Ordering Compatible with Group Operation is Strongly Compatible/Corollary/Proof 1

Theorem
Let $\struct {G, \circ, \preceq}$ be an ordered group with identity $e$.

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

Let $x, y \in G$.

Then the following equivalences hold:

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

Thus by Relation Compatible with Group Operation is Strongly Compatible: Corollary, we obtain the first four results.

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

Again by Relation Compatible with Group Operation is Strongly Compatible: Corollary, we obtain the remaining results.