Ordering Compatible with Group Operation is Strongly Compatible

Theorem
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group whose identity element is $e$.

Let $\prec$ be the reflexive reduction of $\preccurlyeq$.

The following hold:

Proof
By definition of ordered group, $\preccurlyeq$ is a relation compatible with $\circ$.

Thus by Relation Compatible with Group Operation is Strongly Compatible:

By Reflexive Reduction of Relation Compatible with Group Operation is Compatible, $\prec$ is compatible with $\circ$.

Thus again by Relation Compatible with Group Operation is Strongly Compatible:

Hence the result.