theorem extracted_formal_statement_41.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} (n : ℕ+) (x : E₁) :
  ⟪ADMM.y E₁ E₂ n.natPred,
      (OptProblem.A₁ E₂) x + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ := sorry

State before:
case h
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ+
x✝ : E₁
c : F := ADMM.y E₁ E₂ n.natPred
c₁ : ℝ := ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
x : E₁
⊢ ⟪ADMM.y E₁ E₂ n.natPred,
      (OptProblem.A₁ E₂) x + ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)⟫_ℝ =
    ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ +
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ

Tactic: exact
  inner_add_right (ADMM.y E₁ E₂ n.natPred) ((OptProblem.A₁ E₂) x)
    ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)

State after:

Relevant lemmas:
inner_add_right : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_9 : SeminormedAddCommGroup E]
  [inst_10 : InnerProductSpace 𝕜 E] (x y z : E), ⟪x, y + z⟫_𝕜 = ⟪x, y⟫_𝕜 + ⟪x, z⟫_𝕜