Natural Number Multiplication Distributes over Addition

Theorem
The operation of multiplication is distributive over addition on the set of natural numbers $\N$:
 * $\forall x, y, z \in \N:$
 * $\paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$
 * $z \times \paren {x + y} = \paren {z \times x} + \paren {z \times y}$

Proof 3
In the Axiomatization of 1-Based Natural Numbers, this is rendered:
 * $\forall x, y, z \in \N_{> 0}:$
 * $\paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$
 * $z \times \paren {x + y} = \paren {z \times x} + \paren {z \times y}$