Common Factor Cancelling in Congruence/Corollary 1/Warning

Theorem
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 $b$.

Let $a$ not be coprime to $m$.

Then it is not necessarily the case that:
 * $x \equiv y \pmod m$

Proof
Proof by Counterexample:

Let $a = 6, b = 21, x = 7, y = 12, m = 15$.

We note that $\map \gcd {6, 15} = 3$ and so $6$ and $15$ are not coprime.

We have that:

Then:

But: