Power Function is Strictly Increasing on Positive Elements

Theorem
Let $\struct {R, +, \circ, \le}$ be an ordered ring.

Let $x, y \in R$.

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

Let $0 < x < y$.

Then:
 * $0 < \map {\circ^n} x < \map {\circ^n} y$

Proof
The result follows by repeated application of Ring Product preserves Inequalities on Positive Elements.

Also see

 * Odd Power Function is Strictly Increasing