-- Module: Mathlib.Analysis.Convex.Between
Goal:
R : Type u_1
V : Type u_2
P : Type u_4
inst : Ring R
inst_1 : LinearOrder R
inst_2 : IsStrictOrderedRing R
inst_3 : AddCommGroup V
inst_4 : Module R V
inst_5 : AddTorsor V P
inst_6 : IsTorsionFree R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Eq (α := Finset (Fin 3)) Finset.univ {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃) : (a : R) → P) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃) : (a : R) → P) r₂) (t.points i₃)
w : Fin (HAdd.hAdd (α := ℕ) 2 1) → R
hw : Eq.{u_1 + 1} (α := R) (∑ i : Fin (2 + 1), w i) 1
hp :
  Membership.mem (α := P) (γ := AffineSubspace R P) (affineSpan R (Set.range t.points))
    ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P) w : P)
h₁' :
  Membership.mem (α := P) (γ := AffineSubspace R P)
    (affineSpan R
      (insert (α := P)
        ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P)
            (Pi.single (M := fun (a : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ R) i₁ 1) :
          P)
        (singleton (α := P)
          ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P)
              (Finset.affineCombinationLineMapWeights i₂ i₃ r₁) :
            P))))
    ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P) w : P)
h₂' :
  Membership.mem (α := P) (γ := AffineSubspace R P)
    (affineSpan R
      (insert (t.points i₂) (singleton (α := P) ((lineMap (t.points i₁) (t.points i₃) : (a : R) → P) r₂ : P))))
    ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P) w : P)
h₁s :
  Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i₂) : SignType)
    ((SignType.sign : (a : R) → SignType) (w i₃) : SignType)
h₂s :
  Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i₁) : SignType)
    ((SignType.sign : (a : R) → SignType) (w i₃) : SignType)
i : Fin 3
inl :
  ∀ (h₁₂ : i ≠ i₂) (h₁₃ : i ≠ i₃) (h3 : ∀ (i_1 : Fin 3), i_1 = i ∨ i_1 = i₂ ∨ i_1 = i₃)
    (hu : Eq (α := Finset (Fin 3)) Finset.univ {i, i₂, i₃})
    (h₂ : Sbtw R (t.points i) ((lineMap (t.points i) (t.points i₃) : (a : R) → P) r₂) (t.points i₃))
    (h₁' :
      Membership.mem (α := P) (γ := AffineSubspace R P)
        (affineSpan R
          (insert (α := P)
            ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P)
                (Pi.single (M := fun (a : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ R) i 1) :
              P)
            (singleton (α := P)
              ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P)
                  (Finset.affineCombinationLineMapWeights i₂ i₃ r₁) :
                P))))
        ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P) w : P))
    (h₂' :
      Membership.mem (α := P) (γ := AffineSubspace R P)
        (affineSpan R
          (insert (t.points i₂) (singleton (α := P) ((lineMap (t.points i) (t.points i₃) : (a : R) → P) r₂ : P))))
        ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P) w : P))
    (h₂s :
      Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i) : SignType)
        ((SignType.sign : (a : R) → SignType) (w i₃) : SignType)),
    Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i) : SignType)
      ((SignType.sign : (a : R) → SignType) (w i₃) : SignType)
inr :
  ∀ (h₁₂ : i₁ ≠ i) (h₂₃ : i ≠ i₃) (h3 : ∀ (i_1 : Fin 3), i_1 = i₁ ∨ i_1 = i ∨ i_1 = i₃)
    (hu : Eq (α := Finset (Fin 3)) Finset.univ {i₁, i, i₃})
    (h₁ : Sbtw R (t.points i) ((lineMap (t.points i) (t.points i₃) : (a : R) → P) r₁) (t.points i₃))
    (h₁' :
      Membership.mem (α := P) (γ := AffineSubspace R P)
        (affineSpan R
          (insert (α := P)
            ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P)
                (Pi.single (M := fun (a : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ R) i₁ 1) :
              P)
            (singleton (α := P)
              ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P)
                  (Finset.affineCombinationLineMapWeights i i₃ r₁) :
                P))))
        ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P) w : P))
    (h₂' :
      Membership.mem (α := P) (γ := AffineSubspace R P)
        (affineSpan R
          (insert (t.points i) (singleton (α := P) ((lineMap (t.points i₁) (t.points i₃) : (a : R) → P) r₂ : P))))
        ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P) w : P))
    (h₁s :
      Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i) : SignType)
        ((SignType.sign : (a : R) → SignType) (w i₃) : SignType)),
    Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i) : SignType)
      ((SignType.sign : (a : R) → SignType) (w i₃) : SignType)
inr₁ :
  ∀ (h₁₃ : i₁ ≠ i) (h₂₃ : i₂ ≠ i) (h3 : ∀ (i_1 : Fin 3), i_1 = i₁ ∨ i_1 = i₂ ∨ i_1 = i)
    (hu : Eq (α := Finset (Fin 3)) Finset.univ {i₁, i₂, i})
    (h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i) : (a : R) → P) r₁) (t.points i))
    (h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i) : (a : R) → P) r₂) (t.points i))
    (h₁' :
      Membership.mem (α := P) (γ := AffineSubspace R P)
        (affineSpan R
          (insert (α := P)
            ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P)
                (Pi.single (M := fun (a : Fin (HAdd.hAdd (α := ℕ) 2 1)) ↦ R) i₁ 1) :
              P)
            (singleton (α := P)
              ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P)
                  (Finset.affineCombinationLineMapWeights i₂ i r₁) :
                P))))
        ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P) w : P))
    (h₂' :
      Membership.mem (α := P) (γ := AffineSubspace R P)
        (affineSpan R
          (insert (t.points i₂) (singleton (α := P) ((lineMap (t.points i₁) (t.points i) : (a : R) → P) r₂ : P))))
        ((Finset.affineCombination R Finset.univ t.points : (a : Fin (2 + 1) → R) → P) w : P))
    (h₁s :
      Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i₂) : SignType)
        ((SignType.sign : (a : R) → SignType) (w i) : SignType))
    (h₂s :
      Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i₁) : SignType)
        ((SignType.sign : (a : R) → SignType) (w i) : SignType)),
    Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i) : SignType)
      ((SignType.sign : (a : R) → SignType) (w i) : SignType)
⊢ Eq (α := SignType) ((SignType.sign : (a : R) → SignType) (w i) : SignType)
    ((SignType.sign : (a : R) → SignType) (w i₃) : SignType)

Tactic:
rcases h3 i with (rfl | rfl | rfl)

Premises used: rfl