let x1, x2, x3, x4 be set ; :: thesis: card {x1,x2,x3,x4} <= 4
( card {x1} = 1 & {x1,x2,x3,x4} = {x1} \/ {x2,x3,x4} & {x1} is finite & {x2,x3,x4} is finite & card {x2,x3,x4} <= 3 ) by Th70, CARD_1:50, ENUMSET1:44;
then ( 1 + (card {x2,x3,x4}) <= 1 + 3 & card {x1,x2,x3,x4} <= 1 + (card {x2,x3,x4}) ) by Th62, XREAL_1:9;
hence card {x1,x2,x3,x4} <= 4 by XXREAL_0:2; :: thesis: verum