theorem exists_of_linearIndependent_of_finite_span {t : Finset V}
    (hs : LinearIndependent K (fun x => x : s → V)) (hst : s ⊆ (span K ↑t : Submodule K V)) :
    ∃ t' : Finset V, ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ t'.card = t.card := by
  classical
  have :
    ∀ t : Finset V,
      ∀ s' : Finset V,
        ↑s' ⊆ s →
          s ∩ ↑t = ∅ →
            s ⊆ (span K ↑(s' ∪ t) : Submodule K V) →
              ∃ t' : Finset V, ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ t'.card = (s' ∪ t).card :=
    fun t =>
    Finset.induction_on t
      (fun s' hs' _ hss' =>
        have : s = ↑s' := eq_of_linearIndependent_of_span_subtype hs hs' <| by simpa using hss'
        ⟨s', by simp [this]⟩)
      fun b₁ t hb₁t ih s' hs' hst hss' =>
      have hb₁s : b₁ ∉ s := fun h => by
        have : b₁ ∈ s ∩ ↑(insert b₁ t) := ⟨h, Finset.mem_insert_self _ _⟩
        rwa [hst] at this
      have hb₁s' : b₁ ∉ s' := fun h => hb₁s <| hs' h
      have hst : s ∩ ↑t = ∅ :=
        eq_empty_of_subset_empty <|
          -- Porting note: `-inter_subset_left, -subset_inter_iff` required.
          Subset.trans
            (by simp [inter_subset_inter, Subset.refl, -inter_subset_left, -subset_inter_iff])
            (le_of_eq hst)
      Classical.by_cases (p := s ⊆ (span K ↑(s' ∪ t) : Submodule K V))
        (fun this =>
          let ⟨u, hust, hsu, Eq⟩ := ih _ hs' hst this
          have hb₁u : b₁ ∉ u := fun h => (hust h).elim hb₁s hb₁t
          ⟨insert b₁ u, by simp [insert_subset_insert hust], Subset.trans hsu (by simp), by
            simp [Eq, hb₁t, hb₁s', hb₁u]⟩)
        fun this =>
        let ⟨b₂, hb₂s, hb₂t⟩ := not_subset.mp this
        have hb₂t' : b₂ ∉ s' ∪ t := fun h => hb₂t <| subset_span h
        have : s ⊆ (span K ↑(insert b₂ s' ∪ t) : Submodule K V) := fun b₃ hb₃ => by
          have : ↑(s' ∪ insert b₁ t) ⊆ insert b₁ (insert b₂ ↑(s' ∪ t) : Set V) := by
            -- Porting note: Too many theorems to be excluded, so
            --               `simp only` is shorter.
            simp only [insert_eq, union_subset_union, Subset.refl,
              subset_union_right, Finset.union_insert, Finset.coe_insert]
          have hb₃ : b₃ ∈ span K (insert b₁ (insert b₂ ↑(s' ∪ t) : Set V)) :=
            span_mono this (hss' hb₃)
          

Proof state:
ι : Type u'
ι' : Type u_1
R : Type u_2
K : Type u_3
M : Type u_4
M' : Type u_5
M'' : Type u_6
V : Type u
V' : Type u_7
inst✝⁴ : DivisionRing K
inst✝³ : AddCommGroup V
inst✝² : AddCommGroup V'
inst✝¹ : Module K V
inst✝ : Module K V'
v : ι → V
s t✝² : Set V
x y z : V
t✝¹ : Finset V
hs : LinearIndependent K fun (x : ↑s) => ↑x
hst✝¹ : s ⊆ ↑(span K ↑t✝¹)
t✝ : Finset V
b₁ : V
t : Finset V
hb₁t : b₁ ∉ t
ih :
  ∀ (s' : Finset V),
    ↑s' ⊆ s →
      s ∩ ↑t = ∅ → s ⊆ ↑(span K ↑(s' ∪ t)) → ∃ t', ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ Finset.card t' = Finset.card (s' ∪ t)
s' : Finset V
hs' : ↑s' ⊆ s
hst✝ : s ∩ ↑(insert b₁ t) = ∅
hss' : s ⊆ ↑(span K ↑(s' ∪ insert b₁ t))
hb₁s : b₁ ∉ s
hb₁s' : b₁ ∉ s'
hst : s ∩ ↑t = ∅
this✝ : ¬s ⊆ ↑(span K ↑(s' ∪ t))
b₂ : V
hb₂s : b₂ ∈ s
hb₂t : b₂ ∉ ↑(span K ↑(s' ∪ t))
hb₂t' : b₂ ∉ s' ∪ t
b₃ : V
hb₃✝ : b₃ ∈ s
this : ↑(s' ∪ insert b₁ t) ⊆ insert b₁ (insert b₂ ↑(s' ∪ t))
hb₃ : b₃ ∈ span K (insert b₁ (insert b₂ ↑(s' ∪ t)))
⊢ b₃ ∈ ↑(span K ↑(insert b₂ s' ∪ t))

Next tactic:
have : s ⊆ (span K (insert b₁ ↑(s' ∪ t)) : Submodule K V) := by
            simpa [insert_eq, -singleton_union, -union_singleton] using hss'