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

Let $\mathcal{M}$ and $\mathcal{N}$ be two matroids on the same ground set $V$. Let $A_1,\dots ,A_{2n-1}$ be sets which are independent in both $\mathcal{M}$ and $\mathcal{N}$, satisfying $\mathrm{\mid }{A}_i\mathrm{\mid }\ge \text{min}(i,n)$ for all $i$. We show that there exists a partial rainbow set of size $n$, which is independent in both $\mathcal{M}$ and $\mathcal{N}$. 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.