Modulo Addition is Associative

Theorem
Addition modulo $m$ is associative:


 * $$\forall \left[\!\left[{x}\right]\!\right]_m, \left[\!\left[{y}\right]\!\right]_m, \left[\!\left[{z}\right]\!\right]_m \in \R_m: \left({\left[\!\left[{x}\right]\!\right]_m +_m \left[\!\left[{y}\right]\!\right]_m}\right) +_m \left[\!\left[{z}\right]\!\right]_m = \left[\!\left[{x}\right]\!\right]_m +_m \left({\left[\!\left[{y}\right]\!\right]_m +_m \left[\!\left[{z}\right]\!\right]_m}\right)$$

where $$\R_m$$ is the set of all residue classes modulo $m$.

That is:
 * $$\forall x, y, z \in \R: \left({x + y}\right) + z \equiv x + \left({y + z}\right) \pmod m$$

Proof
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