Proof of Nash-Williams' Intersection Conjecture for countable matroids

We prove that if $M$ and $N$ are finitary matroids on a common countable edge set $E$ then they admit a common independent set $I$ such that there is a bipartition $E=E_M\cup E_N$ for which $I\cap E_M$ spans $E_M$ in $M$ and $I\cap E_N$ spans $E_N$ in $N$. It answers positively the Matroid Intersection Conjecture of Nash-Williams in the countable case.