Directed iff Lower Closure Directed

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

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

Then $H$ is directed :
 * $H^\precsim$ is directed

where $H^\precsim$ denotes the lower closure of $H$ with respect to $\precsim$.

Sufficient Condition
Let us assume that $H$ is directed.

Let $x, y \in H^\precsim$.

By definition of lower closure:
 * $\exists x' \in H: x \precsim x'$

and
 * $\exists y' \in H: y \precsim y'$

By definition of directed subset:
 * $\exists z \in H: x' \precsim z \land y' \precsim z$

By definition of reflexivity;
 * $z \precsim z$

By definition of lower closure:
 * $z \in H^\precsim$

Thus by definition of transitivity:
 * $\exists z \in H^\precsim: x \precsim z \land y \precsim z$

Thus by definition:
 * $H^\precsim$ is directed.

Necessary Condition
Let us assume that $H^\precsim$ is directed.

Let $x, y \in H$.

By definition of reflexivity:
 * $x \precsim x \land y \precsim y$

By definition of lower closure:
 * $x, y \in H^\precsim$

By definition of directed subset:
 * $\exists z \in H^\precsim: x \precsim z \land y \precsim z$

By definition of lower closure:
 * $\exists z' \in H: z \precsim z'$

Thus by definition of transitivity
 * $\exists z' \in H: x \precsim z' \land y \precsim z'$

Thus by definition:
 * $H$ is directed.