Principle of Mathematical Induction/Peano Structure

Theorem
Let $\left({P, 0, s}\right)$ be a Peano structure.

Let $Q \left({n}\right)$ be a propositional function depending on $n \in P$.

Suppose that:
 * $(1): \quad Q \left({0}\right)$ is true
 * $(2): \quad \forall n \in P: Q \left({n}\right) \implies Q \left({s \left({n}\right)}\right)$

Then:
 * $\forall n \in P: Q \left({n}\right)$

Proof
Let $A \subseteq P$ be defined by:
 * $A := \left\{{n \in P: Q \left({n}\right)}\right\}$

From $(1)$, $0 \in A$.

From $(2)$:
 * $\forall n \in P: n \in A \implies s \left({n}\right) \in A$

As this holds for all $n \in P$, it holds a fortiori for all $n \in A$.

Thus the condition:
 * $n \in A \implies s \left({n}\right) \in A$

is satisfied.

So by Axiom $(P5)$ of the Peano Axioms:
 * $A = P$

That is:
 * $\forall n \in P: Q \left({n}\right)$