rw [union_diff_distrib, diff_eq_empty.2 (diff_subset_diff_left hX), empty_union, diff_diff,
  diff_diff, ← union_assoc, union_comm, ← diff_diff, diff_diff_cancel_left hBYb.subset_ground,
  ← diff_diff, inter_diff_distrib_left, inter_eq_self_of_subset_left hBYb.subset_ground,
  diff_self_inter]
case intro.intro.intro.intro.intro.intro
α : Type u_1
M✝ : Matroid α
B B' I I' J K X✝¹ Y : Set α
M : Matroid α
X✝ X : Set α
hX : X ⊆ M.E
BX : Set α
hBXb : M.Base BX
hI : M.Basis (BX ∩ X) X
hIBX : BX ∩ X ⊆ BX
hBXdual : M✶.Basis ((M.E \ X) \ BX) (M.E \ X)
BY : Set α
hBYb : M.Base BY
hJ : M.Basis (BY ∩ (M.E \ X)) (M.E \ X)
hBX : M.Basis BX (BX ∩ X ∪ BY ∩ (M.E \ X))
hIJE : M.Spanning (BX ∩ X ∪ BY ∩ (M.E \ X))
hBY : M.Basis BY (BX ∩ X ∪ BY ∩ (M.E \ X))
hJBY : BY ∩ (M.E \ X) ⊆ BY
hBYdual : M✶.Basis (X \ BY) X
⊢ ((BY ∩ (M.E \ X)) \ BX).encard = ((X \ BY ∪ (M.E \ X) \ BX) \ (M.E \ BY)).encard
no goals