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 Th19:
  S c= T & s ==>. t, S implies s ==>.t, T
proof
  assume that
A1: S c= T and
A2: s ==>. t, S;
  consider v, w, s1, t1 such that
A3: s = v^s1^w & t = v^t1^w and
A4: s1 -->. t1, S by A2;
  s1 -->. t1, T by A1,A4;
  hence thesis by A3;
end;
