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 Th44:
  s ==>* t, S & u ==>. v, {[s, t]} implies u ==>* v, S
proof
  assume that
A1: s ==>* t, S and
A2: u ==>. v, {[s, t]};
  consider u1, v1, s1, t1 such that
A3: u = u1^s1^v1 & v = u1^t1^v1 and
A4: s1 -->. t1, {[s, t]} by A2;
  [s1, t1] in {[s, t]} by A4;
  then [s1, t1] = [s, t] by TARSKI:def 1;
  then s1 = s & t1 = t by XTUPLE_0:1;
  hence thesis by A1,A3,Th37;
end;
