Strict Lower Closure is Lower Section/Proof 2

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Theorem

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

Let $p \in S$.

Let $p^\prec$ denote the strict lower closure of $p$.


Then $p^\prec$ is a lower section.


Proof

By Dual Pairs (Order Theory):

strict upper closure is dual to strict lower closure
Upper section is dual to lower section

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

$\blacksquare$