Congruence by Product of Moduli

Theorem
Let $a, b, z \in \R$.

Let $a \equiv b \left({\bmod\, z}\right)$ denote that $a$ is congruent to $b$ modulo $z$.

Then $\forall y \in \R, y \ne 0$:
 * $a \equiv b \left({\bmod\, z}\right) \iff y a \equiv y b \left({\bmod\, y z}\right)$

Proof
Let $y \in \R: y \ne 0$.

Then:

Note the invalidity of the third step when $y = 0$.