Power Function Preserves Ordering in Ordered Group/Proof 1

Proof
By definition of ordered group:
 * $\preceq$ is compatible with $\circ$.

By definition of ordering:
 * $\preceq$ is transitive.

From Reflexive Reduction of Relation Compatible with Group Operation is Compatible:
 * $\prec$ is also compatible with $\circ$.

From Reflexive Reduction of Transitive Antisymmetric Relation is Strict Ordering:
 * $\prec$ is also transitive.

By definition of ordered group:
 * $\struct {S, \circ}$ is a ordered group, and therefore a fortiori a semigroup.

The result follows from Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements.