reserve j, k, m, n for Nat,
  a,b for Int_position,
  k1,k2 for Integer;
reserve P,P1,P2 for Instruction-Sequence of SCMPDS;

theorem
  for s being State of SCMPDS ,iloc being Nat
  , a being Int_position holds s.a = (s +* Start-At(iloc,SCMPDS)).a
proof
  let s be State of SCMPDS, iloc be Nat, a be
  Int_position;
  a in the carrier of SCMPDS;
  then a in dom s by PARTFUN1:def 2;
  then
A1: dom (Start-At(iloc,SCMPDS)) = {IC SCMPDS } &
 a in dom s \/ dom (Start-At(iloc,SCMPDS))
  by XBOOLE_0:def 3;
  a <> IC SCMPDS by SCMPDS_2:43;
  then not a in {IC SCMPDS } by TARSKI:def 1;
  hence thesis by A1,FUNCT_4:def 1;
end;
