have : sgn (sgn q - sgn p) * sgn p ≃ 1 :=
  Rel.trans this ‹sgn (p * sgn q - p * sgn p) ≃ 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 * from_integer (sgn q) - p * from_integer (sgn p)) ≃ 1
this : sgn (sgn q - sgn p) * sgn p ≃ sgn (p * from_integer (sgn q) - p * from_integer (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✝ : sgn (sgn q - sgn p) * sgn p ≃ sgn (p * from_integer (sgn q) - p * from_integer (sgn p))
this : sgn (sgn q - sgn p) * sgn p ≃ 1
⊢ False