theorem extracted_formal_statement_241.{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} [inst_9 : Setting E₁ E₂ F admm admm_kkt] (n : ℕ)
  (h_original_goal :
    (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n) =
      (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - OptProblem.b E₁ E₂) :
  (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n - admm_kkt.x₁) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - admm_kkt.x₂) =
    (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - OptProblem.b E₁ E₂ := sorry

State before:
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
⊢ (OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n) =
    (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - OptProblem.b E₁ E₂

Tactic: rw [e₁, e₂]

State after:
[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
 ⊢ (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n - admm_kkt.x₁) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - admm_kkt.x₂) =
     (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n) - OptProblem.b E₁ E₂]

Relevant lemmas: