Power Function Preserves Ordering in Ordered Group/Corollary

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

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

Let $x \in G$.

Let $n \in \N_{>0}$ be a strictly positive integer.

Then the following hold:
 * $x \le e \implies x^n \le e$
 * $e \le x \implies e \le x^n$
 * $x < e \implies x^n < e$
 * $e < x \implies e < x^n$