This note gives a proof for the familiar elementary theorem that if a permutation of the integers 1, …, n (with n ≥ 2) is expressed as a product π1 of N1 transpositions and also as a product π2 of N2 transpositions, then N1 and N2 are both even or both odd (equivalently: N1 + N2 is even).

Here, a transposition means an interchange of two of the integers. If the interchange is between adjacent integers it is called an adjacent-trans position.