A theorem of the transversal theory for matroids of finite character

Let M = (S, I) be a matroid of finite character on the infinite set S. Let $\text{A}\text{= 〈A}{}_1\text{:i ∈ I〉}$ be any system of subsets of S each having finite rank and let $\text{B}\text{= 〈B}{}_1\text{: j ∈ J〉}$ be a finite system of sets of arbitrary rank. Necessary and sufficient conditions are given for the system $\text{A}$ ? $\text{B}$ to have an independent system of distinct representatives.