Suppose that we have a set of five horses. We wish to prove that they are all the same color. Suppose that we had a proof that all sets of four horses were the same color. If that were true, we could prove that all five horses are the same color by removing a horse to leave a group of four horses. Do this in two ways, and we have two different groups of four horses. By our supposed existing proof, since these are groups of four, all horses in them must be the same color. For example, the first, second, third and fourth horses constitute a group of four, and thus must all be the same color; and the second, third, fourth and fifth horses also constitute a group of four and thus must also all be the same color. For this to occur, all five horses in the group of five must be the same color.
But how are we to get a proof that all sets of four horses are the same color? We apply the same logic again. By the same process, a group of four horses could be broken down into groups of three, and then a group of three horses could be broken down into groups of two, and so on. Eventually we will reach a group size of one, and it is obvious that all horses in a group of one horse must be the same color.
By the same logic we can also increase the group size. A group of five horses can be increased to a group of six, and so on upwards, so that all finite sized groups of horses must be the same color.