obtain ⟨_hx, hx'⟩ := hx
case a
m : ℕ
f : ℕ → ℕ
hf : ∀ (n : ℕ), f (f n) = n + (2 * m + 1)
f_injective : Function.Injective f
NN : Set ℕ := Set.univ
A : Set ℕ := NN \ f '' NN
B : Set ℕ := f '' A
hid : B = f '' NN \ f '' (f '' NN)
ab_disjoint : Disjoint A B
x : ℕ
hx : x ∈ NN \ f '' (f '' NN)
⊢ x ∈ A ∪ f '' NN \ f '' (f '' NN)
case a.intro
m : ℕ
f : ℕ → ℕ
hf : ∀ (n : ℕ), f (f n) = n + (2 * m + 1)
f_injective : Function.Injective f
NN : Set ℕ := Set.univ
A : Set ℕ := NN \ f '' NN
B : Set ℕ := f '' A
hid : B = f '' NN \ f '' (f '' NN)
ab_disjoint : Disjoint A B
x : ℕ
_hx : x ∈ NN
hx' : x ∉ f '' (f '' NN)
⊢ x ∈ A ∪ f '' NN \ f '' (f '' NN)