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 Th5:
  for I being InitHalting really-closed Program of SCM+FSA st
   Initialize ((intloc 0) .--> 1) c= s1 &
   Initialize ((intloc 0) .--> 1) c= s2 &
   I c= p1 & I c= p2 &
    s1 =  s2
 for k being Nat holds
   Comput(p1,s1,k) =  Comput(p2,s2,k) &
  CurInstr(p1,Comput(p1,s1,k)) = CurInstr(p2,Comput(p2,s2,k))
proof
  let I be InitHalting really-closed Program of SCM+FSA;
  assume that
A1: iS c= s1 and
A2: iS c= s2 and
A3: I c= p1 and
A4: I c= p2 and
A5:  s1 =  s2;
    let k be Nat;
    IC s1 = 0 by A1,MEMSTR_0:47;
    then IC s1 in dom I by AFINSQ_1:65;
    then
A6: IC Comput(p1,s1,k) in dom I by AMISTD_1:21,A3;
    IC s2 = 0 by A2,MEMSTR_0:47;
    then
A7:   IC s2 in dom I by AFINSQ_1:65;
    then
A8: IC Comput(p2,s2,k) in dom I by AMISTD_1:21,A4;
    for m being Nat st m < k holds IC(Comput(p2,s2,m)) in dom I
       by AMISTD_1:21,A4,A7;
    hence
A9:   Comput(p1,s1,k) =  Comput(p2,s2,k)
     by A5,A3,A4,AMISTD_2:10;
    thus CurInstr(p2,Comput(p2,s2,k))
       = p2.IC Comput(p2,s2,k) by PBOOLE:143
      .= I.IC Comput(p2,s2,k) by A8,A4,GRFUNC_1:2
      .= p1.IC Comput(p1,s1,k) by A9,A6,A3,GRFUNC_1:2
      .= CurInstr(p1,Comput(p1,s1,k)) by PBOOLE:143;
end;
