/-- Suppose lines from two vertices of a triangle to interior points of the opposite side meet at
`p`. Then `p` lies in the interior of the first (and by symmetry the other) segment from a
vertex to the point on the opposite side. -/
theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
    {t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
    (h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
    (h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i₂, p₂]) :
    Sbtw R (t.points i₁) p p₁ := by
  -- Should not be needed; see comments on local instances in `Data.Sign`.
  letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
  have h₁₃ : i₁ ≠ i₃ := by
    rintro rfl
    simp at h₂
  have h₂₃ : i₂ ≠ i₃ := by
    rintro rfl
    simp at h₁
  have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    intro i
    fin_cases i <;> fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃ <;> simp at h₁₂ h₁₃ h₂₃ ⊢
  have hu : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃} := by
    clear h₁ h₂ h₁' h₂'
    -- Porting note: Originally `decide!`
    fin_cases i₁ <;> fin_cases i₂ <;> fin_cases i₃
      <;> simp (config := {decide := true}) at h₁₂ h₁₃ h₂₃ ⊢
  have hp : p ∈ affineSpan R (Set.range t.points) := by
    have hle : line[R, t.points i₁, p₁] ≤ affineSpan R (Set.range t.points) := by
      refine' affineSpan_pair_le_of_mem_of_mem (mem_affineSpan R (Set.mem_range_self _)) _
      have hle : line[R, t.points i₂, t.points i₃] ≤ affineSpan R (Set.range t.points) := by
        refine' affineSpan_mono R _
        simp [Set.insert_subset_iff]
      rw [AffineSubspace.le_def'] at hle
      exact hle _ h₁.wbtw.mem_affineSpan
    rw [AffineSubspace.le_def'] at hle
    exact hle _ h₁'
  have h₁i := h₁.mem_image_Ioo
  have h₂i := h₂.mem_image_Ioo
  rw [Set.mem_image] at h₁i h₂i
  rcases h₁i with ⟨r₁, ⟨hr₁0, hr₁1⟩, rfl⟩
  rcases h₂i with ⟨r₂, ⟨hr₂0, hr₂1⟩, rfl⟩
  rcases eq_affineCombination_of_mem_affineSpan_of_fintype hp with ⟨w, hw, rfl⟩
  have h₁s :=
    sign_eq_of_affineCombination_mem_affineSpan_single_lineMap t.independent hw (Finset.mem_univ _)
      (Finset.mem_univ _) (Finset.mem_univ _) h₁₂ h₁₃ h₂₃ hr₁0 hr₁1 h₁'
  have h₂s :=
    sign_eq_of_affineCombination_mem_affineSpan_single_lineMap t.independent hw (Finset.mem_univ _)
      (Finset.mem_univ _) (Finset.mem_univ _) h₁₂.symm h₂₃ h₁₃ hr₂0 hr₂1 h₂'
  rw [← Finset.univ.affineCombination_affineCombinationSingleWeights R t.points
      (Finset.mem_univ i₁),
    ← Finset.univ.affineCombination_affineCombinationLineMapWeights t.points (Finset.mem_univ _)
      (Finset.mem_univ _)] at h₁' ⊢
  refine'
    Sbtw.affineCombination_of_mem_affineSpan_pair t.independent hw
      (Finset.univ.sum_affineCombinationSingleWeights R (Finset.mem_univ _))
      (Finset.univ.sum_affineCombinationLineMapWeights (Finset.mem_univ _) (Finset.mem_univ _) _)
      h₁' (Finset.mem_univ i₁) _
  rw [Finset.affineCombinationSingleWeights_apply_self,
    Finset.affineCombinationLineMapWeights_apply_of_ne h₁₂ h₁₃, sbtw_one_zero_iff]
  have hs : ∀ i : Fin 3, SignType.sign (w i) = SignType.sign (w i₃) := by
    intro i
    rcases h3 i with (rfl | rfl | rfl)
    · exact h₂s
    · exact h₁s
    · rfl
  have hss : SignType.sign (∑ i, w i) = 1 := by simp [hw]
  have hs' := sign_sum Finset.univ_nonempty (SignType.sign (w i₃)) fun i _ => hs i
  rw [hs'] at hss
  simp_rw [hss, sign_eq_one_iff] at hs
  refine' ⟨hs i₁, _⟩
  rw [hu] at hw
  rw [Finset.sum_insert, Finset.sum_insert, Finset.sum_singleton] at hw
  · by_contra hle
    rw [not_lt] at hle
    exact (hle.trans_lt (lt_add_of_pos_right _ (Left.add_pos (hs i₂) (hs i₃)))).ne' hw
  · simpa using h₂₃
  · 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
R : Type u_1
V : Type u_2
V' : Type u_3
P : Type u_4
P' : Type u_5
inst✝⁴ : LinearOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : NoZeroSMulDivisors R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
this : DecidableRel fun x x_1 => x < x_1 := LinearOrderedRing.decidableLT
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ :
  Sbtw R (Affine.Simplex.points t i₂) ((lineMap (Affine.Simplex.points t i₂) (Affine.Simplex.points t i₃)) r₁)
    (Affine.Simplex.points t i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ :
  Sbtw R (Affine.Simplex.points t i₁) ((lineMap (Affine.Simplex.points t i₁) (Affine.Simplex.points t i₃)) r₂)
    (Affine.Simplex.points t i₃)
w : Fin (2 + 1) → R
hw : ∑ i in {i₁, i₂, i₃}, w i = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    affineSpan R
      {(Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationSingleWeights R i₁),
        (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)}
h₂' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    affineSpan R {Affine.Simplex.points t i₂, (lineMap (Affine.Simplex.points t i₁) (Affine.Simplex.points t i₃)) r₂}
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i : Fin 3, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
⊢ i₁ ∉ {i₂, i₃}

Next tactic:
simpa [not_or] using ⟨h₁₂, h₁₃⟩