Way Below iff Second Operand Preceding Supremum of Directed Set There Exists Element of Directed Set First Operand Way Below Element

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below continuous lattice.

Let $x, y$ be elements of $S$.

Then
 * $x \ll y$


 * for every directed subset $D$ of $S$ such that $y \preceq \sup D$
 * there exists an element $d$ of $D$: $x \ll d$

Sufficient Condition
Let $x \ll y$.

Let $D$ be a directed subset of $S$ such that
 * $y \preceq \sup D$

By Way Below has Interpolation Property:
 * $\exists x' \in S: x \ll x' \land x' \ll y$

By definition of way below relation:
 * $\exists d \in D: x' \preceq d$

Thus by Preceding and Way Below implies Way Below and definition of reflexivity:
 * $x \ll d$

Necessary Condition
Assume that
 * for every directed subset $D$ of $S$ such that $y \preceq \sup D$
 * there exists an element $d$ of $D$: $x \ll d$

By Way Below implies Preceding:
 * for every directed subset $D$ of $S$ such that $y \preceq \sup D$
 * there exists an element $d$ of $D$: $x \preceq d$

Thus by definition of way below relation: $x \ll y$