Directed in Join Semilattice with Finite Suprema

Theorem
Let $\struct {S, \preceq}$ be a join semilattice.

Let $H$ be a non-empty lower section 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 $\struct {S, \preceq}$

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

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

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

Let $x, y \in H$.


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

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

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

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

Thus by definition
 * $H$ is directed.