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

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

And the second is like it, namely this: