have : AA.OneOfThree (a ≃ 0) (Positive a) (Negative a) :=
  (sign_trichotomy a).atLeastOne
ℕ : Type
inst✝⁵ : Natural ℕ
ℤ : Type
inst✝⁴ : Core ℤ
inst✝³ : Addition ℤ
inst✝² : Multiplication ℤ
inst✝¹ : Negation ℤ
inst✝ : Sign ℤ
a : ℤ
⊢ sgn (sgn a) ≃ sgn a
ℕ : Type
inst✝⁵ : Natural ℕ
ℤ : Type
inst✝⁴ : Core ℤ
inst✝³ : Addition ℤ
inst✝² : Multiplication ℤ
inst✝¹ : Negation ℤ
inst✝ : Sign ℤ
a : ℤ
this : AA.OneOfThree (a ≃ 0) (Positive a) (Negative a)
⊢ sgn (sgn a) ≃ sgn a