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

theorem Th55:
  for s being State of SCM+FSA, I being Program of SCM+FSA, a
being Int-Location st I does not destroy a holds for k being Element of NAT st
IC Comput(P +* I, (Initialize s),k) in dom I
holds Comput(P +* I,
  (Initialize s),k + 1).a =
   Comput(P +* I, (Initialize s),k).a
proof
  let s be State of SCM+FSA;
  let I be Program of SCM+FSA;
  let a be Int-Location;
  assume
A1: I does not destroy a;
  set s1 = Initialize s,
      P1 = P +* I;
A2: I c= P1 by FUNCT_4:25;
  let k be Element of NAT;
  assume
A3: IC Comput(P +* I, (Initialize s),k) in dom I;
  set l = IC Comput(P1, s1,k);
  P1.l = I.l by A3,A2,GRFUNC_1:2;
  then P1.l in rng I by A3,FUNCT_1:def 3;
  then
A4: P1.l does not destroy a by A1;
  thus Comput(P1, s1,k + 1).a = (Following(P1,Comput(P1,s1,k))).a by EXTPRO_1:3
    .= Exec(P1.l, Comput(P1, s1,k)).a by PBOOLE:143
    .= Comput(P1, s1,k).a by A4,SCMFSA7B:20;
end;
