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 Th20:
  not s ==>. t, {}(E^omega, E^omega)
proof
  assume s ==>. t, {}(E^omega, E^omega);
  then consider v, w, s1, t1 such that
  s = v^s1^w and
  t = v^t1^w and
A1: s1 -->. t1, {}(E^omega, E^omega);
  [s1, t1] in {}(E^omega, E^omega) by A1;
  hence contradiction by PARTIT_2:def 1;
end;
