-- Lean 4 proof step from Mathlib.Analysis.Normed.Module.RieszLemma (goal)
Goal:
𝕜 : Type u_1
inst : NormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (↑F : Set E)
hF : ∃ (x : E), x ∉ F
r : ℝ
hr : r < 1
obtain : ∀ (x : E) (hx : x ∉ F), ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
obtain₁ : ∀ (x : E) (hx : x ∉ F), ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
obtain₂ : ∀ (x : E) (hx : x ∉ F) (hFn : (↑F : Set E).Nonempty), ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
obtain₃ obtain₄ :
  ∀ (x : E) (hx : x ∉ F),
    let d := infDist x (↑F : Set E);
    ∀ (hFn : (↑F : Set E).Nonempty) (hdp : 0 < d), ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
obtain₅ :
  ∀ (x : E) (hx : x ∉ F),
    let d := infDist x (↑F : Set E);
    ∀ (hFn : (↑F : Set E).Nonempty) (hdp : 0 < d),
      let r' := max r 2⁻¹;
      ∀ (hr' : r' < 1), ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
obtain₆ :
  ∀ (x : E) (hx : x ∉ F),
    let d := infDist x (↑F : Set E);
    ∀ (hFn : (↑F : Set E).Nonempty) (hdp : 0 < d),
      let r' := max r 2⁻¹;
      ∀ (hr' : r' < 1) (hlt : 0 < r'), ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
obtain₇ :
  ∀ (x : E) (hx : x ∉ F),
    let d := infDist x (↑F : Set E);
    ∀ (hFn : (↑F : Set E).Nonempty) (hdp : 0 < d),
      let r' := max r 2⁻¹;
      ∀ (hr' : r' < 1) (hlt : 0 < r') (hdlt : d < d / r'), ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
obtain₈ :
  ∀ (x : E) (hx : x ∉ F),
    let d := infDist x (↑F : Set E);
    ∀ (hFn : (↑F : Set E).Nonempty) (hdp : 0 < d),
      let r' := max r 2⁻¹;
      ∀ (hr' : r' < 1) (hlt : 0 < r') (hdlt : d < d / r') (y₀ : E) (hy₀F : y₀ ∈ F) (hxy₀ : dist x y₀ < d / r'),
        ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
obtain₉ :
  ∀ (x : E) (hx : x ∉ F),
    let d := infDist x (↑F : Set E);
    ∀ (hFn : (↑F : Set E).Nonempty) (hdp : 0 < d),
      let r' := max r 2⁻¹;
      ∀ (hr' : r' < 1) (hlt : 0 < r') (hdlt : d < d / r') (y₀ : E) (hy₀F : y₀ ∈ F) (hxy₀ : dist x y₀ < d / r')
        (x_ne_y₀ : x - y₀ ∉ F), ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
⊢ ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖
have obtain₁₀ : ∀ (x : E) (hx : x ∉ F),
  let d := infDist x (↑F : Set E);
  ∀ (hFn : (↑F : Set E).Nonempty) (hdp : 0 < d),
    let r' := max r 2⁻¹;
    ∀ (hr' : r' < 1) (hlt : 0 < r') (hdlt : d < d / r') (y₀ : E) (hy₀F : y₀ ∈ F) (hxy₀ : dist x y₀ < d / r')
      (x_ne_y₀ : x - y₀ ∉ F) (y : E) (hy : y ∈ F), r * ‖x - y₀‖ < ‖x - y₀ - y‖
Tactic: classical
    obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF
    let d := Metric.infDist x F
    have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩
    have hdp : 0 < d :=
      lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
        hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)
    let r' := max r 2⁻¹
    have hr' : r' < 1 := by
      simp only [r', max_lt_iff, hr, true_and]
      norm_num
    have hlt : 0 < r' := lt_of_lt_of_le (by simp) (le_max_right r 2⁻¹)
    have hdlt : d < d / r' := (lt_div_iff₀ hlt).mpr ((mul_lt_iff_lt_one_right hdp).2 hr')
    obtain ⟨y₀, hy₀F, hxy₀⟩ : ∃ y ∈ F, dist x y < d / r' := (Metric.infDist_lt_iff hFn).mp hdlt
    have x_ne_y₀ : x - y₀ ∉ F := by
      by_contra h
      have : x - y₀ + y₀ ∈ F := F.add_mem h hy₀F
      simp only [neg_add_cancel_right, sub_eq_add_neg] at this
      exact hx this
    refine ⟨x - y₀, x_ne_y₀, fun y hy => le_of_lt ?_⟩
    have hy₀y : y₀ + y ∈ F := F.add_mem hy₀F hy
    calc
      r * ‖x - y₀‖ ≤ r' * ‖x - y₀‖ := by gcongr; apply le_max_left
      _ < d := by
        rw [← dist_eq_norm]
        exact (lt_div_iff₀' hlt).1 hxy₀
      _ ≤ dist x (y₀ + y) := Metric.infDist_le_dist_of_mem hy₀y
      _ = ‖x - y₀ - y‖ := by rw [sub_sub, dist_eq_norm]
Cleared hypotheses: hF