Proof (by mathematical induction):
In the base case, any group made of a single horse has only 1 color.

In the inductive case, suppose any group made of n horses has only 1 color. We show then that any group made of n+1 horses has only one color. We can then apply the same reasoning repeatedly, starting with a size 1 group, to show any size group of horses has a single color.

For any group with n+1 horses, take the first 1, ..., n of them and form a group. From the above assumption, there's only 1 color in this group. Now take the last 1, ..., n of them, which again has only one color. These groups overlap, so they must be the same single color. These groups also cover the entire group of n+1 horses, so that group must have 1 color. This completes the induction.


So, any finite group of horses has a single color. There are only finitely many horses in the world. Thus there is no horse of a different color. Q.E.D.


As a corollary, yesterday I saw a brown horse, so every horse is brown.