rw [ ← inter_eq_self_of_subset_left Basis.2, inter_right_comm _ R, inter_assoc, R_B'_inter_eq,
  inter_assoc, inter_eq_self_of_subset_right (hB'.2.trans hB'.1.subset_ground),
  inter_right_comm, inter_eq_self_of_subset_left Basis.2]
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.Basis (Xs i ∩ B) (Xs i ∩ M.E) ∧ Xs i ⊆ R
R_B'_inter_eq : R ∩ B' = B
⊢ M.Basis (Xs i ∩ M.E ∩ B') (Xs i ∩ M.E)
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.Basis (Xs i ∩ B) (Xs i ∩ M.E) ∧ Xs i ⊆ R
R_B'_inter_eq : R ∩ B' = B
⊢ M.Basis (Xs i ∩ B) (Xs i ∩ M.E)