Congruent Integers less than Half Modulus are Equal

Theorem
Let $k \in \Z_{>0}$ be a strictly positive integer.

Let $a, b \in \Z$ such that $\size a < \dfrac k 2$ and $\size b < \dfrac k 2$.

Then:
 * $a \equiv b \pmod k \implies a = b$

where $\equiv$ denotes congruence modulo $k$.

Proof
We have that:


 * $-\dfrac k 2 < a < \dfrac k 2$

and:


 * $-\dfrac k 2 < -b < \dfrac k 2$

Thus:
 * $-k < a - b < k$

Let $a \equiv b \pmod k$

Then:
 * $a - b = n k$

for some $n \in \Z$.

But as $-k < n k < k$ it must be the case that $n = 0$.

Thus $a - b = 0$ and the result follows.