Inversion Mapping Reverses Ordering in Ordered Group

Theorem
Let $(G,\circ, \le)$ be an ordered group.

Let $x, y \in G$.

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

Then the following equivalences hold:


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