Strict Lower Closure is Lower Section/Proof 2

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $p \in S$.

Then $p^\prec$, the strict lower closure of $p$, is a lower set.

 By Dual Pairs (Order Theory):
 * strict upper closure is dual to strict lower closure
 * Upper set is dual to lower set

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