Directed Suprema Preserving Mapping at Element is Supremum

Theorem
Let $\struct {S, \vee, \wedge, \preceq}$ and $\struct {T, \vee_2, \wedge_2, \precsim}$ be bounded below continuous lattices.

Let $f: S \to T$ be a mapping such that
 * $f$ preserves directed suprema.

Let $x \in S$.

Then:
 * $\map f x = \sup \set {\map f w: w \in S \land w \ll x}$

Proof
By definition of continuous:
 * $x^\ll$ is directed

and
 * $\struct {S, \vee, \wedge, \preceq}$ is up-complete

and
 * $\struct {S, \vee, \wedge, \preceq}$ satisfies the axiom of approximation.

By definition of mapping preserves directed suprema:
 * $f$ preserves the supremum of $x^\ll$.

By definition of up-complete:
 * $x^\ll$ admits a supremum.

Thus