A chain theorem for 4-connected matroids

A matroid M is said to be k–connected up to separators of size l if whenever A is (k − 1)–separating in M , then either |A| ≤ l or |E(M)− A| ≤ l. We use si(M) and co(M) to denote the simplification and cosimplification of the matroid M . We prove that if a 3–connected matroid M is 4–connected up to separators of size 5, then there is an element x of M such that either co(M\x) or si(M/x) is 3–connected and 4–connected up to separators of size 5, and has a cardinality of |E(M)| − 1 or |E(M)| − 2.