Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence

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

Let $f: S \to T$ be an increasing mapping.

Let $\left({D, \precsim}\right)$ be a directed set.

Let $N: D \to S$ be a Moore-Smith sequence in $S$.

Let $j \in D$.

Then $f\left({\inf\left({N\left[{\precsim \left({j}\right)}\right]}\right)}\right) \preceq_2 \inf \left({\left({f \circ N}\right)\left[{\precsim \left({j}\right)}\right]}\right)$