reserve P,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem Th24:
  for s1,s2 being State of SCM+FSA,
      I being really-closed Program of SCM+FSA, a
being Int-Location st I does not refer a & (for b being Int-Location st a <> b
  holds s1.b = s2.b) & (for f being FinSeq-Location holds s1.f = s2.f) &
:::& Iis_closed_on s1,P1 &
   I is_halting_on s1,P1
 holds
:::I is_closed_on s2,P2 &
   I is_halting_on s2,P2
proof
  let s1,s2 be State of SCM+FSA;
  let I be really-closed Program of SCM+FSA;
  let a be Int-Location;
  assume
A1: I does not refer a;
  set S2 = Initialize s2,
      Q2 = P2 +* I;
  set S1 = Initialize s1,
      Q1 = P1 +* I;
  assume
A2: for b being Int-Location st a <> b holds s1.b = s2.b;
  assume
A3: for f being FinSeq-Location holds s1.f = s2.f;
  assume that
A4: I is_halting_on s1,P1;
  Q1 halts_on S1 by A4,SCMFSA7B:def 7;
  then consider n being Nat such that
A5: CurInstr(Q1,Comput(Q1,S1,n)) = halt SCM+FSA;
  CurInstr(Q2,Comput(Q2,S2,n)) = halt SCM+FSA by A1,A5,Th22,A3,A2;
  then Q2 halts_on S2 by EXTPRO_1:29;
  hence thesis by SCMFSA7B:def 7;
end;
