Congruence by Divisor of Modulus

Theorem
Let $z \in \R$ be a real number.

Let $a, b \in \R$ such that $a$ is congruent modulo $z$ to $b$, that is:
 * $a \equiv b \pmod z$

Let $m \in \R$ such that $z$ is an integral multiple of $m$:
 * $\exists k \in \Z: z = k m$.

(When $z$ is an integer, this can be expressed $m \mathop \backslash z$, i.e. $m$ divides $z$.)

Then:
 * $a \equiv b \pmod m$

Proof
We are given that $\exists k \in \Z: z = k m$.

Thus: