Common Factor Cancelling in Congruence/Corollary 2

Corollary to Common Factor Cancelling in Congruence
Let $a, b, x, y, m \in \Z$. Let:
 * $a x \equiv a y \pmod m$

where $a \equiv b \pmod m$ denotes that $a$ is congruent modulo $m$ to $b$.

If $a$ is coprime to $m$, then:
 * $x \equiv y \pmod m$

Proof
From Common Factor Cancelling in Congruence: Corollary 1:


 * $a x \equiv b y \pmod m$ and $a \equiv b \pmod m$

and:
 * $a$ is coprime to $m$

then:
 * $x \equiv y \pmod m$

The result follows immediately from setting $a = b$.