-- Mathlib/RingTheory/Etale/Field.lean
lemma of_isSeparable [Algebra.IsSeparable K L] : FormallyEtale K L

Proof state:
case comp_bijective
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ ∃ a, (Ideal.Quotient.mkₐ K I).comp a = f

Tactic:
refine ⟨⟨⟨⟨⟨fun x ↦ g x (IntermediateField.AdjoinSimple.gen K x), ?_⟩, ?_⟩, ?_, ?_⟩, ?_⟩, ?_⟩

Result:
case comp_bijective.refine_1
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ (fun x => (g x) (IntermediateField.AdjoinSimple.gen K x)) 1 = 1

case comp_bijective.refine_2
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ ∀ (x y : L),
    { toFun := fun x => (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ?comp_bijective.refine_1 }.toFun
        (x * y) =
      { toFun := fun x => (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ?comp_bijective.refine_1 }.toFun
          x *
        { toFun := fun x => (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ?comp_bijective.refine_1 }.toFun
          y

case comp_bijective.refine_3
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ (↑{ toFun := fun x => (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ?comp_bijective.refine_1,
            map_mul' := ?comp_bijective.refine_2 }).toFun
      0 =
    0

case comp_bijective.refine_4
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ ∀ (x y : L),
    (↑{ toFun := fun x => (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ?comp_bijective.refine_1,
              map_mul' := ?comp_bijective.refine_2 }).toFun
        (x + y) =
      (↑{ toFun := fun x => (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ?comp_bijective.refine_1,
                map_mul' := ?comp_bijective.refine_2 }).toFun
          x +
        (↑{ toFun := fun x => (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ?comp_bijective.refine_1,
                map_mul' := ?comp_bijective.refine_2 }).toFun
          y

case comp_bijective.refine_5
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ ∀ (r : K),
    (↑↑{ toFun := fun x => (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ?comp_bijective.refine_1,
                map_mul' := ?comp_bijective.refine_2, map_zero' := ?comp_bijective.refine_3,
                map_add' := ?comp_bijective.refine_4 }).toFun
        ((algebraMap K L) r) =
      (algebraMap K B) r

case comp_bijective.refine_6
K L : Type u
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
B : Type u
inst✝¹ : CommRing B
inst✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g => (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ (Ideal.Quotient.mkₐ K I).comp
      { toFun := fun x => (g x) (IntermediateField.AdjoinSimple.gen K x), map_one' := ?comp_bijective.refine_1,
        map_mul' := ?comp_bijective.refine_2, map_zero' := ?comp_bijective.refine_3,
        map_add' := ?comp_bijective.refine_4, commutes' := ?comp_bijective.refine_5 } =
    f