∀ (a : ℤ) (a_1 : ℕ) (a_2 : a_1 ≠ 0) (a_3 : a.natAbs.Coprime a_1) (b : ℤ) (b_1 : ℕ) (b_2 : b_1 ≠ 0)
  (b_3 : b.natAbs.Coprime b_1),
  ¬a = b →
    ¬{ num := a, den := a_1, den_nz := a_2, reduced := a_3 } = { num := b, den := b_1, den_nz := b_2, reduced := b_3 }