Finite Infima Set of Coarser Subset is Coarser than Finite Infima Set

Theorem
Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice.

Let $A, B$ be subsets of $S$ such that
 * $A$ is coarser than $B$.

Then $\operatorname{fininfs} \left({A}\right)$ is coarser than $\operatorname{fininfs} \left({B}\right)$

Proof
Let $x \in \operatorname{fininfs} \left({A}\right)$

By definition of finite infima set:
 * $\exists Y \in \mathit{Fin}\left({A}\right): z = \inf Y$ and $Y$ admits an infimum,

where $\mathit{Fin}\left({A}\right)$ denotes the set of all finite subsets of $A$.

By definition of coarset subset:
 * $\forall y \in Y: \exists z \in B: z \preceq y$

By Axiom of Choice:
 * $\exists f:Y \to B: \forall y \in Y: f\left({y}\right) \preceq y$