have : sgn (sgn q - (-1)) ≃ -1 := calc
  sgn (sgn q - (-1))  ≃ _ := Integer.sgn_subst (Integer.sub_substR this)
  sgn (sgn q - sgn p) ≃ _ := ‹sgn (sgn q - sgn p) ≃ sgn p›
  sgn p               ≃ _ := ‹sgn p ≃ -1›
  (-1)                ≃ _ := Rel.refl
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 * from_integer (sgn q) - p * from_integer (sgn p)) ≃ 1
this✝⁴ : sgn (sgn q - sgn p) * sgn p ≃ sgn (p * from_integer (sgn q) - p * from_integer (sgn p))
this✝³ : sgn (sgn q - sgn p) * sgn p ≃ 1
sqrt1_and_eqv : Integer.Sqrt1 (sgn p) ∧ sgn (sgn q - sgn p) ≃ sgn p
this✝² : Integer.Sqrt1 (sgn p)
this✝¹ : sgn (sgn q - sgn p) ≃ sgn p
this✝ : sgn p ≃ 1 ∨ sgn p ≃ -1
h✝ : sgn p ≃ -1
this : -1 ≃ sgn p
⊢ False
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 * from_integer (sgn q) - p * from_integer (sgn p)) ≃ 1
this✝⁵ : sgn (sgn q - sgn p) * sgn p ≃ sgn (p * from_integer (sgn q) - p * from_integer (sgn p))
this✝⁴ : sgn (sgn q - sgn p) * sgn p ≃ 1
sqrt1_and_eqv : Integer.Sqrt1 (sgn p) ∧ sgn (sgn q - sgn p) ≃ sgn p
this✝³ : Integer.Sqrt1 (sgn p)
this✝² : sgn (sgn q - sgn p) ≃ sgn p
this✝¹ : sgn p ≃ 1 ∨ sgn p ≃ -1
h✝ : sgn p ≃ -1
this✝ : -1 ≃ sgn p
this : sgn (sgn q - -1) ≃ -1
⊢ False