Meet of Directed Subsets is Directed

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

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

Then:
 * $\set {x \wedge y: x \in D_1, y \in D_2}$ is a directed subset of $S$.

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

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 $z \wedge t \preceq g \wedge h$
 * $g \wedge h \in \set {x \wedge y: x \in D_1, y \in D_2}$

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

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