exact ⟨B, hB.indep, hB.basis_biInter hA, hB.basis_biInter (by rwa [nonempty_compl])⟩
case intro
ι : Type u_1
α : Type u_2
M : Matroid α
B✝ I J X X' Y Y' F : Set α
Xs Ys Js Is : ι → Set α
i j : ι
e f : α
A : Set ι
hA : A.Nonempty
hA' : A ≠ univ
B : Set α
hB : M.ModularBase B Xs
⊢ ∃ I, M.Indep I ∧ M.Basis ((⋂ i ∈ A, Xs i) ∩ I) (⋂ i ∈ A, Xs i) ∧ M.Basis ((⋂ i ∈ Aᶜ, Xs i) ∩ I) (⋂ i ∈ Aᶜ, Xs i)
no goals