Common Factor Cancelling in Congruence/Corollary 1

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

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

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
Let $a \perp m$.

Then by definition of coprime:
 * $\gcd \set {a, m} = 1$

The result follows immediately from Common Factor Cancelling in Congruence.