I don’t own a horse. I have only ever ridden on one when I was younger, closer to the age of 6 or 7. The horse I rode on was white. Being a mathematician I can prove this to you. So lets go:

Lemma 1. All horses are the same color

Proof.

Clearly one horse is the same color. Thus, lets assume the proposition P(k) that k horses are the same color. By induction we can imply that k+1 horses are the same color. From a set of k+1 horses we can remove one horse, and the remaining k horses will be the same color, by hypothesis.

We remove another horse and replace the first, thus the k horses are the same color, again by hypothesis. By induction we can repeat for the whole set of k horses, thus it follows that all k horses are the same color, asP(k) => P(k+1).

As P(1) is true, by initial conditions (the horse I rode as a child), P holds true for all subsequent values of k. []

Theorem 1. Every horse has an infinite number of legs.

Proof.

The horse I rode as a child had an even number of legs. Of the legs, there were two front legs, with fore legs remaining. That gave it 6 legs, which is an odd number of legs for a horse. The only number that is both odd and even is infinity. Thus, my horse had an infinite number of legs.

To show that all horses have an infinite number of legs, lets assume the inverse. Thus there is a horse somewhere with a finite number of legs. By example, this horse is pictured below:

Clearly this horse does not have an infinite number of legs. But this horse is brown, not white in color, and so, byLemma 1, does not exist. =><= []

Corollary 1. Everything is the same color.

Proof.

Lemma 1 can be generalized as the proof by induction is object independent. Hence, the statement “For all x, if x is a horse, x will be the same color”, can be generalized to “For all x, x will be the same color”. Thus proof of the single color of horses is simply a special case of this. []

Corollary 2. Everything is white.

Proof.

If a sentential formula in x is logically valid, then any particular substitution instance of it, is also a true statement. By personal experience it is evident that white horses exist. Therefore all horses are white. Hence, by corollary 1, everything is white. []

So there you have it. Clear, reasoned proof that all horses have an infinite number of legs and that everything is white.