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

theorem Th4:
  for I being Program of SCM+FSA st
   for s being State of SCM+FSA, P holds I is_halting_on Initialized s,P
  holds Initialize((intloc 0).-->1) is I-halted
proof
  let I be Program of SCM+FSA;
  assume
A1: for s being State of SCM+FSA,P holds I is_halting_on Initialized s,P;
  let s be State of SCM+FSA;
  assume  Initialize((intloc 0).-->1) c= s;
   then Initialize((intloc 0).-->1) c= s;
   then
A2:   s +*Initialize((intloc 0).-->1) = s by FUNCT_4:98;
  let P be Instruction-Sequence of SCM+FSA such that
A3: I c= P;
A4: P +* I = P by A3,FUNCT_4:98;
    I is_halting_on Initialized s,P by A1;
    then P +* I halts_on Initialize Initialized s;
    hence P halts_on s by A2,A4,MEMSTR_0:44;
end;
