theorem extracted_formal_statement_7.{u_2, u_1} {E : Type u_1} [inst : NormedAddCommGroup E] {F : Type u_2}
  [inst_1 : InnerProductSpace ℝ E] [inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace ℝ F] {f : E → ℝ}
  {g : F → ℝ} {x u : E} {y v : F} (hf : u ∈ f_subdifferential f x) (hg : v ∈ f_subdifferential g y)
  (h_original_goal : (u, v) ∈ f_subdifferential (fun z => f z.1 + g z.2) (x, y)) (ε : ℝ) :
  ε > 0 →
    ∀ᶠ (y_1 : WithLp 2 (E × F)) in 𝓝 (x, y),
      f y_1.1 + g y_1.2 - (f (x, y).1 + g (x, y).2) - inner (u, v) (y_1 - (x, y)) ≥ -ε * ‖y_1 - (x, y)‖ := sorry

State before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
F : Type u_2
inst✝² : InnerProductSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
f : E → ℝ
g : F → ℝ
x u : E
y v : F
hf : u ∈ f_subdifferential f x
hg : v ∈ f_subdifferential g y
⊢ (u, v) ∈ f_subdifferential (fun z => f z.1 + g z.2) (x, y)

Tactic: rw [has_f_subdiff_iff] at *

State after:
[E : Type u_1
 inst✝³ : NormedAddCommGroup E
 F : Type u_2
 inst✝² : InnerProductSpace ℝ E
 inst✝¹ : NormedAddCommGroup F
 inst✝ : InnerProductSpace ℝ F
 f : E → ℝ
 g : F → ℝ
 x u : E
 y v : F
 hf : ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner u (y - x) ≥ -ε * ‖y - x‖
 hg : ∀ ε > 0, ∀ᶠ (y_1 : F) in 𝓝 y, g y_1 - g y - inner v (y_1 - y) ≥ -ε * ‖y_1 - y‖
 ⊢ ∀ ε > 0,
     ∀ᶠ (y_1 : WithLp 2 (E × F)) in 𝓝 (x, y),
       f y_1.1 + g y_1.2 - (f (x, y).1 + g (x, y).2) - inner (u, v) (y_1 - (x, y)) ≥ -ε * ‖y_1 - (x, y)‖]

Relevant lemmas:
has_f_subdiff_iff : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_4 : InnerProductSpace ℝ E] {f : E → ℝ}
  {x u : E}, u ∈ f_subdifferential f x ↔ ∀ ε > 0, ∀ᶠ (y : E) in 𝓝 x, f y - f x - inner u (y - x) ≥ -ε * ‖y - x‖