reserve k for Nat,
  da,db for Int-Location,
  fa for FinSeq-Location;

theorem
   for q being non halt-free finite
      (the InstructionsF of SCM+FSA)-valued NAT-defined Function
  for p being q-autonomic non empty FinPartState of SCM+FSA,
      s1, s2
  being State of SCM+FSA st  p c= s1 &  p c= s2
  for P1,P2 being Instruction-Sequence of SCM+FSA
   st q c= P1 & q c= P2
  for i being Nat, da being Int-Location, loc being Nat st
   CurInstr(P1,Comput(P1,s1,i)) = da=0_goto loc &
    loc <> (IC Comput(P1,s1,i))+1 holds (
  Comput(P1,s1,i).da = 0 iff Comput(P2,s2,i).da = 0)
proof
  let q being non halt-free finite
      (the InstructionsF of SCM+FSA)-valued NAT-defined Function;
  let p be q-autonomic non empty FinPartState of SCM+FSA,
      s1, s2 be State
  of SCM+FSA such that
A1:  p c= s1 &  p c= s2;
  let P1,P2 be Instruction-Sequence of SCM+FSA
  such that
A2: q c= P1 & q c= P2;
  let i be Nat, da be Int-Location, loc be Nat;
  set I = CurInstr(P1,Comput(P1,s1,i));
  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: Cs1i1 = Following(P1,Cs1i) by EXTPRO_1:3
    .= Exec (CurInstr(P1, Cs1i), Cs1i);
A4: Cs2i1 = Following(P2,Cs2i) by EXTPRO_1:3
    .= Exec (CurInstr(P2, Cs2i), Cs2i);
   IC SCM+FSA in dom p by AMISTD_5:6;
   then
A5: (Cs1i1|dom  p).IC SCM+FSA = Cs1i1.IC SCM+FSA &
   (Cs2i1|dom  p).IC SCM+FSA =
  Cs2i1.IC SCM+FSA by FUNCT_1:49;
  assume that
A6: I = da=0_goto loc and
A7: loc <> (IC Comput(P1,s1,i))+1;
A8: I = CurInstr(P2,Comput(P2,s2,i)) by A1,A2,AMISTD_5:7;
A9: now
    assume
    Comput(P2,s2,i).da = 0 & Comput(P1,s1,i).da <> 0;
    then
    Cs2i1.IC SCM+FSA = loc & Cs1i1.IC SCM+FSA = IC Cs1i + 1 by A8,A3,A4,A6,
SCMFSA_2:70;
    hence contradiction by A1,A5,A7,A2,EXTPRO_1:def 10;
  end;
A10: (Cs1i1|dom  p) = (Cs2i1|dom  p) by A1,A2,EXTPRO_1:def 10;
  now
    assume
    Comput(P1,s1,i).da = 0 & Comput(P2,s2,i).da <> 0;
    then
    Cs1i1.IC SCM+FSA = loc & Cs2i1.IC SCM+FSA = IC Cs2i + 1 by A8,A3,A4,A6,
SCMFSA_2:70;
    hence contradiction by A1,A5,A10,A7,A2,AMISTD_5:7;
  end;
  hence thesis by A9;
end;
