Strict Lower Closure is Lower Section/Proof 2

Theorem
Let $(S, \preceq)$ be an ordered set.

Let $p \in S$.

Then ${\dot\downarrow} p$, the strict down-set of $p$, is a lower set.

 By Dual Pairs (Order Theory):
 * Strict up-set is dual to strict down-set
 * Upper set is dual to lower set

Thus the theorem holds by Strict Up-Set is Upper Set and the duality principle.

Also see

 * Lower Closure is Lower Set
 * Strict Up-Set is Upper Set