One way to prove the Parity Theorem is first to prove the

*Even Identity Lemma.* If e is the identity permutation in *S*_{n} that maps every element to itself, and if where the t_{i}’s are transpositions, then *k* is an even number.

Before considering a proof of this lemma, let us show how it leads very directly to a proof of the Parity Theorem. Suppose that we have two expressions, and , for a permutation a in terms of transpositions. Then, since the inverse of a composition of a sequence of permutations is the composition of their inverses in the reverse order, and since every transposition is its own inverse, it follows that

This shows that e can be expressed using *k + m* transpositions. Once the lemma is established, we will know that *k + m* is an even number. This assures that *k* and *m* are either both even numbers or they are both odd numbers.

John O. Kiltinen, " Parity Theorem for Permutations - An Enabling Lemma," *Convergence* (December 2004)