The matroid intersection conjecture

We prove the matroid intersection conjecture, due to Nash-Williams \cite{ahaziv}, for infinite matroids: If $M_k=(E,{\mathcal{I}}_k)$ are matroids for $k\in \{0,1\}$, both having the same base set $E$, then there is $J^{\ast }\in {\mathcal{I}}_0\cap {\mathcal{I}}_1$ and disjoint subsets $J_k\subseteq J^{\ast }$ for $k\in \{0,1\}$ such that ${\text{cl}}_{M_0}(J_0)\cup {\text{cl}}_{M_1}(J_1)=E$.