Natural Numbers under Multiplication form Ordered Commutative Semigroup

Theorem
Let $\left({S, \circ, *, \preceq}\right)$ be a Naturally Ordered Semigroup with Product.

Then $\left({S, *, \preceq}\right)$ is an ordered commutative semigroup.

Proof
From Ordering on Naturally Ordered Semigroup Product, we have that:
 * $\forall m, n, p \in S: m \prec p \land 0 \prec n \implies n * m \prec n * p$

So $\left({S, *, \preceq}\right)$ is seen by definition to be an ordered semigroup.

From Multiplication in Naturally Ordered Semigroup is Commutative it follows that $\left({S, *}\right)$ is a commutative semigroup.

Hence the result by definition of ordered commutative semigroup.