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:
 * $$a \equiv b \left({\bmod\, z}\right)$$

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 $$m \backslash z$$, i.e. $$m$$ divides $$z$$.)

Then $$a \equiv b \left({\bmod\, m}\right)$$.

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

Thus:

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