/-- If a Lipschitz functions has line derivatives in a dense set of directions, all of them given by
a single continuous linear map `L`, then it admits `L` as Fréchet derivative. -/
theorem hasFderivAt_of_hasLineDerivAt_of_closure {f : E → F}
    (hf : LipschitzWith C f) {s : Set E} (hs : sphere 0 1 ⊆ closure s)
    {L : E →L[ℝ] F} {x : E} (hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v) :
    HasFDerivAt f L x := by
  rw [hasFDerivAt_iff_isLittleO_nhds_zero, isLittleO_iff]
  intro ε εpos
  obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ (C + ‖L‖ + 1) * δ = ε :=
    ⟨ε / (C + ‖L‖ + 1), by positivity, mul_div_cancel' ε (by positivity)⟩
  obtain ⟨q, hqs, q_fin, hq⟩ : ∃ q, q ⊆ s ∧ q.Finite ∧ sphere 0 1 ⊆ ⋃ y ∈ q, ball y δ := by
    have : sphere 0 1 ⊆ ⋃ y ∈ s, ball y δ := by
      apply hs.trans (fun z hz ↦ ?_)
      obtain ⟨y, ys, hy⟩ : ∃ y ∈ s, dist z y < δ := Metric.mem_closure_iff.1 hz δ δpos
      exact mem_biUnion ys hy
    exact (isCompact_sphere 0 1).elim_finite_subcover_image (fun y _hy ↦ isOpen_ball) this
  have I : ∀ᶠ t in 𝓝 (0 : ℝ), ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖ := by
    apply (Finite.eventually_all q_fin).2 (fun v hv ↦ ?_)
    apply Asymptotics.IsLittleO.def ?_ δpos
    exact hasLineDerivAt_iff_isLittleO_nhds_zero.1 (hL v (hqs hv))
  obtain ⟨r, r_pos, hr⟩ : ∃ (r : ℝ), 0 < r ∧ ∀ (t : ℝ), ‖t‖ < r →
      ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖ := by
    rcases Metric.mem_nhds_iff.1 I with ⟨r, r_pos, hr⟩
    exact ⟨r, r_pos, fun t ht v hv ↦ hr (mem_ball_zero_iff.2 ht) v hv⟩
  apply Metric.mem_nhds_iff.2 ⟨r, r_pos, fun v hv ↦ ?_⟩
  rcases eq_or_ne v 0 with rfl|v_ne
  · simp
  obtain ⟨w, ρ, w_mem, hvw, hρ⟩ : ∃ w ρ, w ∈ sphere 0 1 ∧ v = ρ • w ∧ ρ = ‖v‖ := by
    refine ⟨‖v‖⁻¹ • v, ‖v‖, by simp [norm_smul, inv_mul_cancel (norm_ne_zero_iff.2 v_ne)], ?_, rfl⟩
    simp [smul_smul, mul_inv_cancel (norm_ne_zero_iff.2 v_ne)]
  have norm_rho : ‖ρ‖ = ρ := by rw [hρ, norm_norm]
  have rho_pos : 0 ≤ ρ := by simp [hρ]
  obtain ⟨y, yq, hy⟩ : ∃ y ∈ q, ‖w - y‖ < δ := by simpa [← dist_eq_norm] using hq w_mem
  have : ‖y - w‖ < δ := by rwa [norm_sub_rev]
  calc  ‖f (x + v) - f x - L v‖
      = ‖f (x + ρ • w) - f x - ρ • L w‖ := by simp [hvw]
    _ = ‖(f (x + ρ • w) - f (x + ρ • y)) + (ρ • L y - ρ • L w)
          + (f (x + ρ • y) - f x - ρ • L y)‖ := by congr; abel
    _ ≤ ‖f (x + ρ • w) - f (x + ρ • y)‖ + ‖ρ • L y - ρ • L w‖
          + ‖f (x + ρ • y) - f x - ρ • L y‖ := norm_add₃_le _ _ _
    _ ≤ C * ‖(x + ρ • w) - (x + ρ • y)‖ + ρ * (‖L‖ * ‖y - w‖) + δ * ρ := by
      gcongr
      · exact hf.norm_sub_le _ _
      · rw [← smul_sub, norm_smul, norm_rho]
        gcongr
        

Proof state:
case h₁.h₂.h
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : FiniteDimensional ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
F : Type u_2
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
C D : ℝ≥0
f✝ g : E → ℝ
s✝ : Set E
μ : Measure E
inst✝ : IsAddHaarMeasure μ
f : E → F
hf : LipschitzWith C f
s : Set E
hs : sphere 0 1 ⊆ closure s
L : E →L[ℝ] F
x : E
hL : ∀ v ∈ s, HasLineDerivAt ℝ f (L v) x v
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : 0 < δ
hδ : (↑C + ‖L‖ + 1) * δ = ε
q : Set E
hqs : q ⊆ s
q_fin : Set.Finite q
hq : sphere 0 1 ⊆ ⋃ y ∈ q, ball y δ
I : ∀ᶠ (t : ℝ) in 𝓝 0, ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
r : ℝ
r_pos : 0 < r
hr : ∀ (t : ℝ), ‖t‖ < r → ∀ v ∈ q, ‖f (x + t • v) - f x - t • L v‖ ≤ δ * ‖t‖
v : E
hv : v ∈ ball 0 r
v_ne : v ≠ 0
w : E
ρ : ℝ
w_mem : w ∈ sphere 0 1
hvw : v = ρ • w
hρ : ρ = ‖v‖
norm_rho : ‖ρ‖ = ρ
rho_pos : 0 ≤ ρ
y : E
yq : y ∈ q
hy : ‖w - y‖ < δ
this : ‖y - w‖ < δ
⊢ ‖L y - L w‖ ≤ ‖L‖ * ‖y - w‖

Next tactic:
exact L.lipschitz.norm_sub_le _ _