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 Th57:
  S, ==>.-relation(S) are_equivalent_wrt w
proof
A1: Lang(w, ==>.-relation(S)) c= Lang(w, S)
  proof
    let x be object such that
A2: x in Lang(w, ==>.-relation(S));
    reconsider s = x as Element of E^omega by A2;
    w ==>* s, ==>.-relation(S) by A2,Th46;
    then w ==>* s, S by Th43;
    hence thesis;
  end;
  Lang(w, S) c= Lang(w, ==>.-relation(S)) by Th22,Th48;
  hence Lang(w, S) = Lang(w, ==>.-relation(S)) by A1,XBOOLE_0:def 10;
end;
