General Positivity Rule in Ordered Integral Domain/Corollary

Corollary to General Positivity Rule in Ordered Integral Domain
Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$.

Let $\map P x$ where $x \in D$.

Then:
 * $\map P {n \cdot x}$ and $\map P {x^n}$

Proof
From the definition of power of an element:
 * $\displaystyle n \cdot x = \sum_{i \mathop = 1}^n x$


 * $\displaystyle x^n = \prod_{i \mathop = 1}^n x$

The result then follows directly from General Positivity Rule in Ordered Integral Domain.