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 implies S, (S \/ {[s, t]}) are_equivalent_wrt w
proof
  assume
A1: s ==>* t, S;
A2: Lang(w, S \/ {[s, t]}) c= Lang(w, S)
  proof
    let x be object such that
A3: x in Lang(w, S \/ {[s, t]});
    reconsider u = x as Element of E^omega by A3;
    w ==>* u, S \/ {[s, t]} by A3,Th46;
    then w ==>* u, S by A1,Th45;
    hence thesis;
  end;
  Lang(w, S) c= Lang(w, S \/ {[s, t]}) by Th48,XBOOLE_1:7;
  hence Lang(w, S) = Lang(w, S \/ {[s, t]}) by A2,XBOOLE_0:def 10;
end;
