have : Positive a ∨ Negative a := match this with
| Or.inl (_ : sgn a ≃ 1) =>
  have : Positive a := sgn_positive.mpr ‹sgn a ≃ 1›
  Or.inl this
| Or.inr (_ : sgn a ≃ -1) =>
  have : Negative a := sgn_negative.mpr ‹sgn a ≃ -1›
  Or.inr this
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
x✝ : Sqrt1 (sgn a)
this : sgn a ≃ 1 ∨ sgn a ≃ -1
⊢ Nonzero 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
x✝ : Sqrt1 (sgn a)
this✝ : sgn a ≃ 1 ∨ sgn a ≃ -1
this : Positive a ∨ Negative a
⊢ Nonzero a