One of the earliest results in enumerative combinatorial geometry is the following theorem of de Brujin and Erdős: Every set of points E in a projective plane determines at least |E| lines, unless all the points are contained in a line. Motzkin and others extended the result to higher dimensions, who showed that every set of points E in a projective space determines at least |E| hyperplanes, unless all the points are contained in a hyperplane. Let E be a spanning subset of a d-dimensional vector space. We show that, in the poset of subspaces spanned by subsets of E, there are at least as many (d-k)-dimensional subspaces as there are k-dimensional subspaces, for every k at most d/2. This confirms the “top-heavy” conjecture of Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for l-adic intersection complexes. Joint work with Botong Wang.