Power of Idempotent Element

Theorem
Let $\struct {S, \circ}$ be an algebraic structure.

Let $s \in S$ be an idempotent element with respect to $\circ$.

Then:
 * $\forall n \in \Z_{> 0}: s^n = s$

where $s^n$ is defined as:
 * $s^n = \begin{cases} s & : n = 1 \\

s^{n - 1} \circ s & : n > 1 \end{cases}$

Proof
The proof proceeds by induction.

For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
 * $s^n = s$

$\map P 1$ is the case:
 * $s^1 = s$

which holds by definition.

Thus $\map P 1$ is seen to hold.

Basis for the Induction
$\map P 2$ is the case:

Thus $\map P 2$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * $s^k = s$

from which it is to be shown that:
 * $s^{k + 1} = s$

Induction Step
This is the induction step:

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

Therefore:
 * $\forall n \in \Z_{> 0}: s^n = s$