What do you mean by "the finite lattice", exactly? Is it just a finite subset of the unit lattice in the plane? Anyway, construct a bipartite graph $G$ as follows: its vertices are the rows and columns of points, and its bipartition is $R,C$, where $R$ represents the set of rows, and $C$ the set of columns. Two vertices $r\in R,c\in C$ are connected iff there is a point of $S$ at the intersection between $r$ and $c$. Notice that $A$ is a maximal matching of $G$, i.e. the maximal cardinality of a set of edges such that no two share a vertex, while $B$ is a minimal vertex cover of $G$: a set of vertices of $G$ with minimal cardinality containing at least one endpoint from every edge. The inequality (in fact, it's an equality, because the reversed inequality is easy to prove) is precisely Konig's Theorem: in a bipartite graph, the cardinality of a maximal matching equals the cardinality of a minimal vertex cover. P.S. This was much better suited for the "Combinatorics" section, I think .

A little problem,as an element in B can mean two vertexs,the problem is still a inequality.