theorem extracted_formal_statement_42.{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+) (x : E₁) :
  let c := ADMM.y E₁ E₂ n.natPred;
  let c₁ := ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ;
  ((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) =
      (fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ) + fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ) →
    HasFDerivAt (fun x => ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ + c₁)
        ((toDual ℝ E₁) ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c)) x →
      HasFDerivAt (fun x => ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ)
        ((toDual ℝ E₁) ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c)) x := 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+
x : E₁
c : F := ADMM.y E₁ E₂ n.natPred
c₁ : ℝ := ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
this :
  (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) =
    (fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ) + fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
⊢ HasFDerivAt (fun x => ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ + c₁)
    ((toDual ℝ E₁) ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c)) x

Tactic: apply HasFDerivAt.add_const c₁ _

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+
 x : E₁
 c : F := ADMM.y E₁ E₂ n.natPred
 c₁ : ℝ := ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
 this :
   (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) =
     (fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) x⟫_ℝ) + fun x =>
       ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂⟫_ℝ
 ⊢ HasFDerivAt (fun x => ⟪c, (OptProblem.A₁ E₂) x⟫_ℝ)
     ((toDual ℝ E₁) ((ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) c)) x]

Relevant lemmas:
HasFDerivAt.add_const : ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2}
  [inst_9 : NormedAddCommGroup E] [inst_10 : NormedSpace 𝕜 E] {F : Type u_3} [inst_11 : NormedAddCommGroup F]
  [inst_12 : NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} (c : F),
  HasFDerivAt f f' x → HasFDerivAt (fun x => f x + c) f' x