obtain rfl | hn1 := n.eq_zero_or_pos
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
⊢ False
case inl
A B : Set ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : 0 ≤ countelements A 0 + countelements B 0
h : 0 ∉ A + B
hnA : 0 ∉ A
hnB : 0 ∉ B
hca : countelements A (0 - 1) = countelements A 0
hcb : countelements B (0 - 1) = countelements B 0
⊢ False

case inr
A B : Set ℕ
n : ℕ
hA : 0 ∈ A
hB : 0 ∈ B
hc : n ≤ countelements A n + countelements B n
h : n ∉ A + B
hnA : n ∉ A
hnB : n ∉ B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
⊢ False