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 Th42:
  s ==>* t, {}(E^omega, E^omega) implies s = t
proof
  assume s ==>* t, {}(E^omega, E^omega);
  then ==>.-relation({}(E^omega, E^omega)) reduces s, t;
  then {}(E^omega, E^omega) reduces s, t by Th29;
  then {} reduces s, t by PARTIT_2:def 1;
  hence thesis by REWRITE1:13;
end;
