Hall's theorem revisited

Let $A_{1},\cdots ,A_{n} (n>1)$ be sets. By a simple graph-theoretic argument we show that any set of distinct representatives of $\{A_{i}\}_{i=1}^{n-1}$can be extended to a set of distinct representatives of $\{A_{i}\}_{i=1}^{n}$in more than $\min _{n\in I\subseteq \{1,\cdots ,n\}} (\vert\bigcup _{i\in I}A_{i}\vert-\vert I\vert)$ ways. This yields a natural induction proof of the well-known theorem of P. Hall.