Remove one horse from the set, so that you have k horses. The inductive hypothesis tells us that all these horses are the same colour. Now return the horse you removed and take out a different horse. Again, by the inductive hypothesis, the remaining k horses are all the same colour. The result follows by induction.
All horses have infinitely many legs.
It is also well-known that horses have forelegs in front and two legs at the back. So, since we have shown the number of legs on a horse to be both even and odd, there must be infinitely many of them. A cat has nine tails. No cat has eight tails. A cat has one tail more than no cat.