-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace (goal)
Goal:
E : Type u_1
inst : NormedAddCommGroup E
inst_1 : NormedSpace ℝ E
inst_2 : FiniteDimensional ℝ E
This This₁ This₂ This₃ :
  (¬∃ (δ : ℝ),
        0 < δ ∧
          δ < 1 ∧
            ∀ (s : Finset E),
              (∀ c ∈ s, ‖c‖ ≤ 2) → (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E) →
    False
This₄ This₅ This₆ This₇ This₈ This₉ This₁₀ :
  ∀
    (h :
      ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
        ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card),
    False
This₁₁ :
  ∀
    (h :
      ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
        ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card),
    let N := multiplicity E + 1;
    ∀ (hN : N = multiplicity E + 1), False
This₁₂ :
  ∀
    (h :
      ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
        ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card),
    let N := multiplicity E + 1;
    ∀ (hN : N = multiplicity E + 1)
      (this :
        ∀ (δ : ℝ),
          0 < δ → ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖f i - f j‖),
      False
This₁₃ :
  ∀
    (h :
      ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
        ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card),
    let N := multiplicity E + 1;
    ∀ (hN : N = multiplicity E + 1) (F : (δ : ℝ) → Fin N → E)
      (hF : ∀ (δ : ℝ), 0 < δ → (∀ (i : Fin N), ‖F δ i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖F δ i - F δ j‖),
      False
This₁₄ :
  ∀
    (h :
      ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
        ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card),
    let N := multiplicity E + 1;
    ∀ (hN : N = multiplicity E + 1) (F : (δ : ℝ) → Fin N → E)
      (hF : ∀ (δ : ℝ), 0 < δ → (∀ (i : Fin N), ‖F δ i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖F δ i - F δ j‖)
      (this : ∃ (f : Fin N → E), (∀ (i : Fin N), ‖f i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 ≤ ‖f i - f j‖), False
This₁₅ :
  ∀
    (h :
      ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
        ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card),
    let N := multiplicity E + 1;
    ∀ (hN : N = multiplicity E + 1) (F : (δ : ℝ) → Fin N → E)
      (hF : ∀ (δ : ℝ), 0 < δ → (∀ (i : Fin N), ‖F δ i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖F δ i - F δ j‖)
      (f : Fin N → E) (hf : ∀ (i : Fin N), ‖f i‖ ≤ 2) (h'f : Pairwise fun (i j : Fin N) ↦ 1 ≤ ‖f i - f j‖), False
⊢ ∃ (δ : ℝ),
    0 < δ ∧
      δ < 1 ∧
        ∀ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) → (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) → s.card ≤ multiplicity E
have This₁₆ : ∀
  (h :
    ∀ (δ : ℝ) (_ : 0 < δ) (_ : δ < 1),
      ∃ (s : Finset E), (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card),
  let N := multiplicity E + 1;
  ∀ (hN : N = multiplicity E + 1) (F : (δ : ℝ) → Fin N → E)
    (hF : ∀ (δ : ℝ), 0 < δ → (∀ (i : Fin N), ‖F δ i‖ ≤ 2) ∧ Pairwise fun (i j : Fin N) ↦ 1 - δ ≤ ‖F δ i - F δ j‖)
    (f : Fin N → E) (hf : ∀ (i : Fin N), ‖f i‖ ≤ 2) (h'f : Pairwise fun (i j : Fin N) ↦ 1 ≤ ‖f i - f j‖)
    (finj : Function.Injective f), False
Tactic: classical
  /- This follows from a compactness argument: otherwise, one could extract a converging
    subsequence, to obtain a `1`-separated set in the ball of radius `2` with cardinality
    `N = multiplicity E + 1`. To formalize this, we work with functions `Fin N → E`.
     -/
  by_contra! h
  set N := multiplicity E + 1 with hN
  have :
    ∀ δ : ℝ, 0 < δ → ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧
      Pairwise fun i j => 1 - δ ≤ ‖f i - f j‖ := by
    intro δ hδ
    rcases lt_or_ge δ 1 with (hδ' | hδ')
    · rcases h δ hδ hδ' with ⟨s, hs, h's, s_card⟩
      obtain ⟨f, f_inj, hfs⟩ : ∃ f : Fin N → E, Function.Injective f ∧ range f ⊆ ↑s := by
        have : Fintype.card (Fin N) ≤ s.card := by simp only [Fintype.card_fin]; exact s_card
        rcases Function.Embedding.exists_of_card_le_finset this with ⟨f, hf⟩
        exact ⟨f, f.injective, hf⟩
      simp only [range_subset_iff, Finset.mem_coe] at hfs
      exact ⟨f, fun i => hs _ (hfs i), fun i j hij => h's _ (hfs i) _ (hfs j) (f_inj.ne hij)⟩
    · exact
        ⟨fun _ => 0, by simp, fun i j _ => by
          simpa only [norm_zero, sub_nonpos, sub_self]⟩
  -- For `δ > 0`, `F δ` is a function from `Fin N` to the ball of radius `2` for which two points
  -- in the image are separated by `1 - δ`.
  choose! F hF using this
  -- Choose a converging subsequence when `δ → 0`.
  have : ∃ f : Fin N → E, (∀ i : Fin N, ‖f i‖ ≤ 2) ∧ Pairwise fun i j => 1 ≤ ‖f i - f j‖ := by
    obtain ⟨u, _, zero_lt_u, hu⟩ :
      ∃ u : ℕ → ℝ,
        (∀ m n : ℕ, m < n → u n < u m) ∧ (∀ n : ℕ, 0 < u n) ∧ Filter.Tendsto u Filter.atTop (𝓝 0) :=
      exists_seq_strictAnti_tendsto (0 : ℝ)
    have A : ∀ n, F (u n) ∈ closedBall (0 : Fin N → E) 2 := by
      intro n
      simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right,
        (hF (u n) (zero_lt_u n)).left, forall_const]
    obtain ⟨f, fmem, φ, φ_mono, hf⟩ :
      ∃ f ∈ closedBall (0 : Fin N → E) 2,
        ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto ((F ∘ u) ∘ φ) atTop (𝓝 f) :=
      IsCompact.tendsto_subseq (isCompact_closedBall _ _) A
    refine ⟨f, fun i => ?_, fun i j hij => ?_⟩
    · simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right] at fmem
      exact fmem i
    · have A : Tendsto (fun n => ‖F (u (φ n)) i - F (u (φ n)) j‖) atTop (𝓝 ‖f i - f j‖) :=
        ((hf.apply_nhds i).sub (hf.apply_nhds j)).norm
      have B : Tendsto (fun n => 1 - u (φ n)) atTop (𝓝 (1 - 0)) :=
        tendsto_const_nhds.sub (hu.comp φ_mono.tendsto_atTop)
      rw [sub_zero] at B
      exact le_of_tendsto_of_tendsto' B A fun n => (hF (u (φ n)) (zero_lt_u _)).2 hij
  rcases this with ⟨f, hf, h'f⟩
  -- the range of `f` contradicts the definition of `multiplicity E`.
  have finj : Function.Injective f := by
    intro i j hij
    by_contra h
    have : 1 ≤ ‖f i - f j‖ := h'f h
    simp only [hij, norm_zero, sub_self] at this
    exact lt_irrefl _ (this.trans_lt zero_lt_one)
  let s := Finset.image f Finset.univ
  have s_card : s.card = N := by rw [Finset.card_image_of_injective _ finj]; exact Finset.card_fin N
  have hs : ∀ c ∈ s, ‖c‖ ≤ 2 := by
    simp only [s, hf, forall_apply_eq_imp_iff, forall_const, forall_exists_index, Finset.mem_univ,
      Finset.mem_image, true_and]
  have h's : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 ≤ ‖c - d‖ := by
    simp only [s, forall_apply_eq_imp_iff, forall_exists_index, Finset.mem_univ, Finset.mem_image,
      Ne, forall_apply_eq_imp_iff, true_and]
    intro i j hij
    have : i ≠ j := fun h => by rw [h] at hij; exact hij rfl
    exact h'f this
  have : s.card ≤ multiplicity E := card_le_multiplicity hs h's
  rw [s_card, hN] at this
  exact lt_irrefl _ ((Nat.lt_succ_self (multiplicity E)).trans_le this)