Integer is Congruent to Integer less than Modulus

Theorem
Let $$m \in \Z$$.

Then each integer is congruent (modulo $m$) to precisely one of the integers $$0, 1, \ldots, m - 1$$.

Proof of Existence
Let $$a \in \Z$$.

Then from the Division Theorem: $$\exists r \in \left\{{0, 1, \ldots, m-1}\right\}: a \equiv r \left({\bmod\, m}\right)$$.

Proof of Uniqueness
Suppose that:
 * $$\exists r_1, r_2 \in \left\{{0, 1, \ldots, m-1}\right\}: a \equiv r_1 \left({\bmod\, m}\right) \land a \equiv r_2 \left({\bmod\, m}\right)$$

Then:
 * $$\exists r_1, r_2 \in \Z: a = q_1 m + r_1 = q_2 m + r_2$$

This contradicts the uniqueness clause in the Division Theorem.