Common Factor Cancelling in Congruence/Corollary 1

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

Let:
 * $a x \equiv b y \left({\bmod\, m}\right)$ and $a \equiv b \left({\bmod\, m}\right)$

where $a \equiv b \left({\bmod\, m}\right)$ denotes that $a$ is congruent modulo $m$ to $b$.

If $a$ is coprime to $m$, then:
 * $x \equiv y \left({\bmod\, m}\right)$

Proof
If $a \perp m$ then $\gcd \left\{{a, m}\right\} = 1$ by definition.

The result follows immediately from Common Factor Cancelling in Congruence.