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, db being Int-Location st
   CurInstr(P1,Comput(P1,s1,i)) = Divide(da, db)
    & db in
dom p holds Comput(P1,s1,i).da mod Comput(P1,s1,i).db
= Comput(P2,s2,i).
  da mod Comput(P2,s2,i).db
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, db be Int-Location;
  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: Cs2i1 = Following(P2,Cs2i) by EXTPRO_1:3
    .= Exec (CurInstr(P2, Cs2i), Cs2i);
  assume that
A4: I = Divide(da, db) and
A5: db in dom p and
A6: Comput(P1,s1,i).da mod Comput(P1,s1,i).db <>
Comput(P2,s2,i).
  da mod Comput(P2,s2,i).db;
A7: (Cs1i1|dom  p).db = Cs1i1.db & (Cs2i1|dom  p).db = Cs2i1.db
      by A5,FUNCT_1:49;
  Cs1i1 = Following(P1,Cs1i) by EXTPRO_1:3
    .= Exec (CurInstr(P1, Cs1i), Cs1i);
  then
A8: Cs1i1.db = Cs1i.da mod Cs1i.db by A4,SCMFSA_2:67;
  I = CurInstr(P2,Comput(P2,
s2,i)) by A1,A2,AMISTD_5:7;
  then Cs2i1.db = Cs2i.da mod Cs2i.db by A3,A4,SCMFSA_2:67;
  hence contradiction by A1,A6,A7,A8,A2,EXTPRO_1:def 10;
end;
