Non-Empty Way Below Closure is Directed in Join Semilattice

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

Let $x \in S$ such that:
 * $x^\ll \ne \O$

where $x^\ll$ denotes the way below closure of $x$.

Then $x^\ll$ is directed.

Proof
By assumption:
 * $x^\ll$ is a non-empty set.

Let $y, z \in x^\ll$.

By definition of way below closure:
 * $y \ll x$ and $z \ll x$

By Join is Way Below if Operands are Way Below:
 * $y \vee z \ll x$

By definition of way below closure:
 * $y \vee z \in x^\ll$

By Join Succeeds Operands:
 * $y \preceq y \vee z$ and $z \preceq y \vee z$

Thus by definition:
 * $x^\ll$ is directed.