Rational Multiplication Distributes over Addition

Theorem
Rational multiplication is distributive over addition:


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

Proof
Follows directly from the definition of rational numbers as the quotient field of the integral domain $$\left({\Z, +, \times}\right)$$ of integers.

So $$\left({\Q, +, \times}\right)$$ is a field, and therefore a priori $$\times$$ is distributive over $$+$$ on $$\Q$$.