let X, Y be set ; :: thesis: (proj2_3 X) \ (proj2_3 Y) c= proj2_3 (X \ Y)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in (proj2_3 X) \ (proj2_3 Y) or x in proj2_3 (X \ Y) )
assume A1: x in (proj2_3 X) \ (proj2_3 Y) ; :: thesis: x in proj2_3 (X \ Y)
then x in proj2_3 X by XBOOLE_0:def 5;
then consider y, z being object such that
A2: [y,x,z] in X by Th14;
not x in proj2_3 Y by A1, XBOOLE_0:def 5;
then not [y,x,z] in Y by Th15;
then [y,x,z] in X \ Y by A2, XBOOLE_0:def 5;
hence x in proj2_3 (X \ Y) by Th15; :: thesis: verum