Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset

Theorem
Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be up-complete lattices.

Let $f: S \to T$ be a mapping such that
 * for all directed set $\left({D, \precsim}\right)$ and Moore-Smith sequence $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$

Let $D$ be a directed subset of $S$.

Then $\sup \left({f\left[{D}\right]}\right) = f\left({\sup D}\right)$

where $f\left[{D}\right]$ denotes the image of $D$ under $f$.