Dual of Dual Statement (Order Theory)

Theorem

Let $\Sigma$ be a statement about ordered sets.

Let $\Sigma^*$ be its dual statement.

Then $\Sigma$ is also the dual statement of $\Sigma^*$.

Proof

By definition, the dual statement $\Sigma^*$ is formed by replacing the ordering $\preceq$ with its dual $\succeq$.

By Dual of Dual Ordering, applying this operation twice results in the original sentence $\Sigma$ again.

$\blacksquare$