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 Th49:
  Lang(w, S) = Lang(w, S \/ id (E^omega))
proof
A1: Lang(w, S \/ id (E^omega)) c= Lang(w, S)
  proof
    let x be object;
    assume
A2: x in Lang(w, S \/ id (E^omega));
    then reconsider s = x as Element of E^omega;
    w ==>* s, S \/ id (E^omega) by A2,Th46;
    then w ==>* s, S by Th41;
    hence thesis;
  end;
  Lang(w, S) c= Lang(w, S \/ id (E^omega)) by Th48,XBOOLE_1:7;
  hence thesis by A1,XBOOLE_0:def 10;
end;
