Modulo Addition is Commutative

Theorem
Addition modulo $m$ is commutative:

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

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