# Real Number Ordering is Compatible with Multiplication/Negative Factor

## 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 |

$\blacksquare$

## Sources

- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.4$: Inequalities: $\text{(IV)}$