Integer Multiplication Distributes over Addition

Theorem
Integer multiplication is distributive over addition:


 * $$\forall x, y, z \in \mathbb{Z}: x \times \left({y + z}\right) = \left({x \times y}\right) + \left({x \times z}\right)$$
 * $$\forall x, y, z \in \mathbb{Z}: \left({y + z}\right) \times x = \left({y \times x}\right) + \left({z \times x}\right)$$

Proof
We need to show that, $$\forall a, b, c, d, e, f \in \mathbb{N}$$:


 * $$\left[\left[{a, b}\right]\right]_\boxminus \times \left({\left[\left[{c, d}\right]\right]_\boxminus + \left[\left[{e, f}\right]\right]_\boxminus}\right) = \left({\left[\left[{a, b}\right]\right]_\boxminus \times \left[\left[{c, d}\right]\right]_\boxminus}\right) + \left({\left[\left[{a, b}\right]\right]_\boxminus \times \left[\left[{e, f}\right]\right]_\boxminus}\right)$$


 * $$\left({\left[\left[{c, d}\right]\right]_\boxminus + \left[\left[{e, f}\right]\right]_\boxminus}\right) \times \left[\left[{a, b}\right]\right]_\boxminus = \left({\left[\left[{c, d}\right]\right]_\boxminus \times \left[\left[{a, b}\right]\right]_\boxminus}\right) + \left({\left[\left[{e, f}\right]\right]_\boxminus \times \left[\left[{a, b}\right]\right]_\boxminus}\right)$$

From Natural Numbers form Semiring, we can take it for granted that:
 * addition and multiplication are commutative and associative on the natural numbers $$\mathbb{N}$$;
 * natural number multiplication is distributive over natural number addition.