# Real Number Ordering is Compatible with Multiplication/Negative Factor

 It has been suggested that this page or section be merged into Order of Real Numbers is Dual of Order Multiplied by Negative Number. (Discuss)

## Theorem

$\forall a, b, c \in \R: a < b \land c < 0 \implies a c > b c$

where $\R$ is the set of real numbers.

## Proof

From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.

Thus:

 $\displaystyle a$ $<$ $\displaystyle b$ $\displaystyle \leadsto \ \$ $\displaystyle b - a$ $>$ $\displaystyle 0$ Definition of Positivity Property $\displaystyle \leadsto \ \$ $\displaystyle c \times \paren {b - a}$ $<$ $\displaystyle 0$ Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative $\displaystyle \leadsto \ \$ $\displaystyle b \times c - a \times c$ $<$ $\displaystyle 0$ Ring Axioms: Product is Distributive over Addition $\displaystyle \leadsto \ \$ $\displaystyle a \times c$ $>$ $\displaystyle b \times c$ Definition of Positivity Property

$\blacksquare$