Positive Rational Numbers under Addition form Ordered Semigroup/Proof 1

Proof
It is necessary to ascertain that $\struct {T, \circ {\restriction_T} }$ fulfils the ordered semigroup axioms:

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 $\text {OS} 0$ holds.

From Restriction of Associative Operation is Associative we have that $\text {OS} 1$ holds.

From Restriction of Ordering is Ordering we have that $\text {OS} 2$ holds.

The result follows.