have : sgn (q - p) ≄ 1 := by
  intro (_ : sgn (q - p) ≃ 1)
  show False
  have : sgn (p - q) ≃ -1 := calc
    sgn (p - q)    ≃ _ := sgn_subst (eqv_symm neg_sub)
    sgn (-(q - p)) ≃ _ := sgn_compat_neg
    (-sgn (q - p)) ≃ _ := AA.subst₁ ‹sgn (q - p) ≃ 1›
    (-1)           ≃ _ := Rel.refl
  exact absurd ‹sgn (p - q) ≃ -1› ‹sgn (p - q) ≄ -1›
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : sgn (p - q) ≄ -1
⊢ p ≥ q
ℕ ℤ ℚ : Type
inst✝¹⁰ : Natural ℕ
inst✝⁹ : Integer ℤ
inst✝⁸ : Core ℚ
inst✝⁷ : Addition ℚ
inst✝⁶ : Multiplication ℚ
inst✝⁵ : Negation ℚ
inst✝⁴ : Sign ℚ
inst✝³ : Subtraction ℚ
inst✝² : Reciprocation ℚ
inst✝¹ : Division ℚ
inst✝ : Order ℚ
p q : ℚ
x✝ : sgn (p - q) ≄ -1
this : sgn (q - p) ≄ 1
⊢ p ≥ q