Abstract
We consider the situation that M and N are 3-connected matroids such that | E ( N ) | ⩾ 4 and C ∗ is a cocircuit of M with the property that M / x 0 has an N -minor for some x 0 ∈ C ∗ . We show that either there is an element x ∈ 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.
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