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

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
Each result follows from Properties of Ordered Group/OG1. For example, by Properties of Ordered Group/OG1,


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

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


 * $(\operatorname {OG} 2.1): \quad x \preceq y \iff e \preceq y \circ x^{-1}$