theorem extracted_formal_statement_1.{u_2, u_1} {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E]
  [inst_1 : InnerProductSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace ℝ F]
  {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H) (h_original_goal : Differentiable ℝ fun x => x) :
  IsBoundedLinearMap ℝ fun x => x := sorry

State before:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
⊢ Differentiable ℝ fun x => x

Tactic: apply IsBoundedLinearMap.differentiable

State after:
[case h
 E : Type u_1
 F : Type u_2
 inst✝³ : NormedAddCommGroup E
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : NormedAddCommGroup F
 inst✝ : InnerProductSpace ℝ F
 H : WithLp 2 (E × F) → ℝ
 h : Differentiable ℝ H
 ⊢ IsBoundedLinearMap ℝ fun x => x]

Relevant lemmas:
IsBoundedLinearMap.differentiable : ∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2}
  [inst_4 : NormedAddCommGroup E] [inst_5 : NormedSpace 𝕜 E] {F : Type u_3} [inst_6 : NormedAddCommGroup F]
  [inst_7 : NormedSpace 𝕜 F] {f : E → F}, IsBoundedLinearMap 𝕜 f → Differentiable 𝕜 f