Natural Number Multiplication is Commutative/Proof 3

Proof
Using the following axioms:

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * $\forall a \in \N_{> 0}: a \times n = n \times a$

Basis for the Induction
$\map P 1$ is the case:

and so $\map P 1$ holds.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:
 * $\forall a \in \N: a \times k = k \times a$

Then we need to show:
 * $\forall a \in \N: a \times \paren {k + 1} = \paren {k + 1} \times a$

Induction Step
This is our induction step:

The result follows by the Principle of Mathematical Induction.