y1, n2 being Integer st
( n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & b3 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . y1 & n2 >= b3 ) )
assume that
A46: m >= (t . (intloc (2 + 1))) - (k + 1) and
A47: m <= t . (intloc (2 + 1)) ; ::_thesis: ex y1, n2 being Integer st
( n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & b3 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . y1 & n2 >= b3 )
consider nn being Element of NAT  such that
A48: nn <= (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) and
A49: nn >= ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) - ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (1 + 1))) and
A50: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0)) . nn = ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) . ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) by A13, A16, A17, A22, Lm30;
consider y1, y2 being Integer such that
A51: y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0)) . (((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) - k) and
A52: y2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0)) . nn and
A53: y1 >= y2 by A9, A13, A16, A17, A22, A48, A49;
percases ( m > (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) or m <= (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) ) ;
supposeA54: m > (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) ; ::_thesis: ex y1, n2 being Integer st
( n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & b3 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . y1 & n2 >= b3 )
then  m >= ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) + 1 by INT_1:7;
then A55: m = t . (intloc (2 + 1)) by A16, A47, XXREAL_0:1;
take y1 = y1; ::_thesis: ex n2 being Integer st
( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= n2 )
take n2 = n2; ::_thesis: ( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= n2 )
thus  y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) by A10, A16, A51, Lm22, SCM_HALT:69; ::_thesis: ( n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= n2 )
thus  n2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m by A9, A13, A16, A17, A22, A25, A28, A30, A33, A54, A55; ::_thesis: y1 >= n2
thus  y1 >= n2 by A16, A27, A29, A30, A50, A52, A53, XXREAL_0:2; ::_thesis: verum
end;
suppose m <= (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) ; ::_thesis: ex y1, y2 being Integer st
( y2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & b3 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . y1 & y2 >= b3 )
then consider y1, y2 being Integer such that
A56: y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0)) . (((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) - k) and
A57: y2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0)) . m and
A58: y1 >= y2 by A9, A13, A16, A17, A22, A46;
take y1 = y1; ::_thesis: ex y2 being Integer st
( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & y2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= y2 )
take y2 = y2; ::_thesis: ( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) & y2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= y2 )
thus  y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - (k + 1)) by A10, A16, A56, Lm22, SCM_HALT:69; ::_thesis: ( y2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & y1 >= y2 )
thus  y2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m by A10, A57, Lm22, SCM_HALT:69; ::_thesis: y1 >= y2
thus  y1 >= y2 by A58; ::_thesis: verum
end;
end;
end;
thus  for i being Element of NAT st i >= (t . (intloc (2 + 1))) - (k + 1) & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ::_thesis: verum
proof
let i be Element of NAT ; ::_thesis: ( i >= (t . (intloc (2 + 1))) - (k + 1) & i <= t . (intloc (2 + 1)) implies ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) )
assume that
A59: i >= (t . (intloc (2 + 1))) - (k + 1) and
A60: i <= t . (intloc (2 + 1)) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
percases ( i = t . (intloc (2 + 1)) or i <> t . (intloc (2 + 1)) ) ;
supposeA61: i = t . (intloc (2 + 1)) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
then A62: i > (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) by A23, XREAL_1:29;
A63: i in dom ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A21, A32, A61, FINSEQ_3:25;
percases ( (t . (fsloc 0)) . (t . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . (t . (intloc (2 + 1))) or (t . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . (t . (intloc (2 + 1))) ) by A12, A15, Lm29;
supposeA64: (t . (fsloc 0)) . (t . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . (t . (intloc (2 + 1))) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
reconsider n = t . (intloc (2 + 1)) as Element of NAT by A11, INT_1:3;
take  n ; ::_thesis: ( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  ( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) ) by A59, A61; ::_thesis: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n
thus  ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n by A9, A13, A16, A17, A22, A25, A30, A61, A62, A63, A64; ::_thesis: verum
end;
supposeA65: (t . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . (t . (intloc (2 + 1))) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
 (t . (intloc (2 + 1))) - 1 >= 1 - 1 by A32, XREAL_1:9;
then reconsider n = (t . (intloc (2 + 1))) - 1 as Element of NAT by INT_1:3;
take  n ; ::_thesis: ( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
 n <= n + k by NAT_1:11;
hence  n >= (t . (intloc (2 + 1))) - (k + 1) by A26, XREAL_1:20; ::_thesis: ( n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  n <= t . (intloc (2 + 1)) by XREAL_1:146; ::_thesis: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n
thus  ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n by A9, A13, A16, A17, A22, A25, A30, A61, A62, A63, A65; ::_thesis: verum
end;
end;
end;
suppose i <> t . (intloc (2 + 1)) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
then  i < t . (intloc (2 + 1)) by A60, XXREAL_0:1;
then  i + 1 <= t . (intloc (2 + 1)) by INT_1:7;
then  i <= (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) by A16, XREAL_1:19;
then consider n being Element of NAT  such that
A66: n >= ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) - k and
A67: n <= (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) and
A68: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)))) . (fsloc 0)) . i = ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) . n by A9, A13, A16, A17, A22, A59;
thus  ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ::_thesis: verum
proof
percases ( n = (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) or n <> (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) ) ;
supposeA69: n = (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
percases ( (t . (fsloc 0)) . (t . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) or (t . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) ) by A12, A15, Lm29;
supposeA70: (t . (fsloc 0)) . (t . (intloc (2 + 1))) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
reconsider m = t . (intloc (2 + 1)) as Element of NAT by A11, INT_1:3;
take  m ; ::_thesis: ( m >= (t . (intloc (2 + 1))) - (k + 1) & m <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . m )
 m <= m + (k + 1) by NAT_1:11;
hence  m >= (t . (intloc (2 + 1))) - (k + 1) by XREAL_1:20; ::_thesis: ( m <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . m )
thus  m <= t . (intloc (2 + 1)) ; ::_thesis: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . m
thus  ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . m by A10, A16, A30, A68, A69, A70, Lm22, SCM_HALT:69; ::_thesis: verum
end;
supposeA71: (t . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) = ((IExec ((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - 1) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
take  n ; ::_thesis: ( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  n >= (t . (intloc (2 + 1))) - (k + 1) by A16, A66; ::_thesis: ( n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  n <= t . (intloc (2 + 1)) by A16, A69, XREAL_1:146; ::_thesis: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n
thus  ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n by A10, A16, A30, A68, A69, A71, Lm22, SCM_HALT:69; ::_thesis: verum
end;
end;
end;
supposeA72: n <> (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) ; ::_thesis: ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
A73: (IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1)) < t . (intloc (2 + 1)) by A16, XREAL_1:146;
A74: n < t . (intloc (2 + 1)) by A16, A67, XREAL_1:146, XXREAL_0:2;
 k - k < ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (intloc (2 + 1))) - k by A16, A17, XREAL_1:9;
then A75: n >= 0 + 1 by A66, INT_1:7;
 n <= len ((IExec (((((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A12, A20, A74, XXREAL_0:2;
then A76: n in dom (t . (fsloc 0)) by A20, A75, FINSEQ_3:25;
take  n ; ::_thesis: ( n >= (t . (intloc (2 + 1))) - (k + 1) & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  n >= (t . (intloc (2 + 1))) - (k + 1) by A16, A66; ::_thesis: ( n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  n <= t . (intloc (2 + 1)) by A16, A67, XREAL_1:146, XXREAL_0:2; ::_thesis: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n
thus  ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n by A12, A15, A16, A25, A30, A67, A68, A72, A73, A76, Lm29; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A7, A8);
hence  for k being Element of NAT
for t being State of SCM+FSA
for q being Instruction-Sequence of SCM+FSA st k = t . (intloc (1 + 1)) & k < t . (intloc (2 + 1)) & t . (intloc (2 + 1)) <= len (t . (fsloc 0)) holds
( t . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for m being Element of NAT st ( ( m < (t . (intloc (2 + 1))) - k & m >= 1 ) or ( m > t . (intloc (2 + 1)) & m in dom (t . (fsloc 0)) ) ) holds
(t . (fsloc 0)) . m = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m ) & ( for m being Element of NAT st m >= (t . (intloc (2 + 1))) - k & m <= t . (intloc (2 + 1)) holds
ex x1, x2 being Integer st
( x1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . ((t . (intloc (2 + 1))) - k) & x2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . m & x1 >= x2 ) ) & ( for i being Element of NAT st i >= (t . (intloc (2 + 1))) - k & i <= t . (intloc (2 + 1)) holds
ex n being Element of NAT st
( n >= (t . (intloc (2 + 1))) - k & n <= t . (intloc (2 + 1)) & ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) ) ; ::_thesis: verum
end;

Lm32:  for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA holds
( (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (intloc (1 + 1)) = (s . (intloc (0 + 1))) - 1 & (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (intloc (2 + 1)) = len (s . (fsloc 0)) & (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (fsloc 0) = s . (fsloc 0) )
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA holds
( (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (intloc (1 + 1)) = (s . (intloc (0 + 1))) - 1 & (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (intloc (2 + 1)) = len (s . (fsloc 0)) & (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (fsloc 0) = s . (fsloc 0) )
let s be State of SCM+FSA; ::_thesis: ( (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (intloc (1 + 1)) = (s . (intloc (0 + 1))) - 1 & (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (intloc (2 + 1)) = len (s . (fsloc 0)) & (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (fsloc 0) = s . (fsloc 0) )
set s0 = Initialized s;
set s1 = Exec (((intloc (1 + 1)) := (intloc (0 + 1))),(Initialized s));
set s2 = IExec ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),p,s);
set s3 = IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s);
A1: (Exec (((intloc (1 + 1)) := (intloc (0 + 1))),(Initialized s))) . (intloc (1 + 1)) = (Initialized s) . (intloc (0 + 1)) by SCMFSA_2:63
.= s . (intloc (0 + 1)) by SCMFSA_M:37 ;
A2: (Exec (((intloc (1 + 1)) := (intloc (0 + 1))),(Initialized s))) . (fsloc 0) = (Initialized s) . (fsloc 0) by SCMFSA_2:63
.= s . (fsloc 0) by SCMFSA_M:37 ;
A3: (Exec (((intloc (1 + 1)) := (intloc (0 + 1))),(Initialized s))) . (intloc 0) = (Initialized s) . (intloc 0) by SCMFSA_2:63
.= 1 by SCMFSA_M:9 ;
A4: (IExec ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),p,s)) . (fsloc 0) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(Exec (((intloc (1 + 1)) := (intloc (0 + 1))),(Initialized s))))) . (fsloc 0) by SCMFSA6C:9
.= s . (fsloc 0) by A2, SCMFSA_2:65 ;
A5: (IExec ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),p,s)) . (intloc (1 + 1)) = (Exec ((SubFrom ((intloc (1 + 1)),(intloc 0))),(Exec (((intloc (1 + 1)) := (intloc (0 + 1))),(Initialized s))))) . (intloc (1 + 1)) by SCMFSA6C:8
.= (s . (intloc (0 + 1))) - 1 by A1, A3, SCMFSA_2:65 ;
thus (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (intloc (1 + 1)) = (Exec (((intloc (2 + 1)) :=len (fsloc 0)),(IExec ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),p,s)))) . (intloc (1 + 1)) by SCMFSA6C:6
.= (s . (intloc (0 + 1))) - 1 by A5, Lm10, SCMFSA_2:74 ; ::_thesis: ( (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (intloc (2 + 1)) = len (s . (fsloc 0)) & (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (fsloc 0) = s . (fsloc 0) )
thus (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (intloc (2 + 1)) = (Exec (((intloc (2 + 1)) :=len (fsloc 0)),(IExec ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),p,s)))) . (intloc (2 + 1)) by SCMFSA6C:6
.= len (s . (fsloc 0)) by A4, SCMFSA_2:74 ; ::_thesis: (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (fsloc 0) = s . (fsloc 0)
thus (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),p,s)) . (fsloc 0) = (Exec (((intloc (2 + 1)) :=len (fsloc 0)),(IExec ((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))),p,s)))) . (fsloc 0) by SCMFSA6C:7
.= s . (fsloc 0) by A4, SCMFSA_2:74 ; ::_thesis: verum
end;

Lm33:  for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA st s . (intloc (0 + 1)) = len (s . (fsloc 0)) holds
( s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2 ) )
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA st s . (intloc (0 + 1)) = len (s . (fsloc 0)) holds
( s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2 ) )
let s be State of SCM+FSA; ::_thesis: ( s . (intloc (0 + 1)) = len (s . (fsloc 0)) implies ( s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2 ) ) )
assume A1: s . (intloc (0 + 1)) = len (s . (fsloc 0)) ; ::_thesis: ( s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2 ) )
percases ( len (s . (fsloc 0)) = 0 or len (s . (fsloc 0)) <> 0 ) ;
supposeA2: len (s . (fsloc 0)) = 0 ; ::_thesis: ( s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2 ) )
hence  s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0) are_fiberwise_equipotent by A1, Lm23, SCM_HALT:67; ::_thesis: for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2
thus  for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2 by A2; ::_thesis: verum
end;
supposeA3: len (s . (fsloc 0)) <> 0 ; ::_thesis: ( s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2 ) )
defpred S1[ Nat] means  for t being State of SCM+FSA
for q being Instruction-Sequence of SCM+FSA st t . (intloc (0 + 1)) = $1 + 1 & t . (intloc (0 + 1)) <= len (t . (fsloc 0)) holds
( t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - $1 & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) & ( for i being Element of NAT st i < (len (t . (fsloc 0))) - $1 & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - $1 & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - $1 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) );
set B11 = SubFrom ((intloc (0 + 1)),(intloc 0));
A4: S1[ 0 ]
proof
let t be State of SCM+FSA; ::_thesis: for q being Instruction-Sequence of SCM+FSA st t . (intloc (0 + 1)) = 0 + 1 & t . (intloc (0 + 1)) <= len (t . (fsloc 0)) holds
( t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) & ( for i being Element of NAT st i < (len (t . (fsloc 0))) - 0 & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
let q be Instruction-Sequence of SCM+FSA; ::_thesis: ( t . (intloc (0 + 1)) = 0 + 1 & t . (intloc (0 + 1)) <= len (t . (fsloc 0)) implies ( t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) & ( for i being Element of NAT st i < (len (t . (fsloc 0))) - 0 & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) ) )
assume that
A5: t . (intloc (0 + 1)) = 0 + 1 and
 t . (intloc (0 + 1)) <= len (t . (fsloc 0)) ; ::_thesis: ( t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) & ( for i being Element of NAT st i < (len (t . (fsloc 0))) - 0 & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
set IB = IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t);
A6: (IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (intloc (0 + 1)) = 1 - 1 by A5, Lm23, Lm25, SCM_HALT:66;
A7: (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (1 + 1)) = 1 - 1 by A5, Lm32;
A8: (IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) = (IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,(IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)))) . (fsloc 0) by A5, Lm23, Lm25, SCM_HALT:69
.= (IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0) by A6, Lm23, SCM_HALT:67
.= (Exec ((SubFrom ((intloc (0 + 1)),(intloc 0))),(IExec ((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))),q,t)))) . (fsloc 0) by Lm23, SCM_HALT:24
.= (IExec ((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))),q,t)) . (fsloc 0) by SCMFSA_2:65
.= (IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)))) . (fsloc 0) by Lm23, SCM_HALT:21
.= (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (fsloc 0) by A7, SCM_HALT:67
.= t . (fsloc 0) by Lm32 ;
hence  t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) are_fiberwise_equipotent ; ::_thesis: ( ( for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) & ( for i being Element of NAT st i < (len (t . (fsloc 0))) - 0 & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
thus  for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 by XXREAL_0:2; ::_thesis: ( ( for i being Element of NAT st i < (len (t . (fsloc 0))) - 0 & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
thus  for i being Element of NAT st i < (len (t . (fsloc 0))) - 0 & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i by A8; ::_thesis: for i being Element of NAT st i >= (len (t . (fsloc 0))) - 0 & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
let i be Element of NAT ; ::_thesis: ( i >= (len (t . (fsloc 0))) - 0 & i <= len (t . (fsloc 0)) implies ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) )
assume that
A9: i >= (len (t . (fsloc 0))) - 0 and
A10: i <= len (t . (fsloc 0)) ; ::_thesis: ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
take n = i; ::_thesis: ( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  ( n >= (len (t . (fsloc 0))) - 0 & n <= len (t . (fsloc 0)) ) by A9, A10; ::_thesis: ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n
thus  ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n by A8; ::_thesis: verum
end;
A11: 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 A12: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_t_being_State_of_SCM+FSA
for_q_being_Instruction-Sequence_of_SCM+FSA_st_t_._(intloc_(0_+_1))_=_(k_+_1)_+_1_&_t_._(intloc_(0_+_1))_<=_len_(t_._(fsloc_0))_holds_
(_t_._(fsloc_0),(IExec_((Times_((intloc_(0_+_1)),(((((intloc_(1_+_1))_:=_(intloc_(0_+_1)))_";"_(SubFrom_((intloc_(1_+_1)),(intloc_0))))_";"_((intloc_(2_+_1))_:=len_(fsloc_0)))_";"_(Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA))))))))),q,t))_._(fsloc_0)_are_fiberwise_equipotent_&_(_for_i,_j_being_Element_of_NAT_st_i_>=_(len_(t_._(fsloc_0)))_-_(k_+_1)_&_j_<=_len_(t_._(fsloc_0))_&_i_<_j_holds_
for_x1,_x2_being_Integer_st_x1_=_((IExec_((Times_((intloc_(0_+_1)),(((((intloc_(1_+_1))_:=_(intloc_(0_+_1)))_";"_(SubFrom_((intloc_(1_+_1)),(intloc_0))))_";"_((intloc_(2_+_1))_:=len_(fsloc_0)))_";"_(Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA))))))))),q,t))_._(fsloc_0))_._i_&_x2_=_((IExec_((Times_((intloc_(0_+_1)),(((((intloc_(1_+_1))_:=_(intloc_(0_+_1)))_";"_(SubFrom_((intloc_(1_+_1)),(intloc_0))))_";"_((intloc_(2_+_1))_:=len_(fsloc_0)))_";"_(Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA))))))))),q,t))_._(fsloc_0))_._j_holds_
x1_>=_x2_)_&_(_for_i_being_Element_of_NAT_st_i_<_(len_(t_._(fsloc_0)))_-_(k_+_1)_&_i_>=_1_holds_
((IExec_((Times_((intloc_(0_+_1)),(((((intloc_(1_+_1))_:=_(intloc_(0_+_1)))_";"_(SubFrom_((intloc_(1_+_1)),(intloc_0))))_";"_((intloc_(2_+_1))_:=len_(fsloc_0)))_";"_(Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA))))))))),q,t))_._(fsloc_0))_._i_=_(t_._(fsloc_0))_._i_)_&_(_for_i_being_Element_of_NAT_st_i_>=_(len_(t_._(fsloc_0)))_-_(k_+_1)_&_i_<=_len_(t_._(fsloc_0))_holds_
ex_n_being_Element_of_NAT_st_
(_n_>=_(len_(t_._(fsloc_0)))_-_(k_+_1)_&_n_<=_len_(t_._(fsloc_0))_&_((IExec_((Times_((intloc_(0_+_1)),(((((intloc_(1_+_1))_:=_(intloc_(0_+_1)))_";"_(SubFrom_((intloc_(1_+_1)),(intloc_0))))_";"_((intloc_(2_+_1))_:=len_(fsloc_0)))_";"_(Times_((intloc_(1_+_1)),(((((((intloc_(3_+_1))_:=_(intloc_(2_+_1)))_";"_(SubFrom_((intloc_(2_+_1)),(intloc_0))))_";"_((intloc_(4_+_1))_:=_((fsloc_0),(intloc_(2_+_1)))))_";"_((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1)))))_";"_(SubFrom_((intloc_(5_+_1)),(intloc_(4_+_1)))))_";"_(if>0_((intloc_(5_+_1)),((((intloc_(5_+_1))_:=_((fsloc_0),(intloc_(3_+_1))))_";"_(((fsloc_0),(intloc_(2_+_1)))_:=_(intloc_(5_+_1))))_";"_(((fsloc_0),(intloc_(3_+_1)))_:=_(intloc_(4_+_1)))),(Stop_SCM+FSA))))))))),q,t))_._(fsloc_0))_._i_=_(t_._(fsloc_0))_._n_)_)_)
let t be State of SCM+FSA; ::_thesis: for q being Instruction-Sequence of SCM+FSA st t . (intloc (0 + 1)) = (k + 1) + 1 & t . (intloc (0 + 1)) <= len (t . (fsloc 0)) holds
( t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) & ( for i being Element of NAT st i < (len (t . (fsloc 0))) - (k + 1) & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
let q be Instruction-Sequence of SCM+FSA; ::_thesis: ( t . (intloc (0 + 1)) = (k + 1) + 1 & t . (intloc (0 + 1)) <= len (t . (fsloc 0)) implies ( t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) & ( for i being Element of NAT st i < (len (t . (fsloc 0))) - (k + 1) & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) ) )
set t1 = IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t);
set IB = IExec ((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))),q,t);
set t2 = IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,(IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)));
assume that
A13: t . (intloc (0 + 1)) = (k + 1) + 1 and
A14: t . (intloc (0 + 1)) <= len (t . (fsloc 0)) ; ::_thesis: ( t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) & ( for i being Element of NAT st i < (len (t . (fsloc 0))) - (k + 1) & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
A15: (IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (intloc (0 + 1)) = (Exec ((SubFrom ((intloc (0 + 1)),(intloc 0))),(IExec ((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))),q,t)))) . (intloc (0 + 1)) by Lm23, SCM_HALT:23
.= ((IExec ((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))),q,t)) . (intloc (0 + 1))) - ((IExec ((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))),q,t)) . (intloc 0)) by SCMFSA_2:65
.= ((IExec ((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))),q,t)) . (intloc (0 + 1))) - 1 by Lm23, SCM_HALT:9
.= ((Initialized t) . (intloc (0 + 1))) - 1 by Lm23, Lm25, SCM_HALT:53
.= ((k + 1) + 1) - 1 by A13, SCMFSA_M:37
.= k + 1 ;
then  (IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (intloc (0 + 1)) < t . (intloc (0 + 1)) by A13, XREAL_1:29;
then A16: (IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (intloc (0 + 1)) <= len (t . (fsloc 0)) by A14, XXREAL_0:2;
set Ts = IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t);
A17: (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (1 + 1)) = ((k + 1) + 1) - 1 by A13, Lm32
.= k + 1 ;
A18: (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1)) = len (t . (fsloc 0)) by Lm32;
then A19: (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1)) = len ((IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (fsloc 0)) by Lm32;
A20: k + 1 < (k + 1) + 1 by XREAL_1:29;
A21: k + 1 < t . (intloc (0 + 1)) by A13, XREAL_1:29;
A22: k + 1 < len (t . (fsloc 0)) by A13, A14, A20, XXREAL_0:2;
A23: k + 1 < (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1)) by A14, A18, A21, XXREAL_0:2;
A24: (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (fsloc 0),(IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)))) . (fsloc 0) are_fiberwise_equipotent by A17, A18, A19, A22, Lm31;
A25: (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (fsloc 0) = t . (fsloc 0) by Lm32;
A26: (IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0) = (Exec ((SubFrom ((intloc (0 + 1)),(intloc 0))),(IExec ((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))),q,t)))) . (fsloc 0) by Lm23, SCM_HALT:24
.= (IExec ((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))),q,t)) . (fsloc 0) by SCMFSA_2:65
.= (IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)))) . (fsloc 0) by Lm23, SCM_HALT:21 ;
then A27: t . (fsloc 0),(IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0) are_fiberwise_equipotent by A17, A18, A23, A25, Lm31;
A28: len (t . (fsloc 0)) = len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A24, A25, A26, RFINSEQ:3;
A29: (IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (intloc (0 + 1)) <= len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) by A16, A27, RFINSEQ:3;
A30: (IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,(IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)))) . (fsloc 0) are_fiberwise_equipotent by A12, A15, A16, A28;
A31: (IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) = (IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,(IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)))) . (fsloc 0) by A13, Lm23, Lm25, SCM_HALT:69;
hence  t . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0) are_fiberwise_equipotent by A27, A30, CLASSES1:76; ::_thesis: ( ( for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ) & ( for i being Element of NAT st i < (len (t . (fsloc 0))) - (k + 1) & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
set lk = (len (t . (fsloc 0))) - (k + 1);
A32: ((len (t . (fsloc 0))) - (k + 1)) + 1 = (len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0))) - k by A28;
thus  for i, j being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & j <= len (t . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 ::_thesis: ( ( for i being Element of NAT st i < (len (t . (fsloc 0))) - (k + 1) & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ) & ( for i being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ) )
proof
let i, j be Element of NAT ; ::_thesis: ( i >= (len (t . (fsloc 0))) - (k + 1) & j <= len (t . (fsloc 0)) & i < j implies for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2 )
assume that
A33: i >= (len (t . (fsloc 0))) - (k + 1) and
A34: j <= len (t . (fsloc 0)) and
A35: i < j ; ::_thesis: for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j holds
x1 >= x2
 j > (len (t . (fsloc 0))) - (k + 1) by A33, A35, XXREAL_0:2;
then  j >= (len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0))) - k by A32, INT_1:7;
then consider n being Element of NAT  such that
A36: n >= (len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0))) - k and
A37: n <= len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) and
A38: ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,(IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)))) . (fsloc 0)) . j = ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) . n by A12, A15, A16, A28, A34;
 (len (t . (fsloc 0))) - (k + 1) < ((len (t . (fsloc 0))) - (k + 1)) + 1 by XREAL_1:29;
then A39: n >= ((IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1))) - (k + 1) by A18, A28, A36, XXREAL_0:2;
A40: n <= (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1)) by A28, A37, Lm32;
hereby  ::_thesis: verum
let x1, x2 be Integer; ::_thesis: ( x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j implies b1 >= b2 )
assume that
A41: x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i and
A42: x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . j ; ::_thesis: b1 >= b2
percases ( i = (len (t . (fsloc 0))) - (k + 1) or i <> (len (t . (fsloc 0))) - (k + 1) ) ;
supposeA43: i = (len (t . (fsloc 0))) - (k + 1) ; ::_thesis: b1 >= b2
A44: ex y1, y2 being Integer st
( y1 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)))) . (fsloc 0)) . (((IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1))) - (k + 1)) & y2 = ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)))) . (fsloc 0)) . n & y1 >= y2 ) by A17, A19, A23, A39, A40, Lm31;
A45: i < (len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0))) - k by A32, A43, XREAL_1:29;
A46: 1 <= i by A13, A14, A43, XREAL_1:19;
 i = ((IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1))) - (k + 1) by A43, Lm32;
hence  x1 >= x2 by A12, A15, A16, A26, A28, A31, A38, A41, A42, A44, A45, A46; ::_thesis: verum
end;
suppose i <> (len (t . (fsloc 0))) - (k + 1) ; ::_thesis: b1 >= b2
then  i > (len (t . (fsloc 0))) - (k + 1) by A33, XXREAL_0:1;
then  i >= (len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0))) - k by A32, INT_1:7;
hence  x1 >= x2 by A12, A15, A16, A28, A31, A34, A35, A41, A42; ::_thesis: verum
end;
end;
end;
end;
thus  for i being Element of NAT st i < (len (t . (fsloc 0))) - (k + 1) & i >= 1 holds
((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i ::_thesis: for i being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
proof
let i be Element of NAT ; ::_thesis: ( i < (len (t . (fsloc 0))) - (k + 1) & i >= 1 implies ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i )
assume that
A47: i < (len (t . (fsloc 0))) - (k + 1) and
A48: i >= 1 ; ::_thesis: ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . i
 (len (t . (fsloc 0))) - (k + 1) < ((len (t . (fsloc 0))) - (k + 1)) + 1 by XREAL_1:29;
then  i < (len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0))) - k by A28, A47, XXREAL_0:2;
hence ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) . i by A12, A15, A29, A31, A48
.= (t . (fsloc 0)) . i by A17, A18, A23, A25, A26, A47, A48, Lm31 ;
::_thesis: verum
end;
thus  for i being Element of NAT st i >= (len (t . (fsloc 0))) - (k + 1) & i <= len (t . (fsloc 0)) holds
ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) ::_thesis: verum
proof
let i be Element of NAT ; ::_thesis: ( i >= (len (t . (fsloc 0))) - (k + 1) & i <= len (t . (fsloc 0)) implies ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n ) )
assume that
A49: i >= (len (t . (fsloc 0))) - (k + 1) and
A50: i <= len (t . (fsloc 0)) ; ::_thesis: ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
percases ( i = (len (t . (fsloc 0))) - (k + 1) or i <> (len (t . (fsloc 0))) - (k + 1) ) ;
supposeA51: i = (len (t . (fsloc 0))) - (k + 1) ; ::_thesis: ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
then A52: i < (len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0))) - k by A32, XREAL_1:29;
A53: i >= 1 by A13, A14, A51, XREAL_1:19;
consider n being Element of NAT  such that
A54: n >= ((IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1))) - (k + 1) and
A55: n <= (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1)) and
A56: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)))) . (fsloc 0)) . i = ((IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (fsloc 0)) . n by A17, A18, A19, A22, A49, A50, Lm31;
take  n ; ::_thesis: ( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  n >= (len (t . (fsloc 0))) - (k + 1) by A54, Lm32; ::_thesis: ( n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  n <= len (t . (fsloc 0)) by A55, Lm32; ::_thesis: ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n
thus  ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n by A12, A15, A25, A26, A29, A31, A52, A53, A56; ::_thesis: verum
end;
suppose i <> (len (t . (fsloc 0))) - (k + 1) ; ::_thesis: ex n being Element of NAT st
( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
then  i > (len (t . (fsloc 0))) - (k + 1) by A49, XXREAL_0:1;
then  i >= (len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0))) - k by A32, INT_1:7;
then consider m being Element of NAT  such that
A57: m >= (len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0))) - k and
A58: m <= len ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) and
A59: ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,(IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)))) . (fsloc 0)) . i = ((IExec (((((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))) ";" (SubFrom ((intloc (0 + 1)),(intloc 0)))),q,t)) . (fsloc 0)) . m by A12, A15, A16, A28, A50;
 ((len (t . (fsloc 0))) - (k + 1)) + 1 > (len (t . (fsloc 0))) - (k + 1) by XREAL_1:29;
then  m > ((IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1))) - (k + 1) by A18, A28, A57, XXREAL_0:2;
then consider n being Element of NAT  such that
A60: n >= ((IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1))) - (k + 1) and
A61: n <= (IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (intloc (2 + 1)) and
A62: ((IExec ((Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA)))))),q,(IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)))) . (fsloc 0)) . m = ((IExec (((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))),q,t)) . (fsloc 0)) . n by A17, A18, A19, A22, A28, A58, Lm31;
take  n ; ::_thesis: ( n >= (len (t . (fsloc 0))) - (k + 1) & n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  n >= (len (t . (fsloc 0))) - (k + 1) by A60, Lm32; ::_thesis: ( n <= len (t . (fsloc 0)) & ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n )
thus  n <= len (t . (fsloc 0)) by A61, Lm32; ::_thesis: ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n
thus  ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),q,t)) . (fsloc 0)) . i = (t . (fsloc 0)) . n by A26, A31, A59, A62, Lm32; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A63: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A4, A11);
 s . (intloc (0 + 1)) >= 1 + 0 by A1, A3, INT_1:7;
then reconsider m = (s . (intloc (0 + 1))) - 1 as Element of NAT by INT_1:5;
A64: m + 1 = s . (intloc (0 + 1)) ;
hence  s . (fsloc 0),(IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0) are_fiberwise_equipotent by A1, A63; ::_thesis: for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2
 (len (s . (fsloc 0))) - m = 1 by A1;
hence  for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,s)) . (fsloc 0)) . j holds
x1 >= x2 by A63, A64; ::_thesis: verum
end;
end;
end;

theorem Th41: :: SCMBSORT:41
for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA holds
( s . (fsloc 0),(IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . j holds
x1 >= x2 ) )
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA holds
( s . (fsloc 0),(IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . j holds
x1 >= x2 ) )
let s be State of SCM+FSA; ::_thesis: ( s . (fsloc 0),(IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0) are_fiberwise_equipotent & ( for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . j holds
x1 >= x2 ) )
set W27 = ((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))) ";" ((intloc (0 + 1)) :=len (fsloc 0));
set s0 = Initialized s;
set s1 = Exec (((intloc (1 + 1)) := (intloc 0)),(Initialized s));
set s2 = IExec ((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))),p,s);
set s3 = IExec (((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))),p,s);
set s4 = IExec ((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))),p,s);
set s5 = IExec (((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))),p,s);
set s6 = IExec ((((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))) ";" ((intloc (0 + 1)) :=len (fsloc 0))),p,s);
A1: (IExec (((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))),p,s)) . (fsloc 0) = (Exec (((intloc (5 + 1)) := (intloc 0)),(IExec ((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))),p,s)))) . (fsloc 0) by SCMFSA6C:7
.= (IExec ((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))),p,s)) . (fsloc 0) by SCMFSA_2:63
.= (Exec (((intloc (4 + 1)) := (intloc 0)),(IExec (((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))),p,s)))) . (fsloc 0) by SCMFSA6C:7
.= (IExec (((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))),p,s)) . (fsloc 0) by SCMFSA_2:63
.= (Exec (((intloc (3 + 1)) := (intloc 0)),(IExec ((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))),p,s)))) . (fsloc 0) by SCMFSA6C:7
.= (IExec ((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))),p,s)) . (fsloc 0) by SCMFSA_2:63
.= (Exec (((intloc (2 + 1)) := (intloc 0)),(Exec (((intloc (1 + 1)) := (intloc 0)),(Initialized s))))) . (fsloc 0) by SCMFSA6C:9
.= (Exec (((intloc (1 + 1)) := (intloc 0)),(Initialized s))) . (fsloc 0) by SCMFSA_2:63
.= (Initialized s) . (fsloc 0) by SCMFSA_2:63
.= s . (fsloc 0) by SCMFSA_M:37 ;
A2: (IExec ((((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))) ";" ((intloc (0 + 1)) :=len (fsloc 0))),p,s)) . (fsloc 0) = (Exec (((intloc (0 + 1)) :=len (fsloc 0)),(IExec (((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))),p,s)))) . (fsloc 0) by SCMFSA6C:7
.= s . (fsloc 0) by A1, SCMFSA_2:74 ;
A3: (IExec ((((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))) ";" ((intloc (0 + 1)) :=len (fsloc 0))),p,s)) . (intloc (0 + 1)) = (Exec (((intloc (0 + 1)) :=len (fsloc 0)),(IExec (((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))),p,s)))) . (intloc (0 + 1)) by SCMFSA6C:6
.= len ((IExec ((((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))) ";" ((intloc (0 + 1)) :=len (fsloc 0))),p,s)) . (fsloc 0)) by A1, A2, SCMFSA_2:74 ;
A4: (IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0) = (IExec ((Times ((intloc (0 + 1)),(((((intloc (1 + 1)) := (intloc (0 + 1))) ";" (SubFrom ((intloc (1 + 1)),(intloc 0)))) ";" ((intloc (2 + 1)) :=len (fsloc 0))) ";" (Times ((intloc (1 + 1)),(((((((intloc (3 + 1)) := (intloc (2 + 1))) ";" (SubFrom ((intloc (2 + 1)),(intloc 0)))) ";" ((intloc (4 + 1)) := ((fsloc 0),(intloc (2 + 1))))) ";" ((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1))))) ";" (SubFrom ((intloc (5 + 1)),(intloc (4 + 1))))) ";" (if>0 ((intloc (5 + 1)),((((intloc (5 + 1)) := ((fsloc 0),(intloc (3 + 1)))) ";" (((fsloc 0),(intloc (2 + 1))) := (intloc (5 + 1)))) ";" (((fsloc 0),(intloc (3 + 1))) := (intloc (4 + 1)))),(Stop SCM+FSA))))))))),p,(IExec ((((((((intloc (1 + 1)) := (intloc 0)) ";" ((intloc (2 + 1)) := (intloc 0))) ";" ((intloc (3 + 1)) := (intloc 0))) ";" ((intloc (4 + 1)) := (intloc 0))) ";" ((intloc (5 + 1)) := (intloc 0))) ";" ((intloc (0 + 1)) :=len (fsloc 0))),p,s)))) . (fsloc 0) by Lm26, SCM_HALT:21;
hence  s . (fsloc 0),(IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0) are_fiberwise_equipotent by A2, A3, Lm33; ::_thesis: for i, j being Element of NAT st i >= 1 & j <= len (s . (fsloc 0)) & i < j holds
for x1, x2 being Integer st x1 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . j holds
x1 >= x2
let i, j be Element of NAT ; ::_thesis: ( i >= 1 & j <= len (s . (fsloc 0)) & i < j implies for x1, x2 being Integer st x1 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . j holds
x1 >= x2 )
assume that
A5: i >= 1 and
A6: j <= len (s . (fsloc 0)) and
A7: i < j ; ::_thesis: for x1, x2 being Integer st x1 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . j holds
x1 >= x2
thus  for x1, x2 being Integer st x1 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . i & x2 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . j holds
x1 >= x2 by A2, A3, A4, A5, A6, A7, Lm33; ::_thesis: verum
end;

theorem Th42: :: SCMBSORT:42
for i being Element of NAT
for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P holds
for w being FinSequence of INT st Initialized ((fsloc 0) .--> w) c= s holds
IC (Comput (P,s,i)) in dom Bubble-Sort-Algorithm
proof
set Ba = Bubble-Sort-Algorithm ;
set Ib = ((intloc 0) .--> 1) +* (Start-At (0,SCM+FSA));
let i be Element of NAT ; ::_thesis: for s being State of SCM+FSA
for P being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P holds
for w being FinSequence of INT st Initialized ((fsloc 0) .--> w) c= s holds
IC (Comput (P,s,i)) in dom Bubble-Sort-Algorithm
let s be State of SCM+FSA; ::_thesis: for P being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P holds
for w being FinSequence of INT st Initialized ((fsloc 0) .--> w) c= s holds
IC (Comput (P,s,i)) in dom Bubble-Sort-Algorithm
let P be Instruction-Sequence of SCM+FSA; ::_thesis: ( Bubble-Sort-Algorithm c= P implies for w being FinSequence of INT st Initialized ((fsloc 0) .--> w) c= s holds
IC (Comput (P,s,i)) in dom Bubble-Sort-Algorithm )
assume A1: Bubble-Sort-Algorithm c= P ; ::_thesis: for w being FinSequence of INT st Initialized ((fsloc 0) .--> w) c= s holds
IC (Comput (P,s,i)) in dom Bubble-Sort-Algorithm
let w be FinSequence of INT ; ::_thesis: ( Initialized ((fsloc 0) .--> w) c= s implies IC (Comput (P,s,i)) in dom Bubble-Sort-Algorithm )
set x = (fsloc 0) .--> w;
assume A2: Initialized ((fsloc 0) .--> w) c= s ; ::_thesis: IC (Comput (P,s,i)) in dom Bubble-Sort-Algorithm
set BSA = Bubble-Sort-Algorithm ;
 Initialize ((intloc 0) .--> 1) c= Initialized ((fsloc 0) .--> w) by FUNCT_4:25;
then  Initialize ((intloc 0) .--> 1) c= s by A2, XBOOLE_1:1;
hence  IC (Comput (P,s,i)) in dom Bubble-Sort-Algorithm by Lm26, A1, SCM_HALT:def_1; ::_thesis: verum
end;

theorem Th43: :: SCMBSORT:43
for p being Instruction-Sequence of SCM+FSA
for s being State of SCM+FSA
for t being FinSequence of INT st (Initialize ((intloc 0) .--> 1)) +* ((fsloc 0) .--> t) c= s & Bubble-Sort-Algorithm c= p holds
ex u being FinSequence of REAL st
( t,u are_fiberwise_equipotent & u is non-increasing & u is FinSequence of INT & (Result (p,s)) . (fsloc 0) = u )
proof
let p be Instruction-Sequence of SCM+FSA; ::_thesis: for s being State of SCM+FSA
for t being FinSequence of INT st (Initialize ((intloc 0) .--> 1)) +* ((fsloc 0) .--> t) c= s & Bubble-Sort-Algorithm c= p holds
ex u being FinSequence of REAL st
( t,u are_fiberwise_equipotent & u is non-increasing & u is FinSequence of INT & (Result (p,s)) . (fsloc 0) = u )
let s be State of SCM+FSA; ::_thesis: for t being FinSequence of INT st (Initialize ((intloc 0) .--> 1)) +* ((fsloc 0) .--> t) c= s & Bubble-Sort-Algorithm c= p holds
ex u being FinSequence of REAL st
( t,u are_fiberwise_equipotent & u is non-increasing & u is FinSequence of INT & (Result (p,s)) . (fsloc 0) = u )
let t be FinSequence of INT ; ::_thesis: ( (Initialize ((intloc 0) .--> 1)) +* ((fsloc 0) .--> t) c= s & Bubble-Sort-Algorithm c= p implies ex u being FinSequence of REAL st
( t,u are_fiberwise_equipotent & u is non-increasing & u is FinSequence of INT & (Result (p,s)) . (fsloc 0) = u ) )
set Ba = Bubble-Sort-Algorithm ;
set pp = Initialize ((intloc 0) .--> 1);
set x = (fsloc 0) .--> t;
set z = (IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0);
assume that
A1: (Initialize ((intloc 0) .--> 1)) +* ((fsloc 0) .--> t) c= s and
A2: Bubble-Sort-Algorithm c= p ; ::_thesis: ex u being FinSequence of REAL st
( t,u are_fiberwise_equipotent & u is non-increasing & u is FinSequence of INT & (Result (p,s)) . (fsloc 0) = u )
A3: p +* Bubble-Sort-Algorithm = p by A2, FUNCT_4:98;
 dom ((fsloc 0) .--> t) = {(fsloc 0)} by FUNCOP_1:13;
then A4: fsloc 0 in dom ((fsloc 0) .--> t) by TARSKI:def_1;
then  fsloc 0 in dom ((Initialize ((intloc 0) .--> 1)) +* ((fsloc 0) .--> t)) by FUNCT_4:12;
then A5: s . (fsloc 0) = ((Initialize ((intloc 0) .--> 1)) +* ((fsloc 0) .--> t)) . (fsloc 0) by A1, GRFUNC_1:2
.= ((fsloc 0) .--> t) . (fsloc 0) by A4, FUNCT_4:13
.= t by FUNCOP_1:72 ;
A6: s . (fsloc 0),(IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0) are_fiberwise_equipotent by Th41;
reconsider u = (IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0) as FinSequence of REAL by FINSEQ_3:117;
take  u ; ::_thesis: ( t,u are_fiberwise_equipotent & u is non-increasing & u is FinSequence of INT & (Result (p,s)) . (fsloc 0) = u )
thus  t,u are_fiberwise_equipotent by A5, Th41; ::_thesis: ( u is non-increasing & u is FinSequence of INT & (Result (p,s)) . (fsloc 0) = u )
A7: dom (s . (fsloc 0)) = dom u by A6, RFINSEQ:3;
now__::_thesis:_for_i,_j_being_Element_of_NAT_st_i_in_dom_u_&_j_in_dom_u_&_i_<_j_holds_
u_._i_>=_u_._j
let i, j be Element of NAT ; ::_thesis: ( i in dom u & j in dom u & i < j implies u . i >= u . j )
assume that
A8: i in dom u and
A9: j in dom u and
A10: i < j ; ::_thesis: u . i >= u . j
A11: i >= 1 by A8, FINSEQ_3:25;
A12: j <= len (s . (fsloc 0)) by A7, A9, FINSEQ_3:25;
reconsider y1 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . i as Integer ;
reconsider y2 = ((IExec ((bubble-sort (fsloc 0)),p,s)) . (fsloc 0)) . j as Integer ;
thus  u . i >= u . j by A10, A11, A12, Th41; ::_thesis: verum
end;
hence  u is non-increasing by RFINSEQ:19; ::_thesis: ( u is FinSequence of INT & (Result (p,s)) . (fsloc 0) = u )
thus  u is FinSequence of INT ; ::_thesis: (Result (p,s)) . (fsloc 0) = u
 dom (Initialize ((intloc 0) .--> 1)) misses dom ((fsloc 0) .--> t) by SCMFSA_M:32;
then  Initialize ((intloc 0) .--> 1) c= (Initialize ((intloc 0) .--> 1)) +* ((fsloc 0) .--> t) by FUNCT_4:32;
then  s = s +* (Initialize ((intloc 0) .--> 1)) by A1, FUNCT_4:98, XBOOLE_1:1;
then  s = Initialized s ;
hence  (Result (p,s)) . (fsloc 0) = u by Th33, A3; ::_thesis: verum
end;

theorem Th44: :: SCMBSORT:44
for w being FinSequence of INT holds Initialized ((fsloc 0) .--> w) is Bubble-Sort-Algorithm -autonomic
proof
let w be FinSequence of INT ; ::_thesis: Initialized ((fsloc 0) .--> w) is Bubble-Sort-Algorithm -autonomic
set p = Initialized ((fsloc 0) .--> w);
set q = Bubble-Sort-Algorithm ;
A1: for P, Q being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q holds
for s1, s2 being State of SCM+FSA
for i being Element of NAT st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
proof
let P, Q be Instruction-Sequence of SCM+FSA; ::_thesis: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q implies for s1, s2 being State of SCM+FSA
for i being Element of NAT st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) ) )
assume A2: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q ) ; ::_thesis: for s1, s2 being State of SCM+FSA
for i being Element of NAT st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
let s1, s2 be State of SCM+FSA; ::_thesis: for i being Element of NAT st Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 holds
( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
let i be Element of NAT ; ::_thesis: ( Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 & i <= 10 implies ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) ) )
assume that
A3: Initialized ((fsloc 0) .--> w) c= s1 and
A4: Initialized ((fsloc 0) .--> w) c= s2 and
A5: i <= 10 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
A6: Bubble-Sort-Algorithm c= P by A2;
A7: Bubble-Sort-Algorithm c= Q by A2;
A8: Comput (P,s1,0) = s1 ;
A9: s1 is 0 -started by A3, MEMSTR_0:17;
A10: s2 is 0 -started by A4, MEMSTR_0:17;
A11: s1 . (intloc 0) = 1 by A3, SCMFSA_M:33
.= s2 . (intloc 0) by A4, SCMFSA_M:33 ;
A12: s1 . (fsloc 0) = w by A3, SCMFSA_M:33
.= s2 . (fsloc 0) by A4, SCMFSA_M:33 ;
A13: IC s1 = 0 by A9, MEMSTR_0:def_11
.= IC s2 by A10, MEMSTR_0:def_11 ;
percases ( i = 0 or i = 1 or i = 2 or i = 3 or i = 4 or i = 5 or i = 6 or i = 7 or i = 8 or i = 9 or i = 10 ) by A5, NAT_1:34;
supposeA14: i = 0 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence  (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) by A8, A11, EXTPRO_1:2; ::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus  (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) by A8, A13, A14, EXTPRO_1:2; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus  (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) by A8, A12, A14, EXTPRO_1:2; ::_thesis: verum
end;
supposeA15: i = 1 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A15, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 1 by A2, A9, A15, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A7, A10, A15, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A6, A9, A15, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A15, Lm21 ; ::_thesis: verum
end;
supposeA16: i = 2 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A6, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A16, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 2 by A2, A9, A16, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A10, A16, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A9, A16, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A16, Lm21 ; ::_thesis: verum
end;
supposeA17: i = 3 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A17, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 3 by A2, A9, A17, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A10, A17, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A6, A9, A17, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A17, Lm21 ; ::_thesis: verum
end;
supposeA18: i = 4 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A6, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A18, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 4 by A2, A9, A18, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A10, A18, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A9, A18, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A18, Lm21 ; ::_thesis: verum
end;
supposeA19: i = 5 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A6, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A19, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 5 by A2, A9, A19, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A10, A19, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A9, A19, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A19, Lm21 ; ::_thesis: verum
end;
supposeA20: i = 6 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A20, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 6 by A2, A9, A20, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A10, A20, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A9, A20, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A20, Lm21 ; ::_thesis: verum
end;
supposeA21: i = 7 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A6, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A21, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 7 by A2, A9, A21, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A10, A21, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A9, A21, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A21, Lm21 ; ::_thesis: verum
end;
supposeA22: i = 8 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A22, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 8 by A2, A9, A22, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A10, A22, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A9, A22, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A22, Lm21 ; ::_thesis: verum
end;
supposeA23: i = 9 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A23, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 9 by A2, A9, A23, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A10, A23, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A9, A23, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A23, Lm21 ; ::_thesis: verum
end;
supposeA24: i = 10 ; ::_thesis: ( (Comput (P,s1,i)) . (intloc 0) = (Comput (Q,s2,i)) . (intloc 0) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
hence (Comput (P,s1,i)) . (intloc 0) = s1 . (intloc 0) by A2, A9, Lm21
.= (Comput (Q,s2,i)) . (intloc 0) by A2, A10, A11, A24, Lm21 ;
::_thesis: ( (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) & (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0) )
thus (Comput (P,s1,i)) . (IC ) = 10 by A2, A9, A24, Lm21
.= (Comput (Q,s2,i)) . (IC ) by A2, A10, A24, Lm21 ; ::_thesis: (Comput (P,s1,i)) . (fsloc 0) = (Comput (Q,s2,i)) . (fsloc 0)
thus (Comput (P,s1,i)) . (fsloc 0) = s1 . (fsloc 0) by A2, A9, A24, Lm21
.= (Comput (Q,s2,i)) . (fsloc 0) by A2, A10, A12, A24, Lm21 ; ::_thesis: verum
end;
end;
end;
set UD = {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)};
set Us = (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm);
A25: UsedInt*Loc Bubble-Sort-Algorithm = {(fsloc 0)} by Th35;
A26: UsedIntLoc Bubble-Sort-Algorithm = {(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by Th34;
then A27: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) = {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by A25, ENUMSET1:22;
A28: for P, Q being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q holds
for i being Element of NAT
for s1, s2 being State of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
( (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) )
proof
let P, Q be Instruction-Sequence of SCM+FSA; ::_thesis: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q implies for i being Element of NAT
for s1, s2 being State of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
( (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) ) )
assume A29: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q ) ; ::_thesis: for i being Element of NAT
for s1, s2 being State of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
( (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) )
let i be Element of NAT ; ::_thesis: for s1, s2 being State of SCM+FSA st 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 holds
( (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) )
let s1, s2 be State of SCM+FSA; ::_thesis: ( 11 <= i & Initialized ((fsloc 0) .--> w) c= s1 & Initialized ((fsloc 0) .--> w) c= s2 implies ( (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) ) )
assume that
A30: 11 <= i and
A31: Initialized ((fsloc 0) .--> w) c= s1 and
A32: Initialized ((fsloc 0) .--> w) c= s2 ; ::_thesis: ( (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) )
set Cs11 = Comput (P,s1,11);
set Cs21 = Comput (Q,s2,11);
A33: s1 is 0 -started by A31, MEMSTR_0:17;
A34: s2 is 0 -started by A32, MEMSTR_0:17;
A35: s1 . (intloc 0) = 1 by A31, SCMFSA_M:33
.= s2 . (intloc 0) by A32, SCMFSA_M:33 ;
A36: s1 . (fsloc 0) = w by A31, SCMFSA_M:33
.= s2 . (fsloc 0) by A32, SCMFSA_M:33 ;
A37: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) c= dom (Comput (P,s1,11)) by Th32;
A38: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) c= dom (Comput (Q,s2,11)) by Th32;
now__::_thesis:_for_x_being_set_st_x_in_(UsedInt*Loc_Bubble-Sort-Algorithm)_\/_(UsedIntLoc_Bubble-Sort-Algorithm)_holds_
(Comput_(P,s1,11))_._x_=_(Comput_(Q,s2,11))_._x
let x be set ; ::_thesis: ( x in (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) implies (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1 )
assume  x in (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) ; ::_thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
then A39: x in {(fsloc 0),(intloc 0),(intloc 1),(intloc 2),(intloc 3),(intloc 4),(intloc 5),(intloc 6)} by A25, A26, ENUMSET1:22;
percases ( x = fsloc 0 or x = intloc 0 or x = intloc 1 or x = intloc 2 or x = intloc 3 or x = intloc 4 or x = intloc 5 or x = intloc 6 ) by A39, ENUMSET1:def_6;
supposeA40: x = fsloc 0 ; ::_thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (fsloc 0) by A33, A29, Lm21
.= (Comput (Q,s2,11)) . x by A34, A29, A36, A40, Lm21 ;
::_thesis: verum
end;
supposeA41: x = intloc 0 ; ::_thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A33, A29, Lm21
.= (Comput (Q,s2,11)) . x by A34, A29, A35, A41, Lm21 ;
::_thesis: verum
end;
supposeA42: x = intloc 1 ; ::_thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = len (s1 . (fsloc 0)) by A33, A29, Lm21
.= (Comput (Q,s2,11)) . x by A34, A29, A36, A42, Lm21 ;
::_thesis: verum
end;
supposeA43: x = intloc 2 ; ::_thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A33, A29, Lm21
.= (Comput (Q,s2,11)) . x by A34, A29, A35, A43, Lm21 ;
::_thesis: verum
end;
supposeA44: x = intloc 3 ; ::_thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A33, A29, Lm21
.= (Comput (Q,s2,11)) . x by A34, A29, A35, A44, Lm21 ;
::_thesis: verum
end;
supposeA45: x = intloc 4 ; ::_thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A33, A29, Lm21
.= (Comput (Q,s2,11)) . x by A34, A29, A35, A45, Lm21 ;
::_thesis: verum
end;
supposeA46: x = intloc 5 ; ::_thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A33, A29, Lm21
.= (Comput (Q,s2,11)) . x by A34, A29, A35, A46, Lm21 ;
::_thesis: verum
end;
supposeA47: x = intloc 6 ; ::_thesis: (Comput (P,s1,11)) . b1 = (Comput (Q,s2,11)) . b1
hence (Comput (P,s1,11)) . x = s1 . (intloc 0) by A33, A29, Lm21
.= (Comput (Q,s2,11)) . x by A34, A29, A35, A47, Lm21 ;
::_thesis: verum
end;
end;
end;
then A48: (Comput (P,s1,11)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,11)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) by A37, A38, FUNCT_1:95;
A49: (Comput (P,s1,11)) . (IC ) = 11 by A33, A29, Lm21
.= (Comput (Q,s2,11)) . (IC ) by A34, A29, Lm21 ;
A50: for i being Element of NAT holds IC (Comput (P,s1,i)) in dom Bubble-Sort-Algorithm by A31, Th42, A29;
 for i being Element of NAT holds IC (Comput (Q,s2,i)) in dom Bubble-Sort-Algorithm by A32, Th42, A29;
hence  ( (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) & (Comput (P,s1,i)) . (IC ) = (Comput (Q,s2,i)) . (IC ) ) by A30, A48, A49, A50, Th15, A29; ::_thesis: verum
end;
set DD = {(intloc 0),(IC ),(fsloc 0)};
let P, Q be Instruction-Sequence of SCM+FSA; :: according to EXTPRO_1:def_10 ::_thesis: ( not Bubble-Sort-Algorithm c= P or not Bubble-Sort-Algorithm c= Q or for b1, b2 being set holds
( not Initialized ((fsloc 0) .--> w) c= b1 or not Initialized ((fsloc 0) .--> w) c= b2 or for b3 being Element of NAT holds (Comput (P,b1,b3)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,b2,b3)) | (dom (Initialized ((fsloc 0) .--> w))) ) )
assume A51: ( Bubble-Sort-Algorithm c= P & Bubble-Sort-Algorithm c= Q ) ; ::_thesis: for b1, b2 being set holds
( not Initialized ((fsloc 0) .--> w) c= b1 or not Initialized ((fsloc 0) .--> w) c= b2 or for b3 being Element of NAT holds (Comput (P,b1,b3)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,b2,b3)) | (dom (Initialized ((fsloc 0) .--> w))) )
let s1, s2 be State of SCM+FSA; ::_thesis: ( not Initialized ((fsloc 0) .--> w) c= s1 or not Initialized ((fsloc 0) .--> w) c= s2 or for b1 being Element of NAT holds (Comput (P,s1,b1)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,b1)) | (dom (Initialized ((fsloc 0) .--> w))) )
assume that
A52: Initialized ((fsloc 0) .--> w) c= s1 and
A53: Initialized ((fsloc 0) .--> w) c= s2 ; ::_thesis: for b1 being Element of NAT holds (Comput (P,s1,b1)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,b1)) | (dom (Initialized ((fsloc 0) .--> w)))
let i be Element of NAT ; ::_thesis: (Comput (P,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,i)) | (dom (Initialized ((fsloc 0) .--> w)))
set Cs1i = Comput (P,s1,i);
set Cs2i = Comput (Q,s2,i);
A54: dom (Initialized ((fsloc 0) .--> w)) = {(intloc 0),(IC ),(fsloc 0)} by SCMFSA_M:31;
A55: {(intloc 0),(IC ),(fsloc 0)} c= dom (Comput (P,s1,i)) by SCMFSA_M:34;
A56: {(intloc 0),(IC ),(fsloc 0)} c= dom (Comput (Q,s2,i)) by SCMFSA_M:34;
A57: intloc 0 in (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) by A27, ENUMSET1:def_6;
A58: fsloc 0 in (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) by A27, ENUMSET1:def_6;
A59: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) c= dom (Comput (P,s1,i)) by Th32;
A60: (UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm) c= dom (Comput (Q,s2,i)) by Th32;
A61: ( i > 10 implies 10 + 1 < i + 1 ) by XREAL_1:6;
now__::_thesis:_for_x_being_set_st_x_in_{(intloc_0),(IC_),(fsloc_0)}_holds_
(Comput_(P,s1,i))_._x_=_(Comput_(Q,s2,i))_._x
let x be set ; ::_thesis: ( x in {(intloc 0),(IC ),(fsloc 0)} implies (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1 )
assume A62: x in {(intloc 0),(IC ),(fsloc 0)} ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
percases ( x = intloc 0 or x = IC or x = fsloc 0 ) by A62, ENUMSET1:def_1;
supposeA63: x = intloc 0 ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
percases ( i <= 10 or i > 10 ) ;
suppose i <= 10 ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
hence  (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A1, A52, A53, A63, A51; ::_thesis: verum
end;
suppose i > 10 ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
then  11 <= i by A61, NAT_1:13;
then  (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) by A28, A52, A53, A51;
hence  (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A57, A59, A60, A63, FUNCT_1:95; ::_thesis: verum
end;
end;
end;
supposeA64: x = IC ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
percases ( i <= 10 or i > 10 ) ;
suppose i <= 10 ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
hence  (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A1, A52, A53, A64, A51; ::_thesis: verum
end;
suppose i > 10 ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
then  11 <= i by A61, NAT_1:13;
hence  (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A28, A52, A53, A64, A51; ::_thesis: verum
end;
end;
end;
supposeA65: x = fsloc 0 ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
percases ( i <= 10 or i > 10 ) ;
suppose i <= 10 ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
hence  (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A1, A52, A53, A65, A51; ::_thesis: verum
end;
suppose i > 10 ; ::_thesis: (Comput (P,s1,i)) . b1 = (Comput (Q,s2,i)) . b1
then  11 <= i by A61, NAT_1:13;
then  (Comput (P,s1,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) = (Comput (Q,s2,i)) | ((UsedInt*Loc Bubble-Sort-Algorithm) \/ (UsedIntLoc Bubble-Sort-Algorithm)) by A28, A52, A53, A51;
hence  (Comput (P,s1,i)) . x = (Comput (Q,s2,i)) . x by A58, A59, A60, A65, FUNCT_1:95; ::_thesis: verum
end;
end;
end;
end;
end;
hence  (Comput (P,s1,i)) | (dom (Initialized ((fsloc 0) .--> w))) = (Comput (Q,s2,i)) | (dom (Initialized ((fsloc 0) .--> w))) by A54, A55, A56, FUNCT_1:95; ::_thesis: verum
end;

registration
cluster Bubble-Sort-Algorithm  ->  non halt-free ;
coherence
 not Bubble-Sort-Algorithm is halt-free ;
end;

theorem :: SCMBSORT:45
Bubble-Sort-Algorithm , Initialize ((intloc 0) .--> 1) computes Sorting-Function
proof
let x be set ; :: according to EXTPRO_1:def_14 ::_thesis: ( not x in dom Sorting-Function or ex b1 being set st
( x = b1 & (Initialize ((intloc 0) .--> 1)) +* b1 is Autonomy of Bubble-Sort-Algorithm & Sorting-Function . b1 c= Result (Bubble-Sort-Algorithm,((Initialize ((intloc 0) .--> 1)) +* b1)) ) )
assume  x in dom Sorting-Function ; ::_thesis: ex b1 being set st
( x = b1 & (Initialize ((intloc 0) .--> 1)) +* b1 is Autonomy of Bubble-Sort-Algorithm & Sorting-Function . b1 c= Result (Bubble-Sort-Algorithm,((Initialize ((intloc 0) .--> 1)) +* b1)) )
then consider w being FinSequence of INT  such that
A1: x = (fsloc 0) .--> w by SCMFSA_M:35;
reconsider d = x as FinPartState of SCM+FSA by A1;
set q = Bubble-Sort-Algorithm ;
set p = Initialize ((intloc 0) .--> 1);
A2: dom d = {(fsloc 0)} by A1, FUNCOP_1:13;
take  d ; ::_thesis: ( x = d & (Initialize ((intloc 0) .--> 1)) +* d is Autonomy of Bubble-Sort-Algorithm & Sorting-Function . d c= Result (Bubble-Sort-Algorithm,((Initialize ((intloc 0) .--> 1)) +* d)) )
thus  x = d ; ::_thesis: ( (Initialize ((intloc 0) .--> 1)) +* d is Autonomy of Bubble-Sort-Algorithm & Sorting-Function . d c= Result (Bubble-Sort-Algorithm,((Initialize ((intloc 0) .--> 1)) +* d)) )
A3: dom d misses {(IC )} by A2, SCMFSA_2:57, ZFMISC_1:11;
A4: dom d misses {(intloc 0)} by A2, SCMFSA_2:58, ZFMISC_1:11;
dom (Initialize ((intloc 0) .--> 1)) = (dom ((intloc 0) .--> 1)) \/ {(IC )} by MEMSTR_0:42
.= {(IC )} \/ {(intloc 0)} by FUNCOP_1:13 ;
then A5: dom d misses dom (Initialize ((intloc 0) .--> 1)) by A3, A4, XBOOLE_1:70;
A6: d +* (Initialize ((intloc 0) .--> 1)) = (Initialize ((intloc 0) .--> 1)) +* d by A5, FUNCT_4:35;
Initialized d = d +* (Initialize ((intloc 0) .--> 1))
.= (Initialize ((intloc 0) .--> 1)) +* d by A5, FUNCT_4:35
.= (Initialize ((intloc 0) .--> 1)) +* d
.= (Initialize ((intloc 0) .--> 1)) +* d ;
then A7: (Initialize ((intloc 0) .--> 1)) +* d is Bubble-Sort-Algorithm -autonomic by A1, Th44;
now__::_thesis:_for_t_being_State_of_SCM+FSA_st_(Initialize_((intloc_0)_.-->_1))_+*_d_c=_t_holds_
for_P_being_Instruction-Sequence_of_SCM+FSA_st_Bubble-Sort-Algorithm_c=_P_holds_
P_halts_on_t
let t be State of SCM+FSA; ::_thesis: ( (Initialize ((intloc 0) .--> 1)) +* d c= t implies for P being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P holds
P halts_on t )
assume A8: (Initialize ((intloc 0) .--> 1)) +* d c= t ; ::_thesis: for P being Instruction-Sequence of SCM+FSA st Bubble-Sort-Algorithm c= P holds
P halts_on t
let P be Instruction-Sequence of SCM+FSA; ::_thesis: ( Bubble-Sort-Algorithm c= P implies P halts_on t )
assume A9: Bubble-Sort-Algorithm c= P ; ::_thesis: P halts_on t
set bf = bubble-sort (fsloc 0);
 Initialize ((intloc 0) .--> 1) c= (Initialize ((intloc 0) .--> 1)) +* d by A6, FUNCT_4:25;
then  Initialize ((intloc 0) .--> 1) c= t by A8, XBOOLE_1:1;
hence  P halts_on t by Lm26, A9, SCM_HALT:def_2; ::_thesis: verum
end;
then A10: (Initialize ((intloc 0) .--> 1)) +* d is Bubble-Sort-Algorithm -halted by EXTPRO_1:def_11;
thus A11: (Initialize ((intloc 0) .--> 1)) +* d is Autonomy of Bubble-Sort-Algorithm by A10, A7, EXTPRO_1:def_12; ::_thesis: Sorting-Function . d c= Result (Bubble-Sort-Algorithm,((Initialize ((intloc 0) .--> 1)) +* d))
consider z being FinSequence of REAL  such that
A12: w,z are_fiberwise_equipotent and
A13: z is non-increasing and
 z is FinSequence of INT and
A14: Sorting-Function . d = (fsloc 0) .--> z by A1, SCMFSA_M:36;
consider t being State of SCM+FSA such that
A15: (Initialize ((intloc 0) .--> 1)) +* d c= t by PBOOLE:141;
consider T being Instruction-Sequence of SCM+FSA such that
A16: Bubble-Sort-Algorithm c= T by PBOOLE:145;
consider u being FinSequence of REAL  such that
A17: w,u are_fiberwise_equipotent and
A18: u is non-increasing and
 u is FinSequence of INT and
A19: (Result (T,t)) . (fsloc 0) = u by Th43, A1, A15, A16;
A20: u = z by A12, A13, A17, A18, CLASSES1:76, RFINSEQ:23;
 fsloc 0 in the carrier of SCM+FSA ;
then A21: fsloc 0 in dom (Result (T,t)) by PARTFUN1:def_2;
 d c= (Initialize ((intloc 0) .--> 1)) +* d by FUNCT_4:25;
then A22: dom d c= dom ((Initialize ((intloc 0) .--> 1)) +* d) by RELAT_1:11;
A23: dom ((fsloc 0) .--> z) = {(fsloc 0)} by FUNCOP_1:13;
 Result (Bubble-Sort-Algorithm,((Initialize ((intloc 0) .--> 1)) +* d)) = (Result (T,t)) | (dom ((Initialize ((intloc 0) .--> 1)) +* d)) by A11, A15, A16, EXTPRO_1:def_13;
hence  Sorting-Function . d c= Result (Bubble-Sort-Algorithm,((Initialize ((intloc 0) .--> 1)) +* d)) by A2, A14, A19, A20, A21, A23, A22, FUNCT_4:85, RELAT_1:151; ::_thesis: verum
end;