theorem extracted_formal_statement_0.{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)) (h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1) ⦃c : ℝ⦄ (hc : 0 < c)
  (h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖) (v : Set E)
  (hv1 : v ∈ 𝓝 z.1) (y : E) (yv : y ∈ v)
  (hv2 : ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (y, y') - H z - inner (gradient H z) ((y, y') - z)‖ ≤ c * ‖(y, y') - z‖)
  (hv2' : ‖H (y, z.2) - H z - inner (gradient H z) ((y, z.2) - z)‖ ≤ c * ‖(y, z.2) - z‖) (this : z = (z.1, z.2)) :
  ‖H (y, z.2) - H (z.1, z.2) - inner (gradient H z).1 (y - z.1)‖ ≤ c * ‖y - z.1‖ := sorry

State before:
case h.right
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)
h₁ : HasGradientAt (fun x => H (x, z.2)) (grad_fst H z.2 z.1) z.1
c : ℝ
hc : 0 < c
h₃ : ∀ᶠ (x : WithLp 2 (E × F)) in 𝓝 z, ‖H x - H z - inner (gradient H z) (x - z)‖ ≤ c * ‖x - z‖
v : Set E
hv1 : v ∈ 𝓝 z.1
y : E
yv : y ∈ v
hv2 : ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (y, y') - H z - inner (gradient H z) ((y, y') - z)‖ ≤ c * ‖(y, y') - z‖
hv2' : ‖H (y, z.2) - H z - inner (gradient H z) ((y, z.2) - z)‖ ≤ c * ‖(y, z.2) - z‖
this : z = (z.1, z.2)
⊢ ‖H (y, z.2) - H (z.1, z.2) - inner (gradient H z).1 (y - z.1)‖ ≤ c * ‖y - z.1‖

Tactic: rw [this] at hv2'
    rw [Prod.mk_sub_mk y z.1 z.2 z.2] at hv2'
    simp at hv2'
    rw [norm_prod_right_zero] at hv2'
    exact hv2'

State after:

Relevant lemmas:
Prod.mk_sub_mk : ∀ {G : Type u_6} {H : Type u_7} [inst : Sub G] [inst_6 : Sub H] (x₁ x₂ : G) (y₁ y₂ : H),
  (x₁, y₁) - (x₂, y₂) = (x₁ - x₂, y₁ - y₂)
norm_prod_right_zero : ∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_6 : NormedAddCommGroup F]
  (x : E), ‖(x, 0)‖ = ‖x‖