Modulo Multiplication Distributes over Modulo Addition

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

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


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

where $$\Z_m$$ is the set of integers modulo $m$.

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

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

$$ $$ $$ $$ $$

And the second is like it, namely this:

$$ $$ $$ $$ $$