reserve x for set,
  m,n for Nat,
  a,b,c for Int_position,
  i for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer,
  loc,l1 for Nat,
  I,J for Program of SCMPDS,
  N for with_non-empty_elements set;
reserve P,P1,P2,Q for Instruction-Sequence of SCMPDS;

theorem Th5:
  for s being 0-started State of SCMPDS
  for I being parahalting Program of SCMPDS
    st stop I c= P1 & stop 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 SCMPDS;
  let I be parahalting Program of SCMPDS;
  set SI=stop I;
  assume that
A1: SI c= P1 and
A2: SI c= P2;
    let k be Nat;
A3: IC Comput(P1,s,k) in dom SI by A1,SCMPDS_4:def 6;
A4: IC Comput(P2,s,k) in dom SI by A2,SCMPDS_4:def 6;
    for m being Nat st m < k holds IC(Comput(P2,s,m))
in dom
    SI by A2,SCMPDS_4:def 6;
    hence
A5:   Comput(P1,s,k) =  Comput(P2,s,k) by A1,A2,SCMPDS_4:21;
    thus CurInstr(P2,Comput(P2,s,k))
     = P2.IC Comput(P2,s,k) by PBOOLE:143
      .= SI.IC Comput(P2,s,k) by A2,A4,GRFUNC_1:2
      .= P1.IC Comput(P1,s,k) by A1,A5,A3,GRFUNC_1:2
      .= CurInstr(P1,Comput(P1,s,k)) by PBOOLE:143;
end;
