\begin{definition}[Definition:Power of Mapping]
Let $f: S \to S$ be a mapping from $S$ to itself.
Because the domain of $f$ is equal to the codomain of $f$ (both are $S$), the composite mapping $f \circ f$ is defined.
We define the '''$n$th power of $f$''' as:
:$\forall n \in \N: f^n = \begin{cases}
I_S & : n = 0 \\
f \circ f^{n - 1} & : n > 0
\end{cases}$
where $I_S$ is the identity mapping.
\end{definition}