theorem extracted_formal_statement_25.{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 : ∀ (x y : WithLp 2 (E × F)), ‖gradient H x - gradient H y‖ ≤ ↑l * ‖x - y‖)
  (z z' : WithLp 2 (E × F)) :
  ‖grad_snd H z.1 z.2 - grad_snd H z'.1 z'.2‖ = ‖(gradient H z).2 - (gradient H 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 : ∀ (x y : WithLp 2 (E × F)), ‖gradient H x - gradient H y‖ ≤ ↑l * ‖x - y‖
z z' : WithLp 2 (E × F)
⊢ ‖grad_snd H z.1 z.2 - grad_snd H z'.1 z'.2‖ = ‖(gradient H z).2 - (gradient H z').2‖

Tactic: rw [grad_snd_eq h, grad_snd_eq h]

State after:

Relevant lemmas:
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