∀ {I₁ : Type u_1} {I₂ : Type u_2} {I₁₂ : Type u_3} (c₁ : ComplexShape I₁) (c₂ : ComplexShape I₂)
  (c₁₂ : ComplexShape I₁₂) [inst : TotalComplexShape c₁ c₂ c₁₂] [inst_1 : TotalComplexShape c₂ c₁ c₁₂]
  [inst_2 : TotalComplexShapeSymmetry c₁ c₂ c₁₂] (j : I₁₂),
  Function.LeftInverse
    (fun x =>
      match x with
      | ⟨(i₁, i₂), h⟩ => ⟨(i₂, i₁), ⋯⟩)
    fun x =>
    match x with
    | ⟨(i₂, i₁), h⟩ => ⟨(i₁, i₂), ⋯⟩