∀ {C : Type u_2} [inst : CategoryTheory.Category.{u_1, u_2} C] [inst_1 : CategoryTheory.Preadditive C]
  (P : CategoryTheory.Idempotents.Karoubi C),
  CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.biprod.desc P.decompId_i P.complement.decompId_i)
      (CategoryTheory.Limits.biprod.lift P.decompId_p P.complement.decompId_p) =
    CategoryTheory.CategoryStruct.id (P ⊞ P.complement)