Power Function Preserves Ordering in Ordered Group/Corollary/Proof 1

Corollary to Power Function Preserves Ordering in Ordered Group
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered group with identity $e$.

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

Let $x \in G$.

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

Then the following hold:

Proof
By Power Function Preserves Ordering in Ordered Group:

By Identity Element is Idempotent, $e$ is idempotent with respect to $\circ$.

Therefore by the definition of an idempotent element, $e^n = e$.

Thus the theorem holds.