let X, Y be set ; :: thesis: ( (proj1 X) \+\ (proj1 Y) c= proj1 (X \+\ Y) & (proj2 X) \+\ (proj2 Y) c= proj2 (X \+\ Y) )
( (proj2 X) \ (proj2 Y) c= proj2 (X \ Y) & (proj2 Y) \ (proj2 X) c= proj2 (Y \ X) ) by Th8;
then A1: (proj2 X) \+\ (proj2 Y) c= (proj2 (X \ Y)) \/ (proj2 (Y \ X)) by XBOOLE_1:13;
( (proj1 X) \ (proj1 Y) c= proj1 (X \ Y) & (proj1 Y) \ (proj1 X) c= proj1 (Y \ X) ) by Th8;
then (proj1 X) \+\ (proj1 Y) c= (proj1 (X \ Y)) \/ (proj1 (Y \ X)) by XBOOLE_1:13;
hence ( (proj1 X) \+\ (proj1 Y) c= proj1 (X \+\ Y) & (proj2 X) \+\ (proj2 Y) c= proj2 (X \+\ Y) ) by A1, Th6; :: thesis: verum