∀ {𝕜 : Type u} {A : Type v} [inst : NontriviallyNormedField 𝕜] [inst_1 : NonUnitalNormedRing A]
  [inst_2 : NormedSpace 𝕜 A] [inst_3 : SMulCommClass 𝕜 A A] [inst_4 : IsScalarTower 𝕜 A A]
  {a b : DoubleCentralizer 𝕜 A}, a = b ↔ a.toProd = b.toProd