refine ⟨fun ⟨I,hI,h⟩ ↦ ⟨I,hI,?_⟩, fun ⟨I,hI,h⟩ ↦ ⟨I,hI,?_⟩⟩
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
⊢ (∃ I, M.Indep I ∧ M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y) ↔ ∃ I, M.Indep I ∧ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
case 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
x✝ : ∃ I, M.Indep I ∧ M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y
I : Set α
hI : M.Indep I
h : M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y
⊢ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)

case 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
x✝ : ∃ I, M.Indep I ∧ X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
I : Set α
hI : M.Indep I
h : X ⊆ M.cl (X ∩ I) ∧ Y ⊆ M.cl (Y ∩ I)
⊢ M.Basis (X ∩ I) X ∧ M.Basis (Y ∩ I) Y