Modulo Multiplication is Associative

Theorem
Multiplication modulo $m$ is associative:


 * $\forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m, \left[\!\left[{z}\right]\!\right]_m \in \Z_m: \left({\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m}\right) \times_m \left[\!\left[{z}\right]\!\right]_m = \left[\!\left[{x}\right]\!\right]_m \times_m \left({\left[\!\left[{y}\right]\!\right]_m \times_m \left[\!\left[{z}\right]\!\right]_m}\right)$

That is:
 * $\forall x, y, z \in \Z_m: \left({x \cdot_m y}\right) \cdot_m z = x \cdot_m \left({y \cdot_m z}\right)$