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 Th56:
  S, T are_equivalent_wrt w & S c= U & U c= T implies S, U
  are_equivalent_wrt w & U, T are_equivalent_wrt w
proof
  assume that
A1: Lang(w, S) = Lang(w, T) and
A2: S c= U & U c= T;
  Lang(w, S) c= Lang(w, U) & Lang(w, U) c= Lang(w, T) by A2,Th48;
  hence Lang(w, S) = Lang(w, U) by A1,XBOOLE_0:def 10;
  hence thesis by A1;
end;
