Natural Number Multiplication is Associative/Proof 3

Proof
Using the following axioms:

Let $x, y \in \N_{> 0}$ be arbitrary.

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

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 $\map P {k^+}$ is true.

So this is our induction hypothesis:
 * $\paren {x \times y} \times k = x \times \paren {y \times k}$

Then we need to show:
 * $\paren {x \times y} \times \paren {k + 1} = x \times \paren {y \times \paren {k + 1} }$

Induction Step
This is our induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.