refine' eq_of_spanning_iff_spanning_forall (by simp) (fun S hS ↦ _)
case intro
α✝ : Type u_1
M : Matroid α✝
E✝ I B X C✝ : Set α✝
k✝ : ℕ∞
α : Type u_2
E C : Set α
k : ℕ
⊢ unifOn E ↑k ／ C = unifOn (E \ C) (↑k - (E ∩ C).encard)
case intro
α✝ : Type u_1
M : Matroid α✝
E✝ I B X C✝ : Set α✝
k✝ : ℕ∞
α : Type u_2
E C : Set α
k : ℕ
S : Set α
hS : S ⊆ (unifOn E ↑k ／ C).E
⊢ (unifOn E ↑k ／ C).Spanning S ↔ (unifOn (E \ C) (↑k - (E ∩ C).encard)).Spanning S