Congruence by Product of Moduli

Theorem
Let $$a, b, z \in \R$$ such that $$z \ne 0$$.

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 $$y = 0$$.