Natural Number Addition Commutativity with Successor/Proof 1

Proof
Proof by induction:

From definition of addition:

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

Basis for the Induction
From definition of addition:

Thus $\map P 0$ is seen to be true.

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 $\map P k$:
 * $\forall m \in \N: m^+ + k = \paren {m + k}^+$

Then we need to show that $\map P {k^+}$ follows directly from $\map P k$:
 * $\forall m \in \N: m^+ + k^+ = \paren {m + k^+}^+$

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.

Therefore:
 * $\forall m, n \in \N: m^+ + n = \paren {m + n}^+$