Power Function Preserves Ordering in Ordered Group

Theorem
Let $\struct {S, \circ, \preceq}$ be an ordered group. Let $n \in \N_{>0}$ be a strictly positive integer.

Let $\prec$ be the reflexive reduction of $\preceq$.

Then the following hold:


 * $\forall x, y \in S: x \preceq y \implies x^n \preceq y^n$
 * $\forall x, y \in S: x \prec y \implies x^n \prec y^n$

where $x^n$ denotes the $n$th power of $x$.