A common generalization to strengthenings of Drisko's Theorem for intersections of two matroids

Let M and N be two matroids on the same ground set V. Let A₁,,A₂ₙ₋₁ be sets which are independent in both M and N, satisfying |Aᵢ|≥ min(i,n) for all i. We show that there exists a partial rainbow set of size n, which is independent in both M and N. This is a common generalization of rainbow matching results for bipartite graphs by Aharoni, Berger, Kotlar, and Ziv, and for the intersection of two matroid by Kotlar and Ziv.