Directed in Join Semilattice with Finite Suprema

Theorem
Let $\left({S, \preceq}\right)$ be a join semilattice.

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

Then $H$ is directed
 * for every non-empty finite subset $A$ of $H$, $\sup A \in H$

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

Let $A$ be a non-empty finite subset of $H$.

By Directed iff Finite Subsets have Upper Bounds:
 * $\exists h \in H: \forall a \in A: a \preceq h$

By definition
 * $z$ is upper bound of $A$

By Existence of Non-Empty Finite Suprema in Join Semilattice:
 * $\sup A$ exists in $\left({S, \preceq}\right)$

By definition of supremum:
 * $\sup A \preceq h$

Thus by definition of lower set:
 * $\sup A \in H$

Necessary Condition
Let us assume that
 * for every non-empty finite subset $A$ of $H$, $\sup A \in H$

Let $x, y \in H$.


 * $A := \left\{ {x, y}\right\}$ is a non-empty finite subset of $H$

By assumption:
 * $\sup \left\{ {x, y}\right\} \in H$

By definition of supremum:
 * $\sup A$ is upper bound for $\left\{ {x, y}\right\}$

Thus
 * $x \preceq \sup A \land y \preceq \sup A$

Thus by definition
 * $H$ is directed.