let X, Y be set ; :: thesis: proj2 (X /\ Y) c= (proj2 X) /\ (proj2 Y)
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in proj2 (X /\ Y) or y in (proj2 X) /\ (proj2 Y) )
assume y in proj2 (X /\ Y) ; :: thesis: y in (proj2 X) /\ (proj2 Y)
then consider x being set such that
A3: [x,y] in X /\ Y by Def5;
[x,y] in Y by A3, XBOOLE_0:def 4;
then A4: y in proj2 Y by Def5;
[x,y] in X by A3, XBOOLE_0:def 4;
then y in proj2 X by Def5;
hence y in (proj2 X) /\ (proj2 Y) by A4, XBOOLE_0:def 4; :: thesis: verum