Towards a splitter theorem for internally 4-connected binary matroids II

Let MM and NN be internally 4-connected binary matroids such that MM has a proper NN-minor, and |E(N)|≥7|E(N)|≥7. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M∖eM∖e has no NN-minor whenever ee is in a triangle of MM, and M/eM/e has no NN-minor whenever ee is in a triad of MM, then MM has a minor, M′M′, such that M′M′ is internally 4-connected with an NN- minor, and 1≤|E(M)|−|E(M′)|≤21≤|E(M)|−|E(M′)|≤2.