Natural Numbers under Addition form Commutative Semigroup
Theorem
The algebraic structure $\struct {\N, +}$ consisting of the set of natural numbers $\N$ under addition $+$ is a commutative semigroup.
Proof
By Naturally Ordered Semigroup Exists, there exists a naturally ordered semigroup.
Consider the natural numbers $\N$ defined as the naturally ordered semigroup.
From the definition of the naturally ordered semigroup, $\struct {\N, +}$ is a priori a semigroup.
By Naturally Ordered Semigroup Axioms imply Commutativity, it follows that $\struct {\N, +}$ is a commutative semigroup.
$\blacksquare$