rw [hM, delete_loop_iff, ← singleton_dep, hI.indep.contract_dep_iff, disjoint_singleton_left,
  singleton_union, hI.indep.insert_dep_iff, mem_diff, ← M.cl_inter_ground C, hI.cl_eq_cl,
  and_comm (a := e ∉ I), and_self_right, ← mem_diff, ← mem_diff, diff_diff]
case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : M ／ C = M ／ I ＼ D
⊢ (M ／ C).Loop e ↔ e ∈ M.cl C \ C
case intro.intro.intro.intro.intro
α : Type u_1
M M' N : Matroid α
e f : α
I✝ J R B X Y Z K C C₁ C₂ I D : Set α
hI : M.Basis I (C ∩ M.E)
hM : M ／ C = M ／ I ＼ D
⊢ e ∈ M.cl (C ∩ M.E) \ (I ∪ D) ↔ e ∈ M.cl (C ∩ M.E) \ C