Example 2

Let X be a set and let E be the set of mappings of X into itself. Let $e$ denote the identity mapping of X into itself, and let $f$ be any element of E. Take $S\{u\}$ to be the term $f \circ u$ [^4]. By applying C63 we see that there exists a unique mapping of $\mathbf{N}$ into E, denoted by $n \to f^n$, such that $f^0 = e$ and $f^{n+1} = f \circ f^n$. The mapping $f^n$ is called the nth iterate of the mapping $f$.