Trichotomy Law for Real Numbers

Theorem
The real numbers obey the Trichotomy Law.

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


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