Directed in Join Semilattice

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

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

Then $H$ is directed
 * $\forall x, y \in H: x \vee y \in H$

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

Let $x, y \in H$.

By definition of directed:
 * $\exists z \in H: x \preceq z \land y \preceq z$

By definition
 * $z$ is upper bound of $\set {x, y}$

By definitions of supremum and join:
 * $x \vee y = \sup \set {x, y} \preceq z$

Thus by definition of lower section:
 * $x \vee y \in H$

Necessary Condition
Let us assume that
 * $\forall x, y \in H: x \vee y \in H$

Let $x, y \in H$.

By assumption:
 * $x \vee y \in H$

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

Thus
 * $x \preceq x \vee y \land y \preceq x \vee y$

Thus by definition
 * $H$ is directed.