Well-Ordering Principle/Proof using Naturally Ordered Semigroup

Proof
Consider the natural numbers $\N$ defined as the naturally ordered semigroup $\struct {S, \circ, \preceq}$.

From its definition, $\struct {S, \circ, \preceq}$ is well-ordered by $\preceq$.

The result follows.

As $\N_{\ne 0} = \N \setminus \set 0$, by Set Difference is Subset $\N_{\ne 0} \subseteq \N$.

As $\N$ is well-ordered, by definition, every subset of $\N$ has a smallest element.