Modulo Multiplication Distributes over Modulo Addition

Theorem
Multiplication modulo $m$ is distributive over addition modulo $m$:

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


 * $$\left[\left[{x}\right]\right]_m \times_m \left({\left[\left[{y}\right]\right]_m +_m \left[\left[{z}\right]\right]_m}\right) = \left({\left[\left[{x}\right]\right]_m \times_m \left[\left[{y}\right]\right]_m}\right) +_m \left({\left[\left[{x}\right]\right]_m \times_m \left[\left[{z}\right]\right]_m}\right)$$;
 * $$\left({\left[\left[{x}\right]\right]_m +_m \left[\left[{y}\right]\right]_m}\right) \times_m \left[\left[{z}\right]\right]_m = \left({\left[\left[{x}\right]\right]_m \times_m \left[\left[{z}\right]\right]_m}\right) +_m \left({\left[\left[{y}\right]\right]_m \times_m \left[\left[{z}\right]\right]_m}\right)$$.

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

And the second is like it, namely this: