# Order of Real Numbers is Dual of Order Multiplied by Negative Number

 It has been suggested that this page or section be merged into Real Number Ordering is Compatible with Multiplication/Negative Factor. (Discuss)

## Theorem

$\forall x, y, z \in \R: x > y, z < 0 \implies x \times z < y \times z$

## Proof

Let $z < 0$.

$-z > 0$

and so:

 $\displaystyle x$ $>$ $\displaystyle y$ $\displaystyle \leadsto \ \$ $\displaystyle x \times \paren {-z}$ $>$ $\displaystyle y \times \paren {-z}$ Real Number Axioms: $\R O2$: compatibility with multiplication $\displaystyle \leadsto \ \$ $\displaystyle -\paren {x \times z}$ $>$ $\displaystyle -\paren {y \times z}$ Multiplication by Negative Real Number $\displaystyle \leadsto \ \$ $\displaystyle x \times z$ $<$ $\displaystyle y \times z$ Order of Real Numbers is Dual of Order of their Negatives

$\blacksquare$