-- Module: Mathlib.RingTheory.DedekindDomain.LinearDisjoint
Goal:
A : Type u_1
B : Type u_2
K : Type u_3
L : Type u_4
inst : CommRing A
inst_1 : Field K
inst_2 : Algebra A K
inst_3 : IsFractionRing A K
inst_4 : CommRing B
inst_5 : Field L
inst_6 : Algebra B L
inst_7 : Algebra A L
inst_8 : Algebra K L
inst_9 : FiniteDimensional K L
inst_10 : IsScalarTower A K L
R₁ : Type u_5
R₂ : Type u_6
inst_11 : CommRing R₁
inst_12 : CommRing R₂
inst_13 : IsDomain R₁
inst_14 : Algebra A R₁
inst_15 : Algebra A R₂
inst_16 : Algebra R₁ B
inst_17 : Algebra R₂ B
inst_18 : Algebra R₁ L
inst_19 : Algebra R₂ L
inst_20 : IsScalarTower A R₁ L
inst_21 : IsScalarTower R₁ B L
inst_22 : IsScalarTower R₂ B L
inst_23 : Module.Finite A R₂
F₁ F₂ : IntermediateField K L
inst_24 : Algebra.{u_5, u_4} R₁ ↥F₁
inst_25 : Algebra.{u_6, u_4} R₂ ↥F₂
inst_26 : IsTorsionFree.{u_5, u_4} R₁ ↥F₁
inst_27 : IsScalarTower A (↥F₂) L
inst_28 : IsScalarTower.{u_1, u_6, u_4} A R₂ ↥F₂
inst_29 : IsScalarTower R₁ (↥F₁) L
inst_30 : IsScalarTower R₂ (↥F₂) L
inst_31 : Algebra.IsSeparable.{u_3, u_4} K ↥F₂
inst_32 : Algebra.IsSeparable (↥F₁) L
inst_33 : IsDomain A
inst_34 : IsDedekindDomain B
inst_35 : IsDedekindDomain R₁
inst_36 : IsDedekindDomain R₂
inst_37 : IsFractionRing B L
inst_38 : IsFractionRing.{u_5, u_4} R₁ ↥F₁
inst_39 : IsFractionRing.{u_6, u_4} R₂ ↥F₂
inst_40 : IsIntegrallyClosed A
inst_41 : IsIntegralClosure B R₁ L
inst_42 : IsTorsionFree R₁ B
inst_43 : IsTorsionFree R₂ B
inst_44 : Algebra A B
inst_45 : Module.Finite A B
inst_46 : IsTorsionFree A B
inst_47 : IsTorsionFree A R₁
inst_48 : IsTorsionFree A R₂
inst_49 : Module.Finite A R₁
inst_50 : Module.Finite R₂ B
inst_51 : IsScalarTower A R₂ B
inst_52 : Module.Finite R₁ B
inst_53 : Algebra.IsSeparable.{u_1, u_2} (FractionRing A) (FractionRing B)
inst_54 : IsScalarTower A R₁ B
inst_55 : Free A R₂
h₁ : F₁.LinearDisjoint ↥F₂
h₂ : F₁.toSubalgebra ⊔ F₂.toSubalgebra = ⊤
h₃ :
  IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
    (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂))
ι : Type u_7
b : Basis.{u_7, u_3, u_4} ι K ↥F₂
hb :
  Eq.{u_4 + 1} (α := Submodule A ↥F₂) (span A (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑b : (a : ι) → ↥F₂)))
    (LinearMap.range.{u_1, u_1, u_6, u_4} (M := R₂) (τ₁₂ := RingHom.id A)
      (↑(IsScalarTower.toAlgHom A R₂ ↥F₂) :
        [inst_56 : FunLike (R₂ →ₐ[A] ↥F₂) R₂ ↥F₂] →
          [SemilinearMapClass (R₂ →ₐ[A] ↥F₂) (RingHom.id A) R₂ ↥F₂] → R₂ →ₗ[A] ↥F₂))
rwa :
  ∀ (h₂ : (F₁ ⊔ F₂).toSubalgebra = ⊤)
    (h₃ :
      IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
        (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂)))
    {ι : Type u_7} (b : Basis.{u_7, u_3, u_4} ι K ↥F₂)
    (hb :
      Eq.{u_4 + 1} (α := Submodule A ↥F₂) (span A (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑b : (a : ι) → ↥F₂)))
        (LinearMap.range.{u_1, u_1, u_6, u_4} (M := R₂) (τ₁₂ := RingHom.id A)
          (↑(IsScalarTower.toAlgHom A R₂ ↥F₂) :
            [inst_56 : FunLike (R₂ →ₐ[A] ↥F₂) R₂ ↥F₂] →
              [SemilinearMapClass (R₂ →ₐ[A] ↥F₂) (RingHom.id A) R₂ ↥F₂] → R₂ →ₗ[A] ↥F₂))),
    F₁ ⊔ F₂ = ⊤
rwa₁ :
  ∀ (h₂ : (F₁ ⊔ F₂).toSubalgebra = IntermediateField.toSubalgebra ⊤)
    (h₃ :
      IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
        (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂)))
    {ι : Type u_7} (b : Basis.{u_7, u_3, u_4} ι K ↥F₂)
    (hb :
      Eq.{u_4 + 1} (α := Submodule A ↥F₂) (span A (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑b : (a : ι) → ↥F₂)))
        (LinearMap.range.{u_1, u_1, u_6, u_4} (M := R₂) (τ₁₂ := RingHom.id A)
          (↑(IsScalarTower.toAlgHom A R₂ ↥F₂) :
            [inst_56 : FunLike (R₂ →ₐ[A] ↥F₂) R₂ ↥F₂] →
              [SemilinearMapClass (R₂ →ₐ[A] ↥F₂) (RingHom.id A) R₂ ↥F₂] → R₂ →ₗ[A] ↥F₂))),
    F₁ ⊔ F₂ = ⊤
rwa₂ rwa₃ rwa₄ rwa₅ :
  ∀ (h₂ : F₁ ⊔ F₂ = ⊤)
    (h₃ :
      IsCoprime.{u_2} (R := Ideal B) (Ideal.map (F := R₁ →+* B) (algebraMap R₁ B) (differentIdeal A R₁))
        (Ideal.map (F := R₂ →+* B) (algebraMap R₂ B) (differentIdeal A R₂)))
    {ι : Type u_7} (b : Basis.{u_7, u_3, u_4} ι K ↥F₂)
    (hb :
      Eq.{u_4 + 1} (α := Submodule A ↥F₂) (span A (Set.range.{u_4, u_7 + 1} (ι := ι) (⇑b : (a : ι) → ↥F₂)))
        (LinearMap.range.{u_1, u_1, u_6, u_4} (M := R₂) (τ₁₂ := RingHom.id A)
          (↑(IsScalarTower.toAlgHom A R₂ ↥F₂) :
            [inst_56 : FunLike (R₂ →ₐ[A] ↥F₂) R₂ ↥F₂] →
              [SemilinearMapClass (R₂ →ₐ[A] ↥F₂) (RingHom.id A) R₂ ↥F₂] → R₂ →ₗ[A] ↥F₂))),
    F₁ ⊔ F₂ = ⊤
⊢ F₁ ⊔ F₂ = ⊤

Tactic:
rwa [← sup_toSubalgebra_of_isAlgebraic_right, ← top_toSubalgebra, toSubalgebra_inj] at h₂

Premises used: IntermediateField.top_toSubalgebra, IntermediateField.toSubalgebra_inj, IntermediateField.sup_toSubalgebra_of_isAlgebraic_right