Naturally Ordered Semigroup Axioms imply Commutativity

Theorem
Consider the naturally ordered semigroup axioms:

Axioms $\text {NO} 1$, $\text {NO} 2$ and $\text {NO} 3$ together imply the commutativity of the naturally ordered semigroup $\struct {S, \circ, \preceq}$.

Proof
From, $\struct {S, \circ, \preceq}$ has a smallest element.

This is identified as zero: $0$.

From Zero is Identity in Naturally Ordered Semigroup, $0$ is the identity element of $\struct {S, \circ, \preceq}$,

It may be the case that $S$ is a singleton such that $S = \set 0$.

Then $\struct {S, +}$ degenerates to the trivial group.

From Trivial Group is Abelian, it follows that $+$ is commutative.

Let $S^* := S \setminus \set 0$ denote the complement of $\set 0$ in $S$.

From, $S^*$ also has a smallest element.

This we will call $1$.

It will be shown that $1$ commutes with every element of $S$.