Exposing 3-separations in 3-connected matroids

Let M be a 3-connected matroid other than a wheel or a whirl. In the next paper in this series, we prove that there is an element whose deletion from M or M⁎ is 3-connected and whose only 3-separations are equivalent to those induced by M. The strategy used to prove this theorem involves showing that we can remove some element from a leaf of the tree of 3-separations of M. The main result of this paper is designed to allow us to do this.