General Positivity Rule in Ordered Integral Domain/Corollary
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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:
- $\ds n \cdot x = \sum_{i \mathop = 1}^n x$
- $\ds x^n = \prod_{i \mathop = 1}^n x$
The result then follows directly from General Positivity Rule in Ordered Integral Domain.
$\blacksquare$