Power of Idempotent Element

Theorem
Let $\left({S, \circ}\right)$ 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 $P \left({n}\right)$ be the proposition:
 * $s^n = s$

$P \left({1}\right)$ is the case:
 * $s^1 = s$

which holds by definition.

Thus $P \left({1}\right)$ is seen to hold.

Basis for the Induction
$P \left({2}\right)$ is the case:

Thus $P \left({2}\right)$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k + 1}\right)$ 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 $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

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