Power Function Preserves Ordering in Ordered Group/Corollary/Proof 2

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 $\preccurlyeq$.

Let $x \in G$.

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

Then the following hold:

Proof
By the definition of an ordered group, $\preccurlyeq$ is a transitive relation compatible with $\circ$.

By Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements:
 * $x \preccurlyeq e \implies x^n \preccurlyeq e^n$
 * $e \preccurlyeq x \implies e^n \preccurlyeq x^n$

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

Thus we obtain the first two results:
 * $x \preccurlyeq e \implies x^n \preccurlyeq e$
 * $e \preccurlyeq x \implies e \preccurlyeq x^n$

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

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

Thus by the same method as above, we obtain the remaining results:


 * $x \prec e \implies x^n \prec e$
 * $e \prec x \implies e \prec x^n$