Assume that k horses are the same color. Remove 1 horse. Because k horses are in the stable, then by the assumption, they are all the same color. Now remove another horse and return the original. Repeat the process.
All horses have an infinite number of legs.
First, it is necessary to prove that all horses have the same number of legs. Assume a horse had more or less legs then any other horse. And by the lemma, those cannot exist. Now, one horse has forelegs and two hindlegs. Four plus two is six, which is an odd number of legs for a horse to have.