We will have a matching if the matching condition holds. Given any set of card values (a set \(S \subseteq A\)) we must show that \(|N(S)| \ge |S|\text{.}\) That is, the number of piles that contain those values is at least the number of different values. But what if it wasn't? Say \(|S| = k\text{.}\) If \(|N(S)| \lt k\text{,}\) then we would have fewer than \(4k\) different cards in those piles (since each pile contains 4 cards). But there are \(4k\) cards with the \(k\) different values, so at least one of these cards must be in another pile, a contradiction. Thus the matching condition holds, so there is a matching, as required.
in-context