# 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 $a \in D$:

$\map N a, \map P b \implies \map N {a \times b}$

## Proof

 $\ds \map N a, \map P b$ $\leadsto$ $\ds \map P {-a}, \map P b$ Definition of Strict Negativity Property $\ds$ $\leadsto$ $\ds \map P {\paren {-a} \times b}$ Strict Positivity Property: $(P \, 2)$ $\ds$ $\leadsto$ $\ds \map P {-\paren {a \times b} }$ Product with Ring Negative $\ds$ $\leadsto$ $\ds \map N {a \times b}$ Definition of Strict Negativity Property

$\blacksquare$