Properties of Ordered Group/OG4/Proof 2

Theorem
Let $\left({G, \circ, \preceq}\right)$ be an ordered group with identity $e$.

Let $x \in G$.

Then the following equivalences hold:

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

Thus by Properties of Relation Compatible with Group Operation/CRG4, we obtain the first two results:

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

Thus by again Properties of Relation Compatible with Group Operation/CRG4, we obtain the remaining results: