Positive Rational Numbers under Addition form Ordered Semigroup/Proof 2

Proof
From Rational Numbers form Ordered Field, $\struct {\Q, +, \times, \preceq}$ is an ordered field.

Hence $\struct {\Q, +, \preceq}$ is an ordered group, and so an ordered semigroup.

From Positive Rational Numbers are Closed under Addition we have that $\struct {\Q_{\ge 0}, +}$ is closed.

Hence from Subsemigroup Closure Test, $\struct {\Q_{\ge 0}, +}$ is a subsemigroup of $\struct {\Q, +}$.

From Subsemigroup of Ordered Semigroup is Ordered, $\struct {\Q_{\ge 0}, +, \preceq}$ is an ordered semigroup.