Principle of Mathematical Induction/One-Based/Proof 2

Proof
Let $M$ be the set of all $n \in \N_{>0}$ for which $\map P n$ holds.

By $(1)$ we have that $1 \in M$.

By $(2)$ we have that if $k \in M$ then $k + 1 \in M$.

From the Axiomatization of $1$-Based Natural Numbers, Axiom $(\text F)$, it follows that $M = \N_{>0}$.