Power Function Preserves Ordering in Ordered Group/Corollary

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

Let $x \in \left({G, \circ, \le}\right)$.

Let $n \in \N_{>0}$.

Then the following hold:


 * $x \le I \implies x^n \le I$
 * $I \le x \implies I \le x^n$
 * $x < I \implies x^n < I$
 * $I < x \implies I < x^n$