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 integer multiple of $m$:
 * $\exists k \in \Z: z = k m$

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

Integer Modulus
When $z$ is an integer, and therefore a composite number such that $z = r s$, this result can be expressed as:

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

Thus: