theorem extracted_formal_statement_8.{u_2, u_1} {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E]
  [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F]
  [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] {H : WithLp 2 (E × F) → ℝ} (h : Differentiable ℝ H)
  (z : WithLp 2 (E × F)) : DifferentiableAt ℝ (fun x => H (x, z.2)) z.1 := sorry

State before:
case h
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace ℝ E
inst✝³ : CompleteSpace E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ F
inst✝ : CompleteSpace F
H : WithLp 2 (E × F) → ℝ
h : Differentiable ℝ H
z : WithLp 2 (E × F)
⊢ DifferentiableAt ℝ (fun x => H (x, z.2)) z.1

Tactic: apply diff_prod₁ h

State after:

Relevant lemmas:
diff_prod₁ : ∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_6 : InnerProductSpace ℝ E]
  [inst_7 : NormedAddCommGroup F] [inst_8 : InnerProductSpace ℝ F] {H : WithLp 2 (E × F) → ℝ},
  Differentiable ℝ H → ∀ (y : F), Differentiable ℝ fun x => H (x, y)