reserve m, n for Nat,
  x for set,
  i for Instruction of SCM+FSA,
  I for Program of SCM+FSA,
  a for Int-Location,
  f for FinSeq-Location,
  l, l1 for Nat,
  s,s1,s2 for State of SCM+FSA,
  P,P1,P2 for Instruction-Sequence of SCM+FSA;

theorem Th7:
  for s being 0-started State of SCM+FSA
  for I being parahalting really-closed Program of SCM+FSA st I c= P1 & I c= P2
  for k being Nat holds
    Comput(P1,s,k) = Comput(P2,s,k) &
   CurInstr(P1,Comput(P1,s,k)) = CurInstr(P2,Comput(P2,s,k))
proof
  let s be 0-started State of SCM+FSA;
  let I be parahalting really-closed Program of SCM+FSA;
  assume that
A1: I c= P1 and
A2: I c= P2;
  hereby
    let k be Nat;
    IC s = 0 by MEMSTR_0:def 11;
    then
A3:  IC s in dom I by AFINSQ_1:65;
    then
A4: IC Comput(P1,s,k) in dom I by A1,AMISTD_1:21;
A5: IC Comput(P2,s,k) in dom I by A2,A3,AMISTD_1:21;
    for m being Nat st m < k
     holds IC Comput(P2,s,m) in dom I by A2,AMISTD_1:21,A3;
    hence
A6:  Comput(P1,s,k) = Comput(P2,s,k) by A1,A2,AMISTD_2:10;
    thus CurInstr(P2,Comput(P2,s,k))
     = P2.IC Comput(P2,s,k) by PBOOLE:143
      .= I.IC Comput(P2,s,k) by A5,A2,GRFUNC_1:2
      .= P1.IC Comput(P1,s,k) by A6,A4,A1,GRFUNC_1:2
      .= CurInstr(P1,Comput(P1,s,k)) by PBOOLE:143;
  end;
end;
