Nonzero Natural Number is Successor

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

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

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 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: