have R_B'_inter_eq : R ∩ B' = B := by
  rw [ext_iff]
  refine' fun x ↦ ⟨fun x_mem ↦ _, fun x_mem ↦ ⟨hBb.subset_ground x_mem, hB'.2 x_mem⟩⟩
  by_contra x_nB
  apply (hB'.1.indep.subset (insert_subset x_mem.2 hB'.2)).not_dep
  rw [Dep, and_iff_left ((insert_subset x_mem.2 hB'.2).trans hB'.1.subset_ground)]
  exact (restrict_dep_iff.1 (hBb.insert_dep ⟨x_mem.1, x_nB⟩)).1
case intro.intro.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
R B : Set α
hBb : (M ↾ R).Base B
hB : ∀ (i : ι), (M ↾ R).Basis (Xs i ∩ B) (Xs i)
B' : Set α
hB' : M.Base B' ∧ B ⊆ B'
i : ι
Basis : (M ↾ R).Basis (Xs i ∩ B) (Xs i)
⊢ M.Basis ((fun i => Xs i ∩ M.E) i ∩ B') ((fun i => Xs i ∩ M.E) i)
case intro.intro.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
R B : Set α
hBb : (M ↾ R).Base B
hB : ∀ (i : ι), (M ↾ R).Basis (Xs i ∩ B) (Xs i)
B' : Set α
hB' : M.Base B' ∧ B ⊆ B'
i : ι
Basis : (M ↾ R).Basis (Xs i ∩ B) (Xs i)
R_B'_inter_eq : R ∩ B' = B
⊢ M.Basis ((fun i => Xs i ∩ M.E) i ∩ B') ((fun i => Xs i ∩ M.E) i)