Meet of Directed Subsets is Directed

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

Let $D_1, D_2$ be directed subset of $S$.

Then
 * $\left\{ {x \wedge y: x \in D_1, y \in D_2}\right\}$ is directed subset of $S$

Proof
Let $a, b \in \left\{ {x \wedge y: x \in D_1, y \in D_2}\right\}$.

Then
 * $\exists x \in D_1, y \in D_2: a = x \wedge y$

and
 * $\exists z \in D_1, t \in D_2: b = z \wedge t$

By definition of directed subset:
 * $\exists g \in D_1: x \preceq g \land z \preceq g$

and
 * $\exists h \in D_2: y \preceq h \land t \preceq h$

By Meet Semilattice is Ordered Structure:
 * $x \wedge y \preceq g \wedge h$ and $z \wedge t \preceq g \wedge h$
 * $g \wedge h \in \left\{ {x \wedge y: x \in D_1, y \in D_2}\right\}$

Thus
 * $\exists c \in \left\{ {x \wedge y: x \in D_1, y \in D_2}\right\}: a \preceq c \land b \preceq c$

Hence by definition
 * $\left\{ {x \wedge y: x \in D_1, y \in D_2}\right\}$ is directed.