have hrw : (U.delete {e}).ExtIndep e = U.ExtIndep e := by
  ext; simp [ExtIndep, he.mem_cl_iff_mem, ModularCut.mem_delete_elem_iff]
case inr
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
he : M.Coloop e
⊢ ∃ x ∈ B \ I, U.ExtIndep e (insert x I)
case inr
α : Type u_1
M : Matroid α
I J B F₀ F F' X Y : Set α
e f : α
U✝ U : M.ModularCut
hI : U.ExtIndep e I
hInmax : I ∉ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
hBmax : B ∈ maximals (fun x x_1 => x ⊆ x_1) {I | U.ExtIndep e I}
he : M.Coloop e
hrw : (U.delete {e}).ExtIndep e = U.ExtIndep e
⊢ ∃ x ∈ B \ I, U.ExtIndep e (insert x I)