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

theorem Th54:
  for s being State of SCM+FSA, I being keeping_0 Program of
  SCM+FSA st I is_halting_on Initialized s,P
   holds IExec(I,P,s).intloc 0 = 1 & for k
being Nat holds Comput(P +* I,
(Initialize Initialized s),k).intloc 0 = 1
proof
  set a = intloc 0;
  let s be State of SCM+FSA;
  set D = Data-Locations SCM+FSA;
  let I be keeping_0 Program of SCM+FSA;
  set s0 = Initialized s;
  set s1 = Initialize s0,
      P1 = P +* I;
A1: I c= P1 by FUNCT_4:25;
A2: DataPart s0 = DataPart s1 by MEMSTR_0:79;
A3: now
    let k be Nat;
    thus Comput(P1, s1,k).a = s1.a by A1,SCMFSA6B:def 4
      .= s0.a by A2,SCMFSA_M:2
      .= 1 by SCMFSA_M:9;
  end;
  assume I is_halting_on s0,P;
  then
A4: P1 halts_on s1;
  thus IExec(I,P,s).a = (Result(P1,s1)).a by MEMSTR_0:44
    .= (Result(P1,s1)).a
    .= Comput(P1, s1,LifeSpan(P1,s1)).a by A4,EXTPRO_1:23
    .= 1 by A3;
  let k be Nat;
  thus thesis by A3;
end;
