# 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:

 $\ds a$ $<$ $\ds b$ $\ds \leadsto \ \$ $\ds b - a$ $>$ $\ds 0$ Definition of Positivity Property $\ds \leadsto \ \$ $\ds c \times \paren {b - a}$ $<$ $\ds 0$ Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative $\ds \leadsto \ \$ $\ds b \times c - a \times c$ $<$ $\ds 0$ Ring Axiom $\text D$: Distributivity of Product over Addition $\ds \leadsto \ \$ $\ds a \times c$ $>$ $\ds b \times c$ Definition of Positivity Property

$\blacksquare$

## Examples

### $15 \times 3$ Greater than $12 \times 3$

We have that:

$15 > 12$
$15 \times 3 > 12 \times 3$

That is:

$45 > 36$

### $\dfrac {15} 3$ Greater than $\dfrac {12} 3$

We have that:

$15 > 12$
$\dfrac {15} 3 > \dfrac {12} 3$

That is:

$5 > 4$

### $15 \times \paren {-3}$ Less than $12 \times \paren {-3}$

We have that:

$15 > 12$
$15 \times \paren {-3} < 12 \times \paren {-3}$

That is:

$-45 < -36$

### $\dfrac {15} {-3}$ Less than $\dfrac {12} {-3}$

We have that:

$15 > 12$
$\dfrac {15} {-3} < \dfrac {12} {-3}$

That is:

$-5 < -4$