reserve x for set;
reserve k, l for Nat;
reserve p, q for FinSequence;
reserve R for Relation;
reserve p, q for RedSequence of R;
reserve E for set;
reserve s, t for XFinSequence;
reserve p, q for XFinSequence-yielding FinSequence;
reserve E for set;
reserve S, T, U for semi-Thue-system of E;
reserve s, t, s1, t1, u, v, u1, v1, w for Element of E^omega;
reserve p for FinSequence of E^omega;

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;
