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;
