# Ordering Properties of Real Numbers

## Theorem

### Trichotomy Law

The real numbers obey the Trichotomy Law.

That is, $\forall a, b \in \R$, exactly one of the following holds:

 $(1)$ $:$ $a$ is greater than $b$: $\displaystyle a > b$ $(2)$ $:$ $a$ is equal to $b$: $\displaystyle a = b$ $(3)$ $:$ $a$ is less than $b$: $\displaystyle a < b$

### Ordering is Transitive

Let $a, b, c \in \R$ such that $a > b$ and $b > c$.

Then:

$a > c$

### Ordering is Compatible with Addition

$\forall a, b, c \in \R: a < b \implies a + c < b + c$

### Positive Factor

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

### Negative Factor

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