Positive Rational Numbers under Addition fulfil Naturally Ordered Semigroup Axioms 2 to 4

Theorem
Let $\Q_{\ge 0}$ denote the set of positive rational numbers.

Consider the naturally ordered semigroup axioms:

The algebraic structure:
 * $S := \struct {\Q_{\ge 0}, +, \le}$

is an ordered semigroup which fulfils the axioms:



Proof
First we note that from Positive Rational Numbers under Addition form Ordered Semigroup:
 * $\struct {\Q_{\ge 0}, +, \le}$ is an ordered semigroup.