-- Lean 4 proof step from Mathlib.Algebra.Lie.Weights.RootSystem (goal)
Goal:
K : Type u_1
L : Type u_2
inst : Field K
inst_1 : CharZero K
inst_2 : LieRing L
inst_3 : LieAlgebra K L
inst_4 : IsKilling K L
inst_5 : FiniteDimensional K L
H : LieSubalgebra K L
inst_6 : H.IsCartanSubalgebra
inst_7 : IsTriangularizable K (↥H) L
α_1 β α : Weight K (↥H) L
hα : Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) α
⊢ ∀ (j : ↥(LieSubalgebra.root : Finset (Weight K (↥H) L)))
    (h :
      ¬LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Dual.{u_1, u_2} K ↥H)
          (![(RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
                (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
                  Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α
                  hα),
              (RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H) j] :
            Fin (Nat.succ 0).succ → Dual.{u_1, u_2} K ↥H)),
    Eq (α := Dual.{u_1, u_2} K ↥H)
        ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
          Dual.{u_1, u_2} K ↥H)
        ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H) j : Dual.{u_1, u_2} K ↥H) ∨
      Eq (α := Dual.{u_1, u_2} K ↥H)
        ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
          Dual.{u_1, u_2} K ↥H)
        (-(RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H) j : Dual.{u_1, u_2} K ↥H)
have this : ∀ (β : Weight K (↥H) L)
  (hβ : Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) β)
  (h :
    ¬LinearIndependent (ι := Fin (Nat.succ 0).succ) K (M := Dual.{u_1, u_2} K ↥H)
        (![(RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
              (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
                Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα),
            (RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
              (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
                Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) β
                hβ)] :
          Fin (Nat.succ 0).succ → Dual.{u_1, u_2} K ↥H)),
  Eq (α := Dual.{u_1, u_2} K ↥H)
      ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
          (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
            Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
        Dual.{u_1, u_2} K ↥H)
      ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
          (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
            Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) β hβ) :
        Dual.{u_1, u_2} K ↥H) ∨
    Eq (α := Dual.{u_1, u_2} K ↥H)
      ((RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
          (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
            Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) α hα) :
        Dual.{u_1, u_2} K ↥H)
      (-(RootPairing.root (rootSystem H) : (a : ↥LieSubalgebra.root) → Dual K ↥H)
            (Subtype.mk (p := fun (x : Weight K (↥H) L) ↦
              Membership.mem (γ := Finset (Weight K (↥H) L)) (LieSubalgebra.root : Finset (Weight K (↥H) L)) x) β hβ) :
        Dual.{u_1, u_2} K ↥H)
Tactic: ⟨β, hβ⟩