reserve i, j, k for Nat,
  n for Nat,
  IL for non empty set,
  N for with_non-empty_elements set;
reserve R for non trivial Ring,
  a, b for Data-Location of R,
  loc for Nat,
  I for Instruction of SCM R,
  p for FinPartState of SCM R,
  s, s1, s2 for State of SCM R,
  P,P1,P2 for Instruction-Sequence of SCM R,
  q for FinPartState of SCM;

theorem Th5:
  s.a = (s +* Start-At(loc,SCM R)).a
proof
  a in the carrier of SCM R;
  then a in dom s by PARTFUN1:def 2;
  then
A1: dom (Start-At(loc,SCM R)) = {IC SCM R} & a in dom s \/
dom (Start-At(loc,SCM R)) by XBOOLE_0:def 3;
  a <> IC SCM R by SCMRING3:2;
  then not a in {IC SCM R} by TARSKI:def 1;
  hence thesis by A1,FUNCT_4:def 1;
end;
