Hall's marriage theorem

In honor of Valentine's Day, I will give a loving proof of Hall's marriage theorem. This theorem is equivalent to many important combinatorial theorems, including Dilworth's theorem and max-flow-min-cut. Moreover, it gives precise conditions under which people can marry happily, as long as these people are satisfied with heterosexual marriages. If time permits I will discuss the related algorithm due to Gale and Shapley which can be used to optimize marital bliss.