Natural Number Commutes with 1 under Addition

Theorem
Let $n \in \N_{> 0}$ be a natural number.

Then $n$ commutes with $1$ under the operation of addition:


 * $\forall n \in \N_{> 0}: n + 1 = 1 + n$

Proof
Using the axiomatization:

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * $n + 1 = 1 + n$

Basis for the Induction
Setting $n = 1$ we have that:
 * $1 + 1 = 1 + 1$

and so $\map P 1$ holds trivially.

This is our basis for the induction.

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

So this is our induction hypothesis:
 * $k + 1 = 1 + k$

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

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.