rcases exists_smaller_set 𝒜 𝒞.card h₂ with ⟨𝒜', prop, size⟩
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
⊢ (∂ 𝒞).card ≤ (∂ 𝒜).card
case intro.intro
α : Type u_1
inst✝ : LinearOrder α
s U V : Finset α
n r k i : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h₁ : Set.Sized r ↑𝒜
h₂ : 𝒞.card ≤ 𝒜.card
h₃ : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
prop : 𝒜' ⊆ 𝒜
size : 𝒜'.card = 𝒞.card
⊢ (∂ 𝒞).card ≤ (∂ 𝒜).card