Constancy results for special families of projections

Let {$\mathbb{V}$ = V x R^l : V \in G(n-l,m-l)} be the family of m-dimensional subspaces of R^n containing {0} x R^l, and let \pi_{$\mathbb{V}$} : R^n --> $\mathbb{V}$ be the orthogonal projection onto $\mathbb{V}$. We prove that the mapping V \mapsto Dim \pi_{$\mathbb{V}$}(B) is almost surely constant for any analytic set B \subset R^n, where Dim denotes either Hausdorff or packing dimension.