Power Function Preserves Ordering in Ordered Group

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

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

Then the following hold:

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