theorem extracted_formal_statement_31.{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) :
  (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 := 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]
    simp
    calc
  _ = A₁ (x₁ n) + A₂ (x₂ n) - ((A₁ x₁') + (A₂ x₂')) := by rw [sub_add_sub_comm]
  _ = A₁ (x₁ n) + A₂ (x₂ n) - b + b - ((A₁ x₁') + (A₂ x₂')) := by rw [sub_add_cancel]
  _ = A₁ (x₁ n) + A₂ (x₂ n) - b := by rw [Satisfaction_ofthekkt.eq]; simp

State after:

Relevant lemmas:
sub_add_cancel : ∀ {G : Type u_1} [inst : AddGroup G] (a b : G), a - b + b = a
sub_add_sub_comm : ∀ {α : Type u_1} [inst : SubtractionCommMonoid α] (a b c d : α), a - b + (c - d) = a + c - (b + d)