theorem Th41:
  s ==>* t, S iff s ==>* t, S \/ id (E^omega)
proof
  thus s ==>* t, S implies s ==>* t, S \/ id (E^omega) by Th40,XBOOLE_1:7;
  assume s ==>* t, S \/ id (E^omega);
  then ==>.-relation(S \/ id (E^omega)) reduces s, t;
  then ==>.-relation(S) \/ id (E^omega) reduces s, t by Th28;
  then ==>.-relation(S) reduces s, t by REWRITE1:23;
  hence thesis;
end;
