theorem Zlattice.rank : finrank ℤ L = finrank K E := by
  classical
  have : Module.Finite ℤ L := module_finite K hs
  have : Module.Free ℤ L := module_free K hs
  have : Module ℚ E := Module.compHom E (algebraMap ℚ K)
  let b₀ := Module.Free.chooseBasis ℤ L
  -- Let `b` be a `ℤ`-basis of `L` formed of vectors of `E`
  let b := Subtype.val ∘ b₀
  have : LinearIndependent ℤ b :=
    LinearIndependent.map' b₀.linearIndependent (L.toIntSubmodule.subtype) (ker_subtype _)
  -- We prove some assertions that will be useful later on
  have h_spanL : span ℤ (Set.range b) = AddSubgroup.toIntSubmodule L := by
    convert congrArg (map (Submodule.subtype (AddSubgroup.toIntSubmodule L))) b₀.span_eq
    · rw [map_span, Set.range_comp]
      rfl
    · exact (map_subtype_top _).symm
  have h_spanE : span K (Set.range b) = ⊤ := by rwa [← span_span_of_tower (R := ℤ), h_spanL]
  have h_card : Fintype.card (Module.Free.ChooseBasisIndex ℤ L) =
      (Set.range b).toFinset.card := by
    rw [Set.toFinset_range, Finset.univ.card_image_of_injective]
    rfl
    exact Subtype.coe_injective.comp (Basis.injective _)
  rw [finrank_eq_card_chooseBasisIndex]
    -- We prove that `finrank ℤ L ≤ finrank K E` and `finrank K E ≤ finrank ℤ L`
  refine le_antisymm ?_ ?_
  · -- To prove that `finrank ℤ L ≤ finrank K E`, we proceed by contradiction and prove that, in
    -- this case, there is a ℤ-relation between the vectors of `b`
    obtain ⟨t, ⟨ht_inc, ⟨ht_span, ht_lin⟩⟩⟩ := exists_linearIndependent K (Set.range b)
    -- `e` is a `K`-basis of `E` formed of vectors of `b`
    let e : Basis t K E := Basis.mk ht_lin (by simp [ht_span, h_spanE])
    have : Fintype t := Set.Finite.fintype ((Set.range b).toFinite.subset ht_inc)
    have h : LinearIndependent ℤ (fun x : (Set.range b) => (x : E)) := by
      rwa [linearIndependent_subtype_range (Subtype.coe_injective.comp b₀.injective)]
    contrapose! h
    -- Since `finrank ℤ L > finrank K E`, there exists a vector `v ∈ b` with `v ∉ e`
    obtain ⟨v, hv⟩ : (Set.range b \ Set.range e).Nonempty := by
      rw [Basis.coe_mk, Subtype.range_coe_subtype, Set.setOf_mem_eq, ← Set.toFinset_nonempty]
      contrapose h
      rw [Finset.not_nonempty_iff_eq_empty, Set.toFinset_diff,
        Finset.sdiff_eq_empty_iff_subset] at h
      replace h := Finset.card_le_of_subset h
      rwa [not_lt, h_card, ← topEquiv.finrank_eq, ← h_spanE, ← ht_span,
        finrank_span_set_eq_card _ ht_lin]
    -- Assume that `e ∪ {v}` is not `ℤ`-linear independent then we get the contradiction
    suffices ¬ LinearIndependent ℤ (fun x : ↥(insert v (Set.range e)) => (x : E)) by
      contrapose! this
      refine LinearIndependent.mono ?_ this
      exact Set.insert_subset (Set.mem_of_mem_diff hv) (by simp [ht_inc])
    -- We prove finally that `e ∪ {v}` is not ℤ-linear independent or, equivalently,
    -- not ℚ-linear independent by showing that `v ∈ span ℚ e`.
    rw [LinearIndependent.iff_fractionRing ℤ ℚ,
      (linearIndependent_insert (Set.not_mem_of_mem_diff hv)),  not_and, not_not]
    intro _
    -- But that follows from the fact that there exist `n, m : ℕ`, `n ≠ m`
    -- such that `(n - m) • v ∈ span ℤ e` which is true since `n ↦ Zspan.fract e (n • v)`
    -- takes value into the finite set `fundamentalDomain e ∩ L`
    have h_mapsto : Set.MapsTo (fun n : ℤ => Zspan.fract e (n • v)) Set.univ
        (Metric.closedBall 0 (∑ i, ‖e i‖) ∩ (L : Set E)) := by
      rw [Set.mapsTo_inter, Set.maps_univ_to, Set.maps_univ_to]
      refine ⟨fun _ =>  mem_closedBall_zero_iff.mpr (Zspan.norm_fract_le e _), fun _ => ?_⟩
      · change _ ∈ AddSubgroup.toIntSubmodule L
        rw [← h_spanL]
        refine sub_mem ?_ ?_
        · exact zsmul_mem (subset_span (Set.diff_subset _ _ hv)) _
        · exact span_mono (by simp [ht_inc]) (coe_mem _)
    have h_finite : Set.Finite (Metric.closedBall 0 (∑ i, ‖e i‖) ∩ (L : Set E)) :=
      Metric.finite_isBounded_inter_isClosed Metric.isBounded_closedBall inferInstance
    obtain ⟨n, -, m, -, h_neq, h_eq⟩ := Set.Infinite.exists_ne_map_eq_of_mapsTo
      Set.infinite_univ h_mapsto h_finite
    have h_nz : (-n + m : ℚ) ≠ 0 := by
      rwa [Ne.def, add_eq_zero_iff_eq_neg.not, neg_inj, Rat.coe_int_inj, ← Ne.def]
    apply (smul_mem_iff _ h_nz).mp
    refine span_subset_span ℤ ℚ _ ?_
    rwa [add_smul, neg_smul, SetLike.mem_coe, ← Zspan.fract_eq_fract, ← zsmul_eq_smul_cast ℚ,
      ← zsmul_eq_smul_cast ℚ]
  · -- To prove that `finrank K E ≤ finrank ℤ L`, we use the fact `b` generates `E` over `K`
    -- and thus `finrank K E ≤ card b = finrank ℤ L`
    

Proof state:
case refine_2
K : Type u_1
inst✝⁷ : NormedLinearOrderedField K
inst✝⁶ : HasSolidNorm K
inst✝⁵ : FloorRing K
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace K E
inst✝² : FiniteDimensional K E
inst✝¹ : ProperSpace E
L : AddSubgroup E
inst✝ : DiscreteTopology ↥L
hs : span K ↑L = ⊤
this✝² : Module.Finite ℤ ↥L
this✝¹ : Module.Free ℤ ↥L
this✝ : Module ℚ E
b₀ : Basis (Module.Free.ChooseBasisIndex ℤ ↥L) ℤ ↥L := Module.Free.chooseBasis ℤ ↥L
b : Module.Free.ChooseBasisIndex ℤ ↥L → E := Subtype.val ∘ ⇑b₀
this : LinearIndependent ℤ b
h_spanL : span ℤ (Set.range b) = AddSubgroup.toIntSubmodule L
h_spanE : span K (Set.range b) = ⊤
h_card : Fintype.card (Module.Free.ChooseBasisIndex ℤ ↥L) = Finset.card (Set.toFinset (Set.range b))
⊢ finrank K E ≤ Fintype.card (Module.Free.ChooseBasisIndex ℤ ↥L)

Next tactic:
rw [← topEquiv.finrank_eq, ← h_spanE]