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Thanks,

Jennifer

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In this paper we develop machinery sufficient to prove the following

THEOREM: All horses have an infinite number of legs.

The theorem may seem intuitively obvious to some, but in the interest of rigor we will give a complete proof. We begin with two Lemmas:

Lemma 1: All horses are the same color.

Proof: We use the Principle of Mathematical Induction on the number, n, of horses.

Clearly, one horse is the same color, so the Lemma is true for n=1.

Now assume k horses are the same color, and consider k+1 horses. If we remove any one horse, we are left with k horses, which, by hypothesis, are all the same color. Since we removed an arbitrary horse, all k+1 horses are the same color.

Lemma 2: If a number is both even and odd, then it is infinite.

Proof: Let n be a number which is both even and odd, and assume n is finite.

As an even number, n = 2a for some integer a, and as an odd number, n = 2a+1. Thus 2a = 2a+1, whence 0 = 1. This contradiction establishes Lemma 2.

Proof of Theorem: All horses have forelegs in front and two in back, so that all horses have six legs. Now six is an even number, but six is clearly an odd number of legs for a horse to have. Thus the number of legs on a horse is both even and odd, and so by Lemma 2 it must be infinite.

You say, "But my horse has four legs." That, however, is a horse of a different color, which by Lemma 1 does exist.