have abTwo : AA.TwoOfThree (ab ≃ 0) (Positive ab) (Negative ab) :=
  match ‹AA.TwoOfThree (a < b) (a ≃ b) (a > b)› with
  | AA.TwoOfThree.oneAndTwo (_ : a < b) (_ : a ≃ b) =>
    have : Negative ab := lt_iff_neg_diff.mp ‹a < b›
    have : ab ≃ 0 := zero_diff_iff_eqv.mpr ‹a ≃ b›
    AA.TwoOfThree.oneAndThree ‹ab ≃ 0› ‹Negative ab›
  | AA.TwoOfThree.oneAndThree (_ : a < b) (_ : a > b) =>
    have : Negative ab := lt_iff_neg_diff.mp ‹a < b›
    have : Positive ab := gt_iff_pos_diff.mp ‹a > b›
    AA.TwoOfThree.twoAndThree ‹Positive ab› ‹Negative ab›
  | AA.TwoOfThree.twoAndThree (_ : a ≃ b) (_ : a > b) =>
    have : ab ≃ 0 := zero_diff_iff_eqv.mpr ‹a ≃ b›
    have : Positive ab := gt_iff_pos_diff.mp ‹a > b›
    AA.TwoOfThree.oneAndTwo ‹ab ≃ 0› ‹Positive ab›
ℕ : Type
inst✝⁷ : Natural ℕ
ℤ : Type
inst✝⁶ : Core ℤ
inst✝⁵ : Addition ℤ
inst✝⁴ : Multiplication ℤ
inst✝³ : Negation ℤ
inst✝² : Subtraction ℤ
inst✝¹ : Sign ℤ
inst✝ : Order ℤ
a b : ℤ
abTri : AA.ExactlyOneOfThree (a - b ≃ 0) (Positive (a - b)) (Negative (a - b))
x✝ : AA.TwoOfThree (a < b) (a ≃ b) (a > b)
ab : ℤ := a - b
⊢ False
ℕ : Type
inst✝⁷ : Natural ℕ
ℤ : Type
inst✝⁶ : Core ℤ
inst✝⁵ : Addition ℤ
inst✝⁴ : Multiplication ℤ
inst✝³ : Negation ℤ
inst✝² : Subtraction ℤ
inst✝¹ : Sign ℤ
inst✝ : Order ℤ
a b : ℤ
abTri : AA.ExactlyOneOfThree (a - b ≃ 0) (Positive (a - b)) (Negative (a - b))
x✝ : AA.TwoOfThree (a < b) (a ≃ b) (a > b)
ab : ℤ := a - b
abTwo : AA.TwoOfThree (ab ≃ 0) (Positive ab) (Negative ab)
⊢ False