Proof

Let R be a finite domain and a a nonzero element of R. We need to show that a is invertible. To this end, consider the map

L_a : R -> R     x -> ax.

Since R is a domain, it follows from the cancellation law for domains that L_a is injective. Since R is a finite set, the pigeon hole principle says that the map is necessarily surjective. In particular, there exists y R such that L_a(y) = 1, i.e., ay = 1.