-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.UniversalFactorizationRing (goal)
Goal:
R : Type u_1
inst : CommRing R
n m k : ℕ
hn : n = m + k
h : Nontrivial R
this : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=
  RingHom.toAlgebra (AlgHom.toRingHom (universalFactorizationMap R n m k hn))
⊢ (aeval
          (Algebra.Generators.val
            (Algebra.Presentation.toGenerators
              (Algebra.PreSubmersivePresentation.toPresentation (universalFactorizationMapPresentation R n m k hn)))) :
        (a : MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R)) →
          MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
      (resultant
        (Polynomial.map
          (↑(AlgHom.comp (mapAlgHom (Algebra.ofId R (MvPolynomial (Fin n) R))) (rename Sum.inl)) :
            MvPolynomial (Fin m) R →+* MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R))
          (freeMonic R m))
        (Polynomial.map
          (↑(AlgHom.comp (mapAlgHom (Algebra.ofId R (MvPolynomial (Fin n) R))) (rename Sum.inr)) :
            MvPolynomial (Fin k) R →+* MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R))
          (freeMonic R k))
        m k) =
    resultant (Polynomial.map Algebra.TensorProduct.includeLeftRingHom (freeMonic R m))
      (Polynomial.map
        (↑Algebra.TensorProduct.includeRight :
          MvPolynomial (Fin k) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
        (freeMonic R k))
have inr : resultant
    (Polynomial.map
      (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
        (↑(aeval
              (Algebra.Generators.val
                (Algebra.Presentation.toGenerators
                  (Algebra.PreSubmersivePresentation.toPresentation
                    (universalFactorizationMapPresentation R n m k hn))))) :
          MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
        (↑(AlgHom.comp (mapAlgHom (Algebra.ofId R (MvPolynomial (Fin n) R))) (rename Sum.inl)) :
          MvPolynomial (Fin m) R →+* MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R)))
      (freeMonic R m))
    (Polynomial.map
      (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
        (↑(aeval
              (Algebra.Generators.val
                (Algebra.Presentation.toGenerators
                  (Algebra.PreSubmersivePresentation.toPresentation
                    (universalFactorizationMapPresentation R n m k hn))))) :
          MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
        (↑(AlgHom.comp (mapAlgHom (Algebra.ofId R (MvPolynomial (Fin n) R))) (rename Sum.inr)) :
          MvPolynomial (Fin k) R →+* MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R)))
      (freeMonic R k))
    m k =
  resultant (Polynomial.map Algebra.TensorProduct.includeLeftRingHom (freeMonic R m))
    (Polynomial.map
      (↑Algebra.TensorProduct.includeRight :
        MvPolynomial (Fin k) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
      (freeMonic R k))
Tactic: rw [← (aeval _).coe_toRingHom, ← Polynomial.resultant_map_map,
    Polynomial.map_map, Polynomial.map_map]