Limit Inferior of Inclusion Net is Supremum of Directed Subset

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice.

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

Let $\left({D, \preceq'}\right)$ be a directed ordered subset of $L$.

Let $i_D: D \to S$, the inclusion mapping, be a Moore-Smith sequence in $S$.

Then $\liminf i_D = \sup D$

Proof
Thus