let X, Y be set ; :: thesis: (proj2 X) \ (proj2 Y) c= proj2 (X \ Y)
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in (proj2 X) \ (proj2 Y) or y in proj2 (X \ Y) )
assume A3: y in (proj2 X) \ (proj2 Y) ; :: thesis: y in proj2 (X \ Y)
then y in proj2 X by XBOOLE_0:def 5;
then consider x being set such that
A4: [x,y] in X by Def5;
not y in proj2 Y by A3, XBOOLE_0:def 5;
then not [x,y] in Y by Def5;
then [x,y] in X \ Y by A4, XBOOLE_0:def 5;
hence y in proj2 (X \ Y) by Def5; :: thesis: verum