-- 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 :
  (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)
⊢ Eq (α := MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
    ((aeval fun (x : Fin n) ↦
            (HMul.hMul (α := (MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)[X])
                  (Polynomial.map
                    (↑Algebra.TensorProduct.includeLeft :
                      MvPolynomial (Fin m) S →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
                    (freeMonic S m))
                  (Polynomial.map
                    (↑Algebra.TensorProduct.includeRight :
                      MvPolynomial (Fin k) S →+* MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
                    (freeMonic S k))).coeff
              (↑x : ℕ) :
          (a : MvPolynomial (Fin n) S) → MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        (((map (algebraMap R S) : RingHom.{u_1, u_2} (MvPolynomial (Fin n) R) (MvPolynomial (Fin n) S)) :
            (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin n) S)
          (X i)) :
      MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
    (((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)) :
          (a : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) → MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)
        ((aeval fun (x : Fin n) ↦
              (HMul.hMul (α := (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)[X])
                    (Polynomial.map
                      (↑Algebra.TensorProduct.includeLeft :
                        MvPolynomial (Fin m) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
                      (freeMonic R m))
                    (Polynomial.map
                      (↑Algebra.TensorProduct.includeRight :
                        MvPolynomial (Fin k) R →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
                      (freeMonic R k))).coeff
                (↑x : ℕ) :
            (a : MvPolynomial (Fin n) R) → MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
          (X i)) :
      MvPolynomial (Fin m) S ⊗[S] MvPolynomial (Fin k) S)

Tactic:
simp only [map_X, aeval_X, ← AlgHom.coe_toRingHom, ← Polynomial.coeff_map, Polynomial.map_mul,
      Polynomial.map_map, ← map_map_freeMonic (f := algebraMap R S)]

Premises used: AlgHom.coe_toRingHom, Polynomial.map_mul, Polynomial.coeff_map, MvPolynomial.aeval_X, Algebra.algebraMap, Polynomial.map_map, Polynomial.map_map_freeMonic, MvPolynomial.map_X