This is a slightly strange question perhaps. How many ways are there to prove that if X and Y are two finite orders (total orders) on n elements, then X and Y are isomorphic?
There is a direct proof where you prove the lemma that finite orders have maximum elements, and prove the main result by induction.
You can also prove it directly by using the pigeonhole principle.
You can prove it by deducing it from the fact that directed cycles of equal cardinality are isomorphic and are also arc-transitive i.e. any arc is automorphic to any other.
Are there any other results which have this basic theorem as a corollary? E.g results on graphs or orders/partial orders.