let X, Y be set ; :: thesis: (X \/ Y) \ Y = X \ Y
thus for x being set st x in (X \/ Y) \ Y holds
x in X \ Y :: according to TARSKI:def 3,XBOOLE_0:def 10 :: thesis: X \ Y c= (X \/ Y) \ Y

proof

let x be set ; :: thesis: ( x in (X \/ Y) \ Y implies x in X \ Y )
assume x in (X \/ Y) \ Y ; :: thesis: x in X \ Y
then ( x in X \/ Y & not x in Y ) by XBOOLE_0:def 5;
then ( ( x in X or x in Y ) & not x in Y ) by XBOOLE_0:def 3;
hence x in X \ Y by XBOOLE_0:def 5; :: thesis: verum

end;

thus for x being set st x in X \ Y holds
x in (X \/ Y) \ Y :: according to TARSKI:def 3 :: thesis: verum

proof

let x be set ; :: thesis: ( x in X \ Y implies x in (X \/ Y) \ Y )
assume x in X \ Y ; :: thesis: x in (X \/ Y) \ Y
then ( ( x in X or x in Y ) & not x in Y ) by XBOOLE_0:def 5;
then ( x in X \/ Y & not x in Y ) by XBOOLE_0:def 3;
hence x in (X \/ Y) \ Y by XBOOLE_0:def 5; :: thesis: verum

end;