Case n=1: In a set with only one horse, it is obvious that all horses
in that set are the same color.
Case n=k: Suppose you have a set of k+1 horses. Pull one of these
horses out of the set, so that you have k horses. Suppose that all of
these horses are the same color. Now put back the horse that you took
out, and pull out a different one. Suppose that all of the k horses
now in the set are the same color. Then the set of k+1 horses are all
the same color. We have k true => k+1 true; therefore all horses are
the same color.
Theorem: All horses have an infinite number of legs.
Proof (by intimidation):
Everyone would agree that all horses have an even number of legs. It
is also well-known that horses have forelegs in front and two legs in
back. 4 + 2 = 6 legs, which is certainly an odd number of legs for a
horse to have! Now the only number that is both even and odd is infinity;
therefore all horses have an infinite number of legs.
However, suppose that there is a horse somewhere that does not have an
infinite number of legs. Well, that would be a horse of a different
color; and by the Lemma, it doesn't exist.
