# Greatest Element is Dual to Smallest Element

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## Theorem

Let $\struct {S, \preceq}$ be an ordered set.

Let $a \in S$.

The following are dual statements:

- $a$ is the greatest element of $S$
- $a$ is the smallest element of $S$

## Proof

By definition, $a$ is the greatest element of $S$ if and only if:

- $\forall b \in S: b \preceq a$

The dual of this statement is:

- $\forall b \in S: a \preceq b$

By definition, this means $a$ is the smallest element of $S$.

The converse follows from Dual of Dual Statement (Order Theory).

$\blacksquare$