reserve x,A,B,X,X9,Y,Y9,Z,V for set;

theorem Th4:
  (X \/ Y) \/ Z = X \/ (Y \/ Z)
proof
  thus (X \/ Y) \/ Z c= X \/ (Y \/ Z)
  proof
    let x be object;
    assume x in (X \/ Y) \/ Z;
    then x in X \/ Y or x in Z by XBOOLE_0:def 3;
    then x in X or x in Y or x in Z by XBOOLE_0:def 3;
    then x in X or x in Y \/ Z by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in X \/ (Y \/ Z);
  then x in X or x in Y \/ Z by XBOOLE_0:def 3;
  then x in X or x in Y or x in Z by XBOOLE_0:def 3;
  then x in X \/ Y or x in Z by XBOOLE_0:def 3;
  hence thesis by XBOOLE_0:def 3;
end;
