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
  s ==>* t, S & u ==>* v, S implies s^u ==>* t^v, S & u^s ==>* v^t, S
proof
  assume
A1: s ==>* t, S & u ==>* v, S;
  then s^u ==>* t^u, S & t^u ==>* t^v, S by Th36;
  hence s^u ==>* t^v, S by Th35;
  u^s ==>* u^t, S & u^t ==>* v^t, S by A1,Th36;
  hence thesis by Th35;
end;
