rw [indep_iff_forall_subset_not_circuit hIe]
case refine_2.refine_1
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
hXs : ∀ (i : ι), Xs i ⊆ M.E
hdj : Pairwise (Disjoint on Xs)
h : ∀ (C : Set α), M.Circuit C → C ⊆ ⋃ i, Xs i → ∃ i, C ⊆ Xs i
Is : ι → Set α
hIs : ∀ (i : ι), M.Basis (Is i) (Xs i)
hss : ⋃ i, Is i ⊆ ⋃ i, Xs i
hIe : ⋃ i, Is i ⊆ M.E
h_inter : ∀ (i : ι), Xs i ∩ ⋃ j, Is j = Is i
⊢ M.Indep (⋃ i, Is i)
case refine_2.refine_1
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
hXs : ∀ (i : ι), Xs i ⊆ M.E
hdj : Pairwise (Disjoint on Xs)
h : ∀ (C : Set α), M.Circuit C → C ⊆ ⋃ i, Xs i → ∃ i, C ⊆ Xs i
Is : ι → Set α
hIs : ∀ (i : ι), M.Basis (Is i) (Xs i)
hss : ⋃ i, Is i ⊆ ⋃ i, Xs i
hIe : ⋃ i, Is i ⊆ M.E
h_inter : ∀ (i : ι), Xs i ∩ ⋃ j, Is j = Is i
⊢ ∀ C ⊆ ⋃ i, Is i, ¬M.Circuit C