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
- 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.
$\blacksquare$