Integer is Congruent to Integer less than Modulus

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

Proof

 * Existence:

Let $$a \in \mathbb{Z}$$.

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


 * Uniqueness:

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

Then $$\exists r_1, r_2 \in \mathbb{Z}: a = q_1 m + r_1 = q_2 m + r_2$$.

This contradicts the uniqueness clause in the Division Theorem.