Nonzero Natural Number is Successor

Theorem
Let $\N$ be the $0$-based natural numbers:
 * $\N = \set {0, 1, 2, \ldots}$

Let $s: \N \to \N: \map s n = n + 1$ be the successor mapping.

Then:
 * $\forall n \in \N \setminus \set 0 \paren {\exists m \in \N: \map s m = n}$

Proof
The proof will proceed by the Principle of Finite Induction on $\N \setminus \set 0$.

Basis for the Induction

 * $\map s 0 = 1$

So $1$ is the immediate successor of $0$.

This is our basis for the induction.

Induction Hypothesis
This is our induction hypothesis:
 * $\exists m \in \N: \map s m = k$

Then we need to show:
 * $\exists m' \in \N: \map s {m'} = k + 1$

Induction Step
This is our induction step:

Hence the result by the Principle of Finite Induction.