let X, Y be set ; :: thesis: (proj1 X) \+\ (proj1 Y) c= proj1 (X \+\ Y)
( (proj1 X) \ (proj1 Y) c= proj1 (X \ Y) & (proj1 Y) \ (proj1 X) c= proj1 (Y \ X) ) by Th25;
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) by Th23; :: thesis: verum