-- Lean 4 proof step from Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : NumberField K
I : (FractionalIdeal (R := 𝓞 K) (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ ((volume : Measure (mixedSpace K)) : (a : Set (mixedSpace K)) → ℝ≥0∞) (convexBodySum K B)
hB : 0 ≤ B
h1 : LT.lt (α := ℝ) 0 (↑(finrank ℚ K) : ℝ)⁻¹
h2 : 0 ≤ B / (↑(finrank ℚ K) : ℝ)
h_fund :
  IsAddFundamentalDomain.{u_1, u_1}
    ↥(toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
        (span ℤ
          (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
            (⇑(fractionalIdealLatticeBasis K I :
                  Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                    (mixedSpace K)) :
              (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) → mixedSpace K))))
    (α := mixedSpace K)
    (fundamentalDomain (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) (K := ℝ)
      (fractionalIdealLatticeBasis K I))
    volume
this :
  Countable.{u_1 + 1}
    ↥(toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
        (span ℤ
          (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
            (⇑(fractionalIdealLatticeBasis K I :
                  Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                    (mixedSpace K)) :
              (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) → mixedSpace K))))
⊢ ∃ a ∈ (↑I : FractionalIdeal (𝓞 K)⁰ K),
    a ≠ 0 ∧ (↑|(Algebra.norm ℚ : (a : K) → ℚ) a| : ℝ) ≤ (B / (↑(finrank ℚ K) : ℝ)) ^ finrank ℚ K
have obtain : ∀ (x : mixedSpace K)
  (hx :
    Membership.mem (γ := AddSubgroup (mixedSpace K))
      (toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
        (span ℤ
          (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
            (⇑(fractionalIdealLatticeBasis K I :
                  Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                    (mixedSpace K)) :
              (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) → mixedSpace K))))
      x)
  (h_nz :
    Subtype.mk (p := fun (x : mixedSpace K) ↦
        Membership.mem (γ := AddSubgroup (mixedSpace K))
          (toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
            (span ℤ
              (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
                (⇑(fractionalIdealLatticeBasis K I :
                      Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                        (mixedSpace K)) :
                  (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) →
                    mixedSpace K))))
          x)
        x hx ≠
      0)
  (h_mem :
    Membership.mem (γ := Set (mixedSpace K)) (convexBodySum K B)
      (↑(Subtype.mk (p := fun (x : mixedSpace K) ↦
            Membership.mem (γ := AddSubgroup (mixedSpace K))
              (toAddSubgroup (module_M := AddCommGroup.toIntModule (mixedSpace K))
                (span ℤ
                  (Set.range (ι := Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K))
                    (⇑(fractionalIdealLatticeBasis K I :
                          Basis (Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) ℝ
                            (mixedSpace K)) :
                      (a : Free.ChooseBasisIndex ℤ ↥(↑(↑I : FractionalIdeal (𝓞 K)⁰ K) : Submodule (𝓞 K) K)) →
                        mixedSpace K))))
              x)
            x hx) :
        mixedSpace K)),
  ∃ a ∈ (↑I : FractionalIdeal (𝓞 K)⁰ K),
    a ≠ 0 ∧ (↑|(Algebra.norm ℚ : (a : K) → ℚ) a| : ℝ) ≤ (B / (↑(finrank ℚ K) : ℝ)) ^ finrank ℚ K
Tactic: obtain ⟨⟨x, hx⟩, h_nz, h_mem⟩ := exists_ne_zero_mem_lattice_of_measure_mul_two_pow_le_measure
      h_fund (fun _ ↦ convexBodySum_neg_mem K B) (convexBodySum_convex K B)
      (convexBodySum_compact K B) h