  We consider the situation that M and N are 3-connected matroids such that
|E(N)| > 3 and C* is a cocircuit of M with the property that M/y has an N-minor
for some y in C*. We show that either there is an element x in 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.
