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 .