Choice functions in the intersection of matroids

We prove a common generalization of two results, one on rainbow fractional matchings and one on rainbow sets in the intersection of two matroids: Given $d = r \lceil k \rceil - r + 1$functionsofsize(=sumofvalues)$k$ that are all independent in each of $r$ given matroids, there exists a rainbow set of $supp(f_i)$, $i\le d$, supporting a function with the same properties.