Example 1

Suppose that $a$ is an element of a set $E$ and that $S\{u\}$ is the term $g(u)$, where $g$ is a mapping of $E$ into itself [^3]. Then it is immediately seen by induction on $n$ that for all $n \in \mathbf{N}$ we have $f(n) \in E$; consequently $f$ is a mapping of $\mathbf{N}$ into $E$ such that $f(0) = a$ and $f(n+1) = g(f(n))$ for all integers $n$.

Likewise, let $h$ be a mapping of $\mathbf{N} \times E$ into $E$, and let $\psi$ be the mapping of $\mathbf{N} \times E$ into itself defined by $\psi(n, x) = (n+1, h(n, x))$. By the preceding discussion there exists a unique mapping $g = (\theta, f)$ of $\mathbf{N}$ into $\mathbf{N} \times E$ such that $g(0) = (0, a)$ and $g(n+1) = \psi(g(n))$ for all $n$, from which follows the existence and uniqueness of a mapping $f$ of $\mathbf{N}$ into E such that $f(0) = a$ and $f(n + 1) = h(n, f(n))$ for each integer $n$.