private theorem extend_Z_bilin_key (x₀ : α) (y₀ : γ) : ∃ U ∈ comap e (𝓝 x₀), ∃ V ∈ comap f (𝓝 y₀),
    ∀ (x) (_ : x ∈ U) (x') (_ : x' ∈ U), ∀ (y) (_ : y ∈ V) (y') (_ : y' ∈ V),
    (fun p : β × δ => φ p.1 p.2) (x', y') - (fun p : β × δ => φ p.1 p.2) (x, y) ∈ W' := by
  let ee := fun u : β × β => (e u.1, e u.2)
  let ff := fun u : δ × δ => (f u.1, f u.2)
  have lim_φ : Filter.Tendsto (fun p : β × δ => φ p.1 p.2) (𝓝 (0, 0)) (𝓝 0) := by
    simpa using hφ.tendsto (0, 0)
  have lim_φ_sub_sub :
    Tendsto (fun p : (β × β) × δ × δ => (fun p : β × δ => φ p.1 p.2) (p.1.2 - p.1.1, p.2.2 - p.2.1))
      ((comap ee <| 𝓝 (x₀, x₀)) ×ˢ (comap ff <| 𝓝 (y₀, y₀))) (𝓝 0) := by
    have lim_sub_sub :
      Tendsto (fun p : (β × β) × δ × δ => (p.1.2 - p.1.1, p.2.2 - p.2.1))
        (comap ee (𝓝 (x₀, x₀)) ×ˢ comap ff (𝓝 (y₀, y₀))) (𝓝 0 ×ˢ 𝓝 0) := by
      have := Filter.prod_mono (tendsto_sub_comap_self de x₀) (tendsto_sub_comap_self df y₀)
      rwa [prod_map_map_eq] at this
    rw [← nhds_prod_eq] at lim_sub_sub
    exact Tendsto.comp lim_φ lim_sub_sub
  rcases exists_nhds_zero_quarter W'_nhd with ⟨W, W_nhd, W4⟩
  have :
    ∃ U₁ ∈ comap e (𝓝 x₀), ∃ V₁ ∈ comap f (𝓝 y₀), ∀ (x) (_ : x ∈ U₁) (x') (_ : x' ∈ U₁),
      ∀ (y) (_ : y ∈ V₁) (y') (_ : y' ∈ V₁), (fun p : β × δ => φ p.1 p.2) (x' - x, y' - y) ∈ W := by
    rcases tendsto_prod_iff.1 lim_φ_sub_sub W W_nhd with ⟨U, U_in, V, V_in, H⟩
    rw [nhds_prod_eq, ← prod_comap_comap_eq, mem_prod_same_iff] at U_in V_in
    rcases U_in with ⟨U₁, U₁_in, HU₁⟩
    rcases V_in with ⟨V₁, V₁_in, HV₁⟩
    exists U₁, U₁_in, V₁, V₁_in
    intro x x_in x' x'_in y y_in y' y'_in
    exact H _ _ (HU₁ (mk_mem_prod x_in x'_in)) (HV₁ (mk_mem_prod y_in y'_in))
  rcases this with ⟨U₁, U₁_nhd, V₁, V₁_nhd, H⟩
  obtain ⟨x₁, x₁_in⟩ : U₁.Nonempty := (de.comap_nhds_neBot _).nonempty_of_mem U₁_nhd
  obtain ⟨y₁, y₁_in⟩ : V₁.Nonempty := (df.comap_nhds_neBot _).nonempty_of_mem V₁_nhd
  have cont_flip : Continuous fun p : δ × β => φ.flip p.1 p.2 := by
    show Continuous ((fun p : β × δ => φ p.1 p.2) ∘ Prod.swap)
    exact hφ.comp continuous_swap
  rcases extend_Z_bilin_aux de hφ W_nhd x₀ y₁ with ⟨U₂, U₂_nhd, HU⟩
  rcases extend_Z_bilin_aux df cont_flip W_nhd y₀ x₁ with ⟨V₂, V₂_nhd, HV⟩
  exists U₁ ∩ U₂, inter_mem U₁_nhd U₂_nhd, V₁ ∩ V₂, inter_mem V₁_nhd V₂_nhd
  rintro x ⟨xU₁, xU₂⟩ x' ⟨x'U₁, x'U₂⟩ y ⟨yV₁, yV₂⟩ y' ⟨y'V₁, y'V₂⟩
  have key_formula : φ x' y' - φ x y
    = φ (x' - x) y₁ + φ (x' - x) (y' - y₁) + φ x₁ (y' - y) + φ (x - x₁) (y' - y) := by simp; abel
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
G : Type u_5
inst✝¹⁶ : TopologicalSpace α
inst✝¹⁵ : AddCommGroup α
inst✝¹⁴ : TopologicalAddGroup α
inst✝¹³ : TopologicalSpace β
inst✝¹² : AddCommGroup β
inst✝¹¹ : TopologicalAddGroup β
inst✝¹⁰ : TopologicalSpace γ
inst✝⁹ : AddCommGroup γ
inst✝⁸ : TopologicalAddGroup γ
inst✝⁷ : TopologicalSpace δ
inst✝⁶ : AddCommGroup δ
inst✝⁵ : TopologicalAddGroup δ
inst✝⁴ : UniformSpace G
inst✝³ : AddCommGroup G
inst✝² : UniformAddGroup G
inst✝¹ : SeparatedSpace G
inst✝ : CompleteSpace G
e : β →+ α
de : DenseInducing ⇑e
f : δ →+ γ
df : DenseInducing ⇑f
φ : β →+ δ →+ G
hφ : Continuous fun p => (φ p.1) p.2
W' : Set G
W'_nhd : W' ∈ 𝓝 0
x₀ : α
y₀ : γ
ee : β × β → α × α := fun u => (e u.1, e u.2)
ff : δ × δ → γ × γ := fun u => (f u.1, f u.2)
lim_φ : Tendsto (fun p => (φ p.1) p.2) (𝓝 (0, 0)) (𝓝 0)
lim_φ_sub_sub :
  Tendsto (fun p => (fun p => (φ p.1) p.2) (p.1.2 - p.1.1, p.2.2 - p.2.1))
    (comap ee (𝓝 (x₀, x₀)) ×ˢ comap ff (𝓝 (y₀, y₀))) (𝓝 0)
W : Set G
W_nhd : W ∈ 𝓝 0
W4 : ∀ {v w s t : G}, v ∈ W → w ∈ W → s ∈ W → t ∈ W → v + w + s + t ∈ W'
U₁ : Set β
U₁_nhd : U₁ ∈ comap (⇑e) (𝓝 x₀)
V₁ : Set δ
V₁_nhd : V₁ ∈ comap (⇑f) (𝓝 y₀)
H : ∀ x ∈ U₁, ∀ x' ∈ U₁, ∀ y ∈ V₁, ∀ y' ∈ V₁, (fun p => (φ p.1) p.2) (x' - x, y' - y) ∈ W
x₁ : β
x₁_in : x₁ ∈ U₁
y₁ : δ
y₁_in : y₁ ∈ V₁
cont_flip : Continuous fun p => ((AddMonoidHom.flip φ) p.1) p.2
U₂ : Set β
U₂_nhd : U₂ ∈ comap (⇑e) (𝓝 x₀)
HU : ∀ x ∈ U₂, ∀ x' ∈ U₂, (fun p => (φ p.1) p.2) (x' - x, y₁) ∈ W
V₂ : Set δ
V₂_nhd : V₂ ∈ comap (⇑f) (𝓝 y₀)
HV : ∀ x ∈ V₂, ∀ x' ∈ V₂, (fun p => ((AddMonoidHom.flip φ) p.1) p.2) (x' - x, x₁) ∈ W
x : β
xU₁ : x ∈ U₁
xU₂ : x ∈ U₂
x' : β
x'U₁ : x' ∈ U₁
x'U₂ : x' ∈ U₂
y : δ
yV₁ : y ∈ V₁
yV₂ : y ∈ V₂
y' : δ
y'V₁ : y' ∈ V₁
y'V₂ : y' ∈ V₂
key_formula : (φ x') y' - (φ x) y = (φ (x' - x)) y₁ + (φ (x' - x)) (y' - y₁) + (φ x₁) (y' - y) + (φ (x - x₁)) (y' - y)
⊢ (fun p => (φ p.1) p.2) (x', y') - (fun p => (φ p.1) p.2) (x, y) ∈ W'

Next tactic:
rw [key_formula]