theorem extracted_formal_statement_24.{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))
  (h_original_goal : LipschitzWith l fun z => grad_snd H z.1 z.2) (x y : WithLp 2 (E × F)) :
  ‖grad_snd H x.1 x.2 - grad_snd H y.1 y.2‖ ≤ ↑l * ‖x - y‖ := 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 *

State after:
[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 : ∀ (x y : WithLp 2 (E × F)), ‖gradient H x - gradient H y‖ ≤ ↑l * ‖x - y‖
 ⊢ ∀ (x y : WithLp 2 (E × F)), ‖grad_snd H x.1 x.2 - grad_snd H y.1 y.2‖ ≤ ↑l * ‖x - y‖]

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‖