Square of Modulo less One equals One

Theorem
Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:
 * $\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \ldots, \eqclass {m - 1} m}$

Then:
 * $\eqclass {m - 1} m \times_m \eqclass {m - 1} m = \eqclass 1 m$

where $\times_m$ denotes multiplication modulo $m$.