Strict Lower Closure is Lower Set/Proof 1

Proof
Let $l \in p^\prec$.

Let $s \in S$ with $s \preceq l$.

Then by the definition of strict lower closure:
 * $l \prec p$

Thus by Extended Transitivity:
 * $s \prec p$

So by the definition of strict lower closure:
 * $s \in p^\prec$

Since this holds for all such $l$ and $s$, $p^\prec$ is a lower set.