Properties of Strict Negativity

Theorem
Let $\left({D, +, \times}\right)$ be an ordered integral domain, whose 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 negativity property on $D$:


 * $\forall a \in D: N \left({a}\right) \iff P \left({-a}\right)$

Then the following properties apply for all $a, b \in D$:
 * $(1): \quad N \left({a}\right) \iff a < 0$
 * $(2): \quad P \left({a}\right) \iff N \left({-a}\right)$
 * $(3): \quad N \left({a}\right), N \left({b}\right) \implies N \left({a + b}\right)$
 * $(4): \quad N \left({a}\right), N \left({b}\right) \implies P \left({a \times b}\right)$
 * $(5): \quad N \left({a}\right), P \left({b}\right) \implies N \left({a \times b}\right)$