theorem extracted_formal_statement_45.{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+)
  (h_original_goal :
    SubderivAt
        (fun x =>
          ⟪ADMM.y E₁ E₂ n.natPred,
            (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
        (ADMM.x₁ E₂ F ↑n) =
      {(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)}) :
  HasGradientAt
    (fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred,
        (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
    ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)) (ADMM.x₁ E₂ F ↑n) := 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+
⊢ SubderivAt
      (fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred,
          (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
      (ADMM.x₁ E₂ F ↑n) =
    {(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)}

Tactic: apply SubderivAt_eq_gradient (inner_convex1 n)

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+
 ⊢ HasGradientAt
     (fun x =>
       ⟪ADMM.y E₁ E₂ n.natPred,
         (OptProblem.A₁ E₂) x + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ)
     ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) (ADMM.y E₁ E₂ n.natPred)) (ADMM.x₁ E₂ F ↑n)]

Relevant lemmas:
SubderivAt_eq_gradient : ∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_9 : InnerProductSpace ℝ E]
  [inst_10 : CompleteSpace E] {f : E → ℝ} {x f'x : E},
  ConvexOn ℝ univ f → HasGradientAt f f'x x → SubderivAt f x = {f'x}