Square of Modulo less One equals One

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

Let $\Z_m$ be the set of integers modulo $m$:
 * $\Z_m = \left\{{\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m - 1}\right]\!\right]_m}\right\}$

Then:
 * $\left[\!\left[{m - 1}\right]\!\right]_m \times_m \left[\!\left[{m - 1}\right]\!\right]_m = \left[\!\left[{1}\right]\!\right]_m$

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