Modulo Addition is Associative

Theorem
Addition modulo $m$ is associative:

$$\forall \left[\left[{x}\right]\right]_m, \left[\left[{y}\right]\right]_m, \left[\left[{z}\right]\right]_m \in \mathbb{Z}_m: \left({\left[\left[{x}\right]\right]_m +_m \left[\left[{y}\right]\right]_m}\right) +_m \left[\left[{z}\right]\right]_m = \left[\left[{x}\right]\right]_m +_m \left({\left[\left[{y}\right]\right]_m +_m \left[\left[{z}\right]\right]_m}\right)$$.

Proof
Follows directly from the definition of addition modulo $m$: