reserve j, k, m, n for Nat,
  a,b for Int_position,
  k1,k2 for Integer;
reserve P,P1,P2 for Instruction-Sequence of SCMPDS;

theorem
  for q be non halt-free finite
   (the InstructionsF of SCMPDS)-valued NAT-defined Function
  for p being q-autonomic non empty FinPartState of SCMPDS, s1, s2
being State of SCMPDS st  p c= s1 &  p c= s2 &
 q c= P1 & q c= P2
for i,m being Nat,a being
Int_position,k1,k2 be Integer st CurInstr(P1,
Comput(P1,s1,i)) = (a,k1)<=0_goto
  k2 & m= IC Comput(P1,s1,i) & m+k2 >= 0 & k2 <> 1 holds (Comput(
P1,s1,i).
DataLoc(Comput(P1,s1,i).a,k1) > 0 iff Comput(P2,s2,i)
.DataLoc(Comput(P2,
  s2,i).a,k1) > 0 )
proof
   let q be non halt-free finite
    (the InstructionsF of SCMPDS)-valued NAT-defined Function;
  let p be q-autonomic non empty FinPartState of SCMPDS,
      s1, s2 be State
  of SCMPDS such that
A1:  p c= s1 &  p c= s2 and
A2: q c= P1 & q c= P2;
  let i,m be Nat,a be Int_position,k1,k2 be Integer;
  set Cs1i = Comput(P1,s1,i);
  set Cs2i = Comput(P2,s2,i);
  set Cs1i1 = Comput(P1,s1,i+1);
  set Cs2i1 = Comput(P2,s2,i+1);
A3: IC Cs1i = IC Cs2i & (Cs1i1|dom  p) = (Cs2i1|dom  p)
           by A1,A2,AMISTD_5:7,EXTPRO_1:def 10;
  set I = CurInstr(P1,Comput(P1,s1,i));
A4: Cs1i1 = Following(P1,Cs1i) by EXTPRO_1:3
    .= Exec (CurInstr(P1,Cs1i), Cs1i);
A5: m+1>=0;
 IC SCMPDS in dom p by AMISTD_5:6;
 then IC SCMPDS in dom  p;
 then
A6: (Cs1i1|dom  p).IC SCMPDS =
 Cs1i1.IC SCMPDS & (Cs2i1|dom  p).IC SCMPDS =
  Cs2i1. IC SCMPDS by FUNCT_1:49;
A7: Cs2i1 = Following(P2,Cs2i) by EXTPRO_1:3
    .= Exec (CurInstr(P2,Cs2i), Cs2i);
  assume that
A8: I = (a,k1)<=0_goto k2 and
A9: m= IC Cs1i & m+k2 >= 0 & k2 <> 1;
A10: I = CurInstr(P2,Comput(P2,s2,i))
 by A1,A2,AMISTD_5:7;
A11: now
    assume that
A12: Comput(P2,s2,i).DataLoc(Cs2i.a,k1) > 0 and
A13: Comput(P1,s1,i).DataLoc(Cs1i.a,k1) <= 0;
A14: Cs1i1.IC SCMPDS = ICplusConst(Cs1i,k2) by A4,A8,A13,SCMPDS_2:56;
    Cs2i1.IC SCMPDS = IC Cs2i + 1 by A10,A7,A8,A12,SCMPDS_2:56
      .=ICplusConst(Cs2i,1) by Th9;
    hence contradiction by A6,A3,A9,A5,A14,Th7;
  end;
  now
    assume that
A15: Comput(P1,s1,i).DataLoc(Cs1i.a,k1) > 0 and
A16: Comput(P2,s2,i).DataLoc(Cs2i.a,k1) <= 0;
A17: Cs2i1.IC SCMPDS = ICplusConst(Cs2i,k2) by A10,A7,A8,A16,SCMPDS_2:56;
    Cs1i1.IC SCMPDS = IC Cs1i + 1 by A4,A8,A15,SCMPDS_2:56
      .=ICplusConst(Cs1i,1) by Th9;
    hence contradiction by A6,A3,A9,A5,A17,Th7;
  end;
  hence thesis by A11;
end;
