Modulo Multiplication is Well-Defined/Proof 2

Proof
The equivalence class $\eqclass a m$ is defined as:
 * $\eqclass a m = \set {x \in \Z: x = a + k m: k \in \Z}$

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

Thus the notation for multiplication of two residue classes modulo $z$ is not usually $\eqclass a m \times_m \eqclass b m$.

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

Using this notation:

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