-- Lean 4 proof step from Mathlib.RingTheory.Etale.QuasiFinite (goal)
Goal:
R : Type u_2
S : Type u_3
R' : Type u_4
R'' : Type u_5
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : FiniteType R S
inst_4 : CommRing R'
inst_5 : Algebra R R'
inst_6 : CommRing R''
inst_7 : Algebra R R''
inst_8 : Algebra R'' S
inst_9 : Algebra.IsIntegral R R''
inst_10 : IsScalarTower R R'' S
q : Ideal S
P : Ideal R'
inst_11 : P.IsPrime
e : R' ⊗[R] S
e₀ : R' ⊗[R] R''
he₀ : IsIdempotentElem e₀
he₀e :
  ((TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) : R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S) :
        (a : R' ⊗[R] R'') → R' ⊗[R] S)
      e₀ =
    e
P' : Ideal (R' ⊗[R] S)
hP'q : Ideal.comap (F := S →+* R' ⊗[R] S) (AlgHom.toRingHom TensorProduct.includeRight) P' = q
H :
  ∀ (P'' : Ideal (R' ⊗[R] R'')),
    P''.IsPrime →
      P''.LiesOver P →
        e₀ ∉ P'' →
          P'' =
            Ideal.comap (F := R' ⊗[R] R'' →+* R' ⊗[R] S)
              (AlgHom.toRingHom
                (TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) : R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S))
              P'
g : R''
hgq : ((algebraMap R'' S : (a : R'') → S) g : S) ∉ q
hg :
  Function.Surjective.{u_5 + 1, u_3 + 1} (α := Localization.Away g) (β :=
    Localization.Away (M := S) ((algebraMap R'' S : (a : R'') → S) g : S))
    (⇑(Localization.awayMap (algebraMap R'' S) g) :
      (a : Localization.Away g) → Localization.Away ((algebraMap R'' S : (a : R'') → S) g))
this : Algebra.IsIntegral R' (Localization.Away e₀)
a : R' ⊗[R] S
H₁ :
  ∃ (b : R' ⊗[R] R'') (m : ℕ),
    ((AlgHom.toRingHom
              (TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) : R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S) :
            R' ⊗[R] R'' →+* R' ⊗[R] S) :
          (a : R' ⊗[R] R'') → R' ⊗[R] S)
        b =
      HPow.hPow (α := R' ⊗[R] S)
          (((AlgHom.toRingHom
                    (TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) :
                      R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S) :
                  R' ⊗[R] R'' →+* R' ⊗[R] S) :
                (a : R' ⊗[R] R'') → R' ⊗[R] S)
              (e₀ * 1 ⊗ₜ[R] g) :
            R' ⊗[R] S)
          m *
        0
⊢ ∃ (b : R' ⊗[R] R'') (m : ℕ),
    ((AlgHom.toRingHom
              (TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) : R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S) :
            R' ⊗[R] R'' →+* R' ⊗[R] S) :
          (a : R' ⊗[R] R'') → R' ⊗[R] S)
        b =
      HPow.hPow (α := R' ⊗[R] S)
          (((AlgHom.toRingHom
                    (TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) :
                      R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S) :
                  R' ⊗[R] R'' →+* R' ⊗[R] S) :
                (a : R' ⊗[R] R'') → R' ⊗[R] S)
              (e₀ * 1 ⊗ₜ[R] g) :
            R' ⊗[R] S)
          m *
        a
have H₂ : ∀ (a : R') (b : S),
  ∃ (b_1 : R' ⊗[R] R'') (m : ℕ),
    ((AlgHom.toRingHom
              (TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) : R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S) :
            R' ⊗[R] R'' →+* R' ⊗[R] S) :
          (a : R' ⊗[R] R'') → R' ⊗[R] S)
        b_1 =
      HPow.hPow (α := R' ⊗[R] S)
          (((AlgHom.toRingHom
                    (TensorProduct.map (AlgHom.id R' R') (IsScalarTower.toAlgHom R R'' S) :
                      R' ⊗[R] R'' →ₐ[R'] R' ⊗[R] S) :
                  R' ⊗[R] R'' →+* R' ⊗[R] S) :
                (a : R' ⊗[R] R'') → R' ⊗[R] S)
              (e₀ * 1 ⊗ₜ[R] g) :
            R' ⊗[R] S)
          m *
        a ⊗ₜ[R] b
Tactic: induction a with
    | zero => use 0; simp
    | tmul a b =>
      obtain ⟨b', m, e : _ = _⟩ := Localization.awayMap_surjective_iff.mp hg b
      refine ⟨e₀ ^ m * a ⊗ₜ b', m, ?_⟩
      simp [e, mul_pow, mul_assoc]
    | add a b ha hb =>
      obtain ⟨xa, ma, ea⟩ := ha
      obtain ⟨xb, mb, eb⟩ := hb
      refine ⟨(e₀ * 1 ⊗ₜ[R] g) ^ mb * xa + (e₀ * 1 ⊗ₜ[R] g) ^ ma * xb, ma + mb, ?_⟩
      dsimp at ea eb ⊢
      simp only [map_add, map_mul, map_pow, Algebra.TensorProduct.map_tmul, AlgHom.coe_id, id_eq,
        ea, eb, pow_add]
      ring
Cleared hypotheses: a