Hall's theorem revisited

Let A(1), ... , A(n) (n > 1) be sets. By a simple graph-theoretic argument we show that any set of distinct representatives of {Ai}(i=1)(n-1) can be e xtended to a set of distinct representatives of {A(i)}(i=1)(n) in more than min(n is an element ofI subset of or equal to {1, ..., n}) (\U-i is an ele ment ofI A(i)\ - \I\) ways. This yields a natural induction proof of the we ll-known theorem of P. Hall.