Filtered iff Finite Subsets have Lower Bounds

Theorem
Let $\struct {S, \precsim}$ be a preordered set.

Let $H$ be a non-empty subset of $S$.

Then $H$ is filtered :


 * for every finite subset $A$ of $H$:
 * $\exists h \in H: \forall a \in A: h \precsim a$

Proof
This follows by mutatis mutandis of the proof of Directed iff Finite Subsets have Upper Bounds.