That's the base case. If one horse is removed from that set, then all remaining horses have the same color, by the induction assumption. Put the horse back and remove a different horse. Once again, we have a set of horses all the same color, obviously the very color found in the previous set since there are horses in common between the sets. Therefore, by induction, all horses are of the same color.
Horses have hindlegs.
They also have forelegs, which makes six legs. Now, six is an odd number of legs for a horse to have. Six is also an even number. And the only number that is both odd and even is infinity. Therefore, horses have an infinite number of legs.