theorem extracted_formal_statement_3.{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} {l : NNReal}
  (h : Differentiable ℝ H) (hl : LipschitzWith l (gradient H)) : LipschitzWith l fun z => grad_snd H z.1 z.2 := sorry

State before:
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
l : NNReal
h : Differentiable ℝ H
hl : LipschitzWith l (gradient H)
⊢ LipschitzWith l fun z => grad_snd H z.1 z.2

Tactic: rw [lipschitzWith_iff_norm_sub_le] at *
    intro z z'
    calc
  _ = ‖(gradient H z).2 - (gradient H z').2‖ := by rw [grad_snd_eq h, grad_snd_eq h]
  _ = ‖(gradient H z - gradient H z').2‖ := rfl
  _ ≤ ‖(gradient H z - gradient H z')‖ := (snd_norm_le_prod_L2 _)
  _ ≤ _ := hl z z'

State after:

Relevant lemmas:
lipschitzWith_iff_norm_sub_le : ∀ {E : Type u_2} {F : Type u_3} [inst : SeminormedAddGroup E]
  [inst_6 : SeminormedAddGroup F] {f : E → F} {C : NNReal}, LipschitzWith C f ↔ ∀ (x y : E), ‖f x - f y‖ ≤ ↑C * ‖x - y‖
grad_snd_eq : ∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_6 : InnerProductSpace ℝ E]
  [inst_7 : CompleteSpace E] [inst_8 : NormedAddCommGroup F] [inst_9 : InnerProductSpace ℝ F]
  [inst_10 : CompleteSpace F] {H : WithLp 2 (E × F) → ℝ},
  Differentiable ℝ H → ∀ (z : WithLp 2 (E × F)), (gradient H z).2 = grad_snd H z.1 z.2
snd_norm_le_prod_L2 : ∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_6 : NormedAddCommGroup F]
  (z : WithLp 2 (E × F)), ‖z.2‖ ≤ ‖z‖