have hKE : K ∩ M.E = K := by
  rw [inter_eq_left, ← dual_ground]; exact hK.trans (cl_subset_ground _ _)
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X✝ Y Z F C K X : Set α
hK : K ⊆ M✶.cl ∅
⊢ M.cl X ∩ K = X ∩ K
α : Type u_1
β : Type u_2
M N : Matroid α
e f : α
B L L' I X✝ Y Z F C K X : Set α
hK : K ⊆ M✶.cl ∅
hKE : K ∩ M.E = K
⊢ M.cl X ∩ K = X ∩ K