let X, Y be set ; :: thesis: (X \/ Y) \ Y = X \ Y
thus for x being object 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 object ; :: thesis: ( x in (X \/ Y) \ Y implies x in X \ Y )
assume A1: x in (X \/ Y) \ Y ; :: thesis: x in X \ 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 x in X \ Y by A2, XBOOLE_0:def 5; :: thesis: verum

end;

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

proof

let x be object ; :: thesis: ( x in X \ Y implies x in (X \/ Y) \ Y )
assume A3: x in X \ Y ; :: thesis: x in (X \/ Y) \ 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 x in (X \/ Y) \ Y by A4, XBOOLE_0:def 5; :: thesis: verum

end;