Composition of Product Mappings on Natural Numbers

Theorem
Let $a \in \N$ be a natural number.

Let $\mu_a: \N \to \N$ be the mapping defined as:
 * $\forall x \in \N: \map {\mu_a} x = x a$

Then:
 * $\mu_{a b} = \mu_b \circ \mu_a$