Directed Suprema Preserving Mapping at Element is Supremum

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

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

Let $x \in S$.

Then $f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$

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

and
 * $\left({S, \vee, \wedge, \preceq}\right)$ is up-complete

and
 * $\left({S, \vee, \wedge, \preceq}\right)$ satisfies 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