-- Module: Mathlib.RingTheory.Polynomial.UniversalFactorizationRing
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
n m k : ℕ
hn : n = m + k
i : Fin n
hX :
  Eq (α := (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)[X])
    (Polynomial.map (R := MvPolynomial (Fin m) R)
      (RingHom.comp
        (↑Algebra.TensorProduct.includeLeft :
          MvPolynomial (Fin m) S →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        (map (algebraMap R S)))
      (freeMonic R m))
    (Polynomial.map
      (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
        (↑(Algebra.TensorProduct.lift (AlgHom.comp Algebra.TensorProduct.includeLeft (mapAlgHom (Algebra.ofId R S)))
                (AlgHom.comp (AlgHom.restrictScalars R Algebra.TensorProduct.includeRight)
                  (mapAlgHom (Algebra.ofId R S)))
                ⋯ :
              AlgHom.{u_1, u_1, u_2} R (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
                (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)) :
          MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        (↑Algebra.TensorProduct.includeLeft :
          MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R))
      (freeMonic R m))
hX₁ :
  Eq (α := (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)[X])
    (Polynomial.map (R := MvPolynomial (Fin k) R)
      (RingHom.comp
        (↑Algebra.TensorProduct.includeRight :
          MvPolynomial (Fin k) S →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        (map (algebraMap R S)))
      (freeMonic R k))
    (Polynomial.map
      (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
        (↑(Algebra.TensorProduct.lift (AlgHom.comp Algebra.TensorProduct.includeLeft (mapAlgHom (Algebra.ofId R S)))
                (AlgHom.comp (AlgHom.restrictScalars R Algebra.TensorProduct.includeRight)
                  (mapAlgHom (Algebra.ofId R S)))
                ⋯ :
              AlgHom.{u_1, u_1, u_2} R (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
                (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)) :
          MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        (↑Algebra.TensorProduct.includeRight :
          MvPolynomial (Fin k) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R))
      (freeMonic R k))
⊢ (HMul.hMul (α := (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)[X])
          (Polynomial.map (R := MvPolynomial (Fin m) R)
            (RingHom.comp
              (↑Algebra.TensorProduct.includeLeft :
                MvPolynomial (Fin m) S →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
              (map (algebraMap R S)))
            (freeMonic R m))
          (Polynomial.map (R := MvPolynomial (Fin k) R)
            (RingHom.comp
              (↑Algebra.TensorProduct.includeRight :
                MvPolynomial (Fin k) S →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
              (map (algebraMap R S)))
            (freeMonic R k))).coeff
      (↑i : ℕ) =
    (HMul.hMul (α := (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)[X])
          (Polynomial.map
            (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
              (↑(Algebra.TensorProduct.lift
                      (AlgHom.comp Algebra.TensorProduct.includeLeft (mapAlgHom (Algebra.ofId R S)))
                      (AlgHom.comp (AlgHom.restrictScalars R Algebra.TensorProduct.includeRight)
                        (mapAlgHom (Algebra.ofId R S)))
                      ⋯ :
                    AlgHom.{u_1, u_1, u_2} R (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
                      (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)) :
                MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →+*
                  MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
              (↑Algebra.TensorProduct.includeLeft :
                MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R))
            (freeMonic R m))
          (Polynomial.map
            (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
              (↑(Algebra.TensorProduct.lift
                      (AlgHom.comp Algebra.TensorProduct.includeLeft (mapAlgHom (Algebra.ofId R S)))
                      (AlgHom.comp (AlgHom.restrictScalars R Algebra.TensorProduct.includeRight)
                        (mapAlgHom (Algebra.ofId R S)))
                      ⋯ :
                    AlgHom.{u_1, u_1, u_2} R (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
                      (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)) :
                MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →+*
                  MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
              (↑Algebra.TensorProduct.includeRight :
                MvPolynomial (Fin k) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R))
            (freeMonic R k))).coeff
      (↑i : ℕ)

Tactic:
congr 2