reserve m,n for Nat,
  I for Program of SCM+FSA,
  s,s1,s2 for State of SCM+FSA,
  a for Int-Location,
  f for FinSeq-Location,
  p,p1,p2 for Instruction-Sequence of SCM+FSA;

theorem
  for I being InitHalting really-closed Program of SCM+FSA,
      f being FinSeq-Location
  holds not f in UsedI*Loc I implies (IExec(I,p,s)).f = s.f
proof
  let I be InitHalting really-closed Program of SCM+FSA,
      f be FinSeq-Location;
    f <> intloc 0 & f <> IC SCM+FSA by SCMFSA_2:57,58;
    then
A1: not f in dom iS by SCMFSA_M:11,TARSKI:def 2;
A2: (IExec(I,p,s)).f = (Result(p+*I,Initialized s)).f
       by SCMFSA6B:def 1;
A3:  iS c= Initialized s by FUNCT_4:25;
A4: I c= p+*I by FUNCT_4:25;
  then p+*I halts_on Initialized s by Def1,A3;
  then consider n such that
A5: Result(p+*I,Initialized s) = Comput(p +* I,Initialized s,n) and
  CurInstr(p+*I,Result(p+*I,Initialized s))
   = halt SCM+FSA by EXTPRO_1:def 9;
  IC Initialized s = 0 by MEMSTR_0:def 11;
  then IC Initialized s in dom I by AFINSQ_1:65;
  then
A6: for m st m < n holds IC
   Comput(p+*I,Initialized s,m) in dom I by AMISTD_1:21,A4;
  assume not f in UsedI*Loc I;
  hence (IExec(I,p,s)).f = (Initialized s).f by A2,A5,A6,FUNCT_4:25,SF_MASTR:63
    .= s.f by A1,FUNCT_4:11;
end;
