theorem Th57:
  S, ==>.-relation(S) are_equivalent_wrt w
proof
A1: Lang(w, ==>.-relation(S)) c= Lang(w, S)
  proof
    let x be object such that
A2: x in Lang(w, ==>.-relation(S));
    reconsider s = x as Element of E^omega by A2;
    w ==>* s, ==>.-relation(S) by A2,Th46;
    then w ==>* s, S by Th43;
    hence thesis;
  end;
  Lang(w, S) c= Lang(w, ==>.-relation(S)) by Th22,Th48;
  hence Lang(w, S) = Lang(w, ==>.-relation(S)) by A1,XBOOLE_0:def 10;
end;
