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 $a x \equiv b y \pmod m$.

Warning
This result does not hold when $x, y, m \notin \Z$.

We get to this stage in the above proof:
 * $x y = \left({x' + k_1 m}\right) \left({y' + k_2 m}\right) = x' y' + \left({x' k_2 + y' k_1}\right) m + k_1 k_2 m^2$

and we note that:
 * $\left({x' k_2 + y' k_1}\right) m + k_1 k_2 m^2$

is not necessarily an integer.

In fact, $\left({x' k_2 + y' k_1}\right) m + k_1 k_2 m^2$ can only be guaranteed to be an integer if each of $x', y', m \in \Z$.

Hence $x' y'$ is not necessarily congruent to $x y$.