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, f being FinSeq-Location st
 CurInstr(P1,Comput(P1,s1,i)
) =
  da := (f,db) & da in dom p for k1,k2 being Element of NAT st k1 = |.
Comput(P1,s1,i).db.| & k2 = |. Comput(P2,s2,i).db.|
holds ( Comput(P1,s1,
  i).f)/.k1 = ( Comput(P2,s2,i).f)/.k2
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, f be FinSeq-Location;
  set Cs1i1 = Comput(P1,s1,i+1);
  set Cs2i1 = Comput(P2,s2,i+1);
A3: Cs1i1|dom  p = Cs2i1|dom  p by A1,A2,EXTPRO_1:def 10;
  set Cs2i = Comput(P2,s2,i);
  set Cs1i = Comput(P1,s1,i);
  set I = CurInstr(P1,Comput(P1,s1,i));
A4: Cs1i1 = Following(P1,Cs1i) by EXTPRO_1:3
    .= Exec (CurInstr(P1, Cs1i), Cs1i);
A5: da in dom  p implies
 (Cs1i1|dom  p).da = Cs1i1.da & (Cs2i1|dom  p).da =
  Cs2i1.da by FUNCT_1:49;
A6: Cs2i1 = Following(P2,Cs2i) by EXTPRO_1:3
    .= Exec (CurInstr(P2, Cs2i), Cs2i);
  assume that
A7: I = da := (f,db) and
A8: da in dom p;
A9: (ex k1 being Nat st k1 = |.Cs1i.db.| & Exec(I, Cs1i).da = (
Cs1i .f)/.k1 )& ex k2 being Nat st k2 = |.Cs2i.db.| &
 Exec(I, Cs2i
  ). da = (Cs2i.f)/.k2 by A7,SCMFSA_2:72;
  let i1,i2 be Element of NAT;
  assume
  i1 = |. Comput(P1,s1,i).db.| & i2 = |. Comput(P2
,s2,i).db.| &
  ( Comput(P1,s1,i).f)/.i1 <> ( Comput(P2,s2,i).f)/.
i2;
  hence contradiction by A1,A4,A6,A5,A3,A9,A8,A2,AMISTD_5:7;
end;
