Inversion Mapping Reverses Ordering in Ordered Group/Corollary/Proof 2

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

Let $x \in G$.

Then the following equivalences hold:

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

Thus by Inverses of Elements Related by Compatible Relation: Corollary:

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

Thus again by Inverses of Elements Related by Compatible Relation: Corollary: