Principle of Finite Induction/Peano Structure

Theorem
Let $\struct {P, s, 0}$ be a Peano structure.

Let $S \subseteq P$.

Suppose that:


 * $(1): \quad 0 \in S$


 * $(2): \quad \forall n: n \in S \implies \map s n \in S$

Then:


 * $S = P$

Proof
This is nothing but a reformulation of Axiom $(P5)$ of the Peano Axioms.