Trichotomy Law for Real Numbers/Proof 2

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


$\le$ is a total ordering on $\R$.

The trichotomy follows directly from Trichotomy Law.