let X, Y be set ; :: thesis: ( not X is finite & ( card Y in card X or card Y c= card X ) implies ( X \/ Y,X are_equipotent & card (X \/ Y) = card X ) )
assume B1: ( not X is finite & ( card Y in card X or card Y c= card X ) ) ; :: thesis: ( X \/ Y,X are_equipotent & card (X \/ Y) = card X )
X c= X \/ Y by XBOOLE_1:7;
then ( (card X) +` (card Y) = card X & card (X \/ Y) c= (card X) +` (card Y) & card X c= card (X \/ Y) ) by B1, Th34, CARD_1:27, CARD_2:47;
then card X = card (X \/ Y) by XBOOLE_0:def 10;
hence ( X \/ Y,X are_equipotent & card (X \/ Y) = card X ) by CARD_1:21; :: thesis: verum