Strict Negativity is equivalent to Strictly Preceding Zero

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

Let $\le$ be the total ordering induced by $P$, and let $<$ be its strict total ordering counterpart.

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


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

Then for all $a \in D$:
 * $\map N a \iff a < 0$