refine ⟨?_, by simpa using hB.basis_inter true, by simpa using hB.basis_inter false, ?_⟩
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
B : Set α
hB : M.ModularBase B fun i => bif i then X else Y
⊢ M.Basis ((X ∪ Y) ∩ B) (X ∪ Y) ∧ M.Basis (X ∩ B) X ∧ M.Basis (Y ∩ B) Y ∧ M.Basis (X ∩ Y ∩ B) (X ∩ Y)
case intro.refine_1
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
B : Set α
hB : M.ModularBase B fun i => bif i then X else Y
⊢ M.Basis ((X ∪ Y) ∩ B) (X ∪ Y)

case intro.refine_2
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
B : Set α
hB : M.ModularBase B fun i => bif i then X else Y
⊢ M.Basis (X ∩ Y ∩ B) (X ∩ Y)