∀ {R : Type u_3} {L : Type u_1} {M : Type u_2} [inst : CommRing R] [inst_1 : LieRing L] [inst_2 : LieAlgebra R L]
  [inst_3 : AddCommGroup M] [inst_4 : Module R M] [inst_5 : LieRingModule L M] [inst_6 : LieModule R L M]
  (c c' : ↥(LieModule.Cohomology.twoCocycle R L M)) (x : LieModule.Cohomology.oneCochain R L M),
  ↑c' = ↑c + (LieModule.Cohomology.d₁₂ R L M) x →
    ∀ {x_1 y : LieAlgebra.ofTwoCocycle c},
      (((LieAlgebra.ofProd c).symm ⁅x_1, y⁆).1,
          ((LieAlgebra.ofProd c).symm ⁅x_1, y⁆).2 - x ((LieAlgebra.ofProd c).symm ⁅x_1, y⁆).1) =
        (LieAlgebra.ofProd c').symm
          ⁅(LieAlgebra.ofProd c')
              (((LieAlgebra.ofProd c).symm x_1).1,
                ((LieAlgebra.ofProd c).symm x_1).2 - x ((LieAlgebra.ofProd c).symm x_1).1),
            (LieAlgebra.ofProd c')
              (((LieAlgebra.ofProd c).symm y).1, ((LieAlgebra.ofProd c).symm y).2 - x ((LieAlgebra.ofProd c).symm y).1)⁆