Ordering Compatible with Group Operation is Strongly Compatible/Corollary/Proof 2

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

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

Let $x, y \in G$.

Then the following equivalences hold:

Proof
Each result follows from Ordering Compatible with Group Operation is Strongly Compatible.

For example, by Ordering Compatible with Group Operation is Strongly Compatible:


 * $x \preccurlyeq y \iff x \circ x^{-1} \preccurlyeq y \circ x^{-1}$

Since $x \circ x^{-1} = e$:


 * $x \preccurlyeq y \iff e \preccurlyeq y \circ x^{-1}$