Modulo Multiplication is Well-Defined/Proof 2

Proof
The equivalence class $\left[\!\left[{a}\right]\!\right]_m$ is defined as:
 * $\left[\!\left[{a}\right]\!\right]_m = \left\{{x \in \Z: x = a + k m: k \in \Z}\right\}$

that is, the set of all integers which differ from $a$ by an integer multiple of $m$.

Thus the notation for addition of two residue classes modulo $z$ is not usually $\left[\!\left[{a}\right]\!\right]_m +_m \left[\!\left[{b}\right]\!\right]_m$.

What is more normally seen is:
 * $a b \pmod m$

Using this notation:

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