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

theorem
  (X \/ Y) \ Y = X \ Y
proof
  thus for x being object holds x in (X \/ Y) \ Y implies x in X \ Y
  proof let x be object;
    assume
A1: x in (X \/ Y) \ Y;
    then x in (X \/ Y) by XBOOLE_0:def 5;
    then
A2: x in X or x in Y by XBOOLE_0:def 3;
    not x in Y by A1,XBOOLE_0:def 5;
    hence thesis by A2,XBOOLE_0:def 5;
  end;
  thus for x being object holds x in X \ Y implies x in (X \/ Y) \ Y
  proof let x be object;
    assume
A3: x in X \ Y;
    then x in X or x in Y by XBOOLE_0:def 5;
    then
A4: x in (X \/ Y) by XBOOLE_0:def 3;
    not x in Y by A3,XBOOLE_0:def 5;
    hence thesis by A4,XBOOLE_0:def 5;
  end;
end;
