Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative

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 $a \in D$:
 * $\map N a, \map P b \implies \map N {a \times b}$