theorem extracted_formal_statement_33.{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 :
    -ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
      ADMM.ρ E₁ E₂ F *
        ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ) :
  ADMM.ρ E₁ E₂ F * ⟪-(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F *
      ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ := 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
⊢ -ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
    ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ

Tactic: rw [neg_mul (ρ) (inner (A₂ (x₂ (n + 1) - x₂ n)) (A₂ (e₂ (n + 1))))]
    rw [← mul_neg]
    rw [← inner_neg_left (A₂ (x₂ (n + 1) - x₂ n)) (A₂ (e₂ (n + 1)))]

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
 ⊢ ADMM.ρ E₁ E₂ F * ⟪-(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - ADMM.x₂ E₁ F n), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ =
     ADMM.ρ E₁ E₂ F * ⟪(OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n - ADMM.x₂ E₁ F (n + 1)), (OptProblem.A₂ E₁) (e₂ (n + 1))⟫_ℝ]

Relevant lemmas:
mul_neg : ∀ {α : Type u} [inst : Mul α] [inst_10 : HasDistribNeg α] (a b : α), a * -b = -(a * b)
neg_mul : ∀ {α : Type u} [inst : Mul α] [inst_10 : HasDistribNeg α] (a b : α), -a * b = -(a * b)
inner_neg_left : ∀ {𝕜 : Type u_1} {E : Type u_2} [inst : RCLike 𝕜] [inst_10 : SeminormedAddCommGroup E]
  [inst_11 : InnerProductSpace 𝕜 E] (x y : E), ⟪-x, y⟫_𝕜 = -⟪x, y⟫_𝕜