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

Theorem
Let $\left({G, \circ, \le}\right)$ be an ordered group with identity $e$.

Let $<$ be the reflexive reduction of $\le$.

Let $x, y \in G$.

Then the following equivalences hold:


 * $(\operatorname{OG}2.1):\quad x \le y \iff e \le y \circ x^{-1}$
 * $(\operatorname{OG}2.2):\quad x \le y \iff e \le x^{-1} \circ y$


 * $(\operatorname{OG}2.3):\quad x \le y \iff x \circ y^{-1} \le e$
 * $(\operatorname{OG}2.4):\quad x \le y \iff y^{-1} \circ x \le e$


 * $(\operatorname{OG}2.1'):\quad x < y \iff e < y \circ x^{-1}$
 * $(\operatorname{OG}2.2'):\quad x < y \iff e < x^{-1} \circ y$


 * $(\operatorname{OG}2.3'):\quad x < y \iff x \circ y^{-1} < e$
 * $(\operatorname{OG}2.4'):\quad x < y \iff y^{-1} \circ x < e$

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


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

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


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