 Solution

Hall's theorem states that a finite bipartite graph with classes $L,R$ has a matching covering $L$ exactly when $|N(S)|\ge |S|$ for every $S\subseteq L$. Necessity follows because the distinct matched partners of $S$ belong to $N(S)$.

For sufficiency take a maximum matching. If it leaves $x\in L$ unmatched, explore alternating paths starting at $x$, traversing unmatched edges from left to right and matched edges back to the left. No reached right vertex can be unmatched, since flipping the edges of such an alternating path would augment the matching. Let $S,T$ be the reached left and right sets. Every neighbour of $S$ lies in $T$: unmatched edges are explored, and a matched edge from a reached left vertex leads to its already reached predecessor. Conversely every member of $T$ is a neighbour of $S$. Their matched partners are exactly $S\setminus \{x\}$, so $|N(S)|=|T|=|S|-1$, contradicting the hypothesis. Thus a matching covering $L$ exists.