Coarser Between Generator Set and Filter is Generator Set of Filter

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

Let $F$ be a filter on $L$.

Let $G$ be a generator set of $F$.

Let $A$ be a subset of $S$ such that
 * $G$ is coarser than $A$ and $A$ is coarser than $F$.

Then $A$ is generator set of $F$.