Composition of Addition Mappings on Natural Numbers

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

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

Then:
 * $\alpha_{a + b} = \alpha_b \circ \alpha_a$