Total Ordering is Total Relation

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Theorem

Let $S$ be a set.

Let $\mathcal R \subseteq S \times S$ be a total ordering.


Then $\mathcal R$ is a total relation.


Proof

By definition of total ordering:

$\mathcal R$ is a reflexive relation on the strength of being an ordering
$\mathcal R$ is a connected relation on the strength of being a total ordering.

$\blacksquare$