theorem extracted_formal_statement_99.{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+) :
  let S := SubderivAt (OptProblem.f₂ E₁ F) (ADMM.x₂ E₁ F ↑n);
  let p := (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) (ADMM.y E₁ E₂ (↑n - 1));
  let q :=
    ADMM.ρ E₁ E₂ F •
      (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
        ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂);
  0 ∈ S + {p} + {q} → -p - q ∈ S := 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+
S : Set E₂ := SubderivAt (OptProblem.f₂ E₁ F) (ADMM.x₂ E₁ F ↑n)
p : E₂ := (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) (ADMM.y E₁ E₂ (↑n - 1))
q : E₂ :=
  ADMM.ρ E₁ E₂ F •
    (ContinuousLinearMap.adjoint (OptProblem.A₂ E₁))
      ((OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑n) - OptProblem.b E₁ E₂)
this : 0 ∈ S + {p} + {q}
⊢ -p - q ∈ S

Tactic: apply change_item this

State after:

Relevant lemmas: