Natural Number Multiplication Distributes over Addition/Proof 3

Proof
Using the following axioms:

Left Distributive Law for Natural Numbers
First we show that:
 * $n \times \paren {x + y} = \paren {n \times x} + \paren {n \times y}$

Right Distributive Law for Natural Numbers
Then we show that:
 * $\paren {x + y} \times n = \paren {x \times n} + \paren {y \times n}$

The result follows.