Let \(G=(V, E)\) be a bipartite graph with sides \(V_{1}\) and \(V_{2}\), each of \(n\) vertices. Show that if there is a complete matching of \(V_{1}\) to \(V_{2}\), then for every subset \(S\) of \(V_{1}\),

\[ |S| \leq|A(S)| \]

where \(A(S)\) is the set of all vertices in \(V_{2}\) adjacent to some vertex in \(S\). (This is one half of a double implication called Hall's Theorem.)