rw [restrict_indep_iff, hJ.contract_indep_iff, hJ'.contract_indep_iff, restrict_indep_iff,
  union_subset_iff, and_iff_left (hJ.subset.trans subset_union_right), union_comm R C,
  ← diff_subset_iff, and_assoc, and_assoc, and_congr_right_iff, and_comm, and_congr_left_iff]
case intro
α : Type u_1
M✝ M' N : Matroid α
e f : α
I✝ J✝ R✝ B X Y Z K : Set α
M₀ M₁ M₂ M : Matroid α
C R : Set α
h : Disjoint C R
I : Set α
x✝ : I ⊆ (M ／ C ↾ R).E
J : Set α
hJ : (M ↾ (R ∪ C)).Basis' J C
hJ' : M.Basis' J C
⊢ (M ／ C ↾ R).Indep I ↔ ((M ↾ (R ∪ C)) ／ C).Indep I
case intro
α : Type u_1
M✝ M' N : Matroid α
e f : α
I✝ J✝ R✝ B X Y Z K : Set α
M₀ M₁ M₂ M : Matroid α
C R : Set α
h : Disjoint C R
I : Set α
x✝ : I ⊆ (M ／ C ↾ R).E
J : Set α
hJ : (M ↾ (R ∪ C)).Basis' J C
hJ' : M.Basis' J C
⊢ M.Indep (I ∪ J) → Disjoint C I → (I ⊆ R ↔ I \ C ⊆ R)