let X, Y be set ; :: thesis: ( not X is finite & ( X,Y are_equipotent or Y,X are_equipotent ) implies ( X \/ Y,X are_equipotent & card (X \/ Y) = card X ) )
assume that
A1: not X is finite and
A2: ( X,Y are_equipotent or Y,X are_equipotent ) ; :: thesis: ( X \/ Y,X are_equipotent & card (X \/ Y) = card X )
A3: card X = card Y by A2, CARD_1:5;
A4: card X c= card (X \/ Y) by CARD_1:11, XBOOLE_1:7;
A5: card (X \/ Y) c= (card X) +` (card Y) by Th33;
(card X) +` (card Y) = card X by A1, A3, Th74;
then card X = card (X \/ Y) by A4, A5;
hence ( X \/ Y,X are_equipotent & card (X \/ Y) = card X ) by CARD_1:5; :: thesis: verum