Strict Negativity is equivalent to Strict Positivity of Negative

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Theorem

Let $\struct {D, +, \times}$ be an ordered integral domain, whose (strict) positivity property is denoted $P$.

Let $N$ be the (strict) negativity property on $D$:

$\forall a \in D: \map N a \iff \map P {-a}$


Then for all $a \in D$:

$\map P a \iff \map N {-a}$


Proof

\(\ds \map P a\) \(\leadstoandfrom\) \(\ds \map P {-\paren {-a} }\)
\(\ds \) \(\leadstoandfrom\) \(\ds \map N {-a}\) Definition of Strict Negativity Property

$\blacksquare$


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