Power Function Preserves Ordering in Ordered Group

Theorem
Let $\struct {S, \circ, \preceq}$ be an ordered group.

Let $x,y \in S$. Let $n \in \N_{>0}$ be a strictly positive integer. Let $\prec$ be the reflexive reduction of $\preceq$.

Then the following hold:
 * If $x \preceq y$ then $x^n \preceq y^n$
 * If $x \prec y$ then $x^n \prec y^n$

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

Proof
By the definition of an ordered group, $\preceq$ is compatible with $\circ$.

By the definition of an ordering, $\preceq$ is transitive.

By Reflexive Reduction of Relation Compatible with Group Operation is Compatible, $\prec$ is also compatible with $\circ$.

By Reflexive Reduction of Transitive Antisymmetric Relation is Strict Ordering, $\prec$ is also transitive.

By the definition of an ordered group, $\struct {S, \circ}$ is a group, and therefore a semigroup.

Thus the theorem holds by Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements.