Power Function Preserves Ordering in Ordered Semigroup

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

Let $x, y \in S$, and suppose that $x \preceq y$.

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

Then:
 * $x^n \preceq y^n$

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

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

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

Thus by Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements, $x^n \preceq y^n$.