theorem extracted_formal_statement_146.{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) :
  (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1)) - (OptProblem.A₁ E₂) admm_kkt.x₁ +
      ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - (OptProblem.A₂ E₁) admm_kkt.x₂) =
    (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1) - admm_kkt.x₁) +
      (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - admm_kkt.x₂) := 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) (ADMM.x₁ E₂ F (n + 1)) - (OptProblem.A₁ E₂) admm_kkt.x₁ +
      ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1)) - (OptProblem.A₂ E₁) admm_kkt.x₂) =
    (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (n + 1) - admm_kkt.x₁) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (n + 1) - admm_kkt.x₂)

Tactic: simp only [map_sub]

State after:

Relevant lemmas:
map_sub : ∀ {G : Type u_7} {H : Type u_8} {F : Type u_9} [inst : FunLike F G H] [inst_9 : AddGroup G]
  [inst_10 : SubtractionMonoid H] [inst_11 : AddMonoidHomClass F G H] (f : F) (a b : G), f (a - b) = f a - f b