Inversion Mapping Reverses Ordering in Ordered Group/Corollary/Proof 1

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 Group Inverse Reverses Ordering in Ordered Group:

Since $e^{-1} = e$, the theorem holds.