theorem extracted_formal_statement_10.{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‖)
  (h₃' :
    ∀ᶠ (x' : E) in 𝓝 z.1,
      ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖)
  (h_original_goal :
    ∀ᶠ (x : E) in 𝓝 z.1, ‖H (x, z.2) - H (z.1, z.2) - inner (gradient H z).1 (x - z.1)‖ ≤ c * ‖x - z.1‖) :
  ∃ v ∈ 𝓝 z.1, ∀ y ∈ v, ‖H (y, z.2) - H (z.1, z.2) - inner (gradient H z).1 (y - z.1)‖ ≤ c * ‖y - z.1‖ := 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
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‖
h₃' :
  ∀ᶠ (x' : E) in 𝓝 z.1,
    ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (x', y') - H z - inner (gradient H z) ((x', y') - z)‖ ≤ c * ‖(x', y') - z‖
⊢ ∀ᶠ (x : E) in 𝓝 z.1, ‖H (x, z.2) - H (z.1, z.2) - inner (gradient H z).1 (x - z.1)‖ ≤ c * ‖x - z.1‖

Tactic: rw [Filter.eventually_iff_exists_mem] at h₃' ⊢

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
 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‖
 h₃' :
   ∃ v ∈ 𝓝 z.1,
     ∀ y ∈ v, ∀ᶠ (y' : F) in 𝓝 z.2, ‖H (y, y') - H z - inner (gradient H z) ((y, y') - z)‖ ≤ c * ‖(y, y') - z‖
 ⊢ ∃ v ∈ 𝓝 z.1, ∀ y ∈ v, ‖H (y, z.2) - H (z.1, z.2) - inner (gradient H z).1 (y - z.1)‖ ≤ c * ‖y - z.1‖]

Relevant lemmas:
Filter.eventually_iff_exists_mem : ∀ {α : Type u} {p : α → Prop} {f : Filter α},
  (∀ᶠ (x : α) in f, p x) ↔ ∃ v ∈ f, ∀ y ∈ v, p y