Natural Number Addition is Associative/Proof 2

Proof
Consider the von Neumann construction of natural numbers $\N$, as elements of the minimal infinite successor set $\omega$.

We are to show that:
 * $\paren {x + y} + n = x + \paren {y + n}$

for all $x, y, n \in \N$.

From the definition of addition, we have that:

Let $x, y \in \N$ be arbitrary.

For all $n \in \N$, let $\map P n$ be the proposition:
 * $\paren {x + y} + n = x + \paren {y + n}$

Basis for the Induction
$\map P 0$ is the case:

and so $\map P 0$ holds.

This is our basis for the induction.

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

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

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

Induction Step
This is our induction step:

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