Modulo Multiplication is Well-Defined

Theorem
The multiplication modulo $m$ operation on $\Z_m$, the set of integers modulo $m$, defined by the rule:


 * $\left[\!\left[{x}\right]\!\right]_m \times_m \left[\!\left[{y}\right]\!\right]_m = \left[\!\left[{x y}\right]\!\right]_m$

is a well-defined operation.

That is:
 * If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a x \equiv b y \pmod m$.