reserve s, s1, s2 for State of SCM+FSA,
  p, p1 for Instruction-Sequence of SCM+FSA,
  a, b for Int-Location,
  d for read-write Int-Location,
  f for FinSeq-Location,
  I for MacroInstruction of SCM+FSA,
  J for good MacroInstruction of SCM+FSA,
  k, m for Nat;

theorem
  for I being really-closed Program of SCM+FSA holds
I is_halting_on Initialized s,p & not f in
  UsedI*Loc I implies IExec(I,p,s).f = (Initialized s).f
proof let I be really-closed Program of SCM+FSA;
  set a = f;
  assume that
A1: I is_halting_on Initialized s,p and
A2: not a in UsedI*Loc I;
  set Is = Initialized s, pI = p +* I;
A3: p+*I halts_on Initialize Is by A1,SCMFSA7B:def 7;
A4: Is = Initialize Is by MEMSTR_0:44;
A5: I c= pI by FUNCT_4:25;
   IC Is = 0 by MEMSTR_0:def 11;
  then IC Is in dom I by AFINSQ_1:65;
  then for m st m < (LifeSpan(pI,Is)) holds IC Comput(pI,Is,m) in dom
I by A5,AMISTD_1:21;
  then
A6: Comput(pI,Is,LifeSpan(pI,Is)).a = Is.a by A2,FUNCT_4:25,SF_MASTR:63;
  DataPart IExec(I,p,s) = DataPart(Result(pI,Is)) by SCMFSA6B:def 1
    .= DataPart(Result(pI,Is))
    .= DataPart Comput(pI,Is,LifeSpan(pI,Is)) by A4,A3,EXTPRO_1:23;
  hence thesis by A6,SCMFSA_M:2;
end;
