Principle of Finite Induction/One-Based

Theorem
Let $S \subseteq \N_{>0}$ be a subset of the $1$-based natural numbers.

Suppose that:


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


 * $(2): \quad \forall n \in \N_{>0} : n \in S \implies n + 1 \in S$

Then:


 * $S = \N_{>0}$

Proof
Consider $\N$ defined as a naturally ordered semigroup.

The result follows directly from Principle of Mathematical Induction for Naturally Ordered Semigroup: General Result.