Contracting an element from a cocircuit

We consider the situation that M and N are 3-connected matroids such that |E(N)|⩾4 and C∗ is a cocircuit of M with the property that M/x0 has an N-minor for some x0∈C∗. We show that either there is an element x∈C∗ such that si(M/x) or co(si(M/x)) is 3-connected with an N-minor, or there is a four-element fan of M that contains two elements of C∗ and an element x such that si(M/x) is 3-connected with an N-minor.