-- Lean 4 proof step from Mathlib.RingTheory.Polynomial.UniversalFactorizationRing (goal)
Goal:
R : Type u_1
S : Type u_2
T : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : CommRing T
inst_3 : Algebra R S
inst_4 : Algebra R T
n m k : ℕ
hn : n = m + k
p : MonicDegreeEq S n
f :
  { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
    f.comp (universalFactorizationMap R n m k hn) =
      (Equiv.symm (mapEquivMonic R S n) : (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S) p }
⊢ (fun (q : { q : MonicDegreeEq S m × MonicDegreeEq S k // (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑p : S[X]) }) ↦
        (Subtype.mk (p := fun (f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S) ↦
            f.comp (universalFactorizationMap R n m k hn) =
              (Equiv.symm (mapEquivMonic R S n) : (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S) p)
            (Algebra.TensorProduct.lift
              ((Equiv.symm (mapEquivMonic R S m) : (a : MonicDegreeEq S m) → MvPolynomial (Fin m) R →ₐ[R] S)
                (↑q : MonicDegreeEq S m × MonicDegreeEq S k).1)
              ((Equiv.symm (mapEquivMonic R S k) : (a : MonicDegreeEq S k) → MvPolynomial (Fin k) R →ₐ[R] S)
                (↑q : MonicDegreeEq S m × MonicDegreeEq S k).2)
              ⋯)
            ⋯ :
          { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
            f.comp (universalFactorizationMap R n m k hn) =
              (Equiv.symm (mapEquivMonic R S n) : (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S) p }))
      ((fun
            (f :
              { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
                f.comp (universalFactorizationMap R n m k hn) =
                  (Equiv.symm (mapEquivMonic R S n) : (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S) p }) ↦
          (Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
              (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑p : S[X]))
              ((mapEquivMonic R S m : (a : MvPolynomial (Fin m) R →ₐ[R] S) → MonicDegreeEq S m)
                  ((↑f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S).comp
                    Algebra.TensorProduct.includeLeft),
                (mapEquivMonic R S k : (a : MvPolynomial (Fin k) R →ₐ[R] S) → MonicDegreeEq S k)
                  ((↑f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S).comp
                    Algebra.TensorProduct.includeRight))
              ⋯ :
            { q : MonicDegreeEq S m × MonicDegreeEq S k // (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑p : S[X]) }))
        f) =
    f
have a : ∀ (i : Fin m),
  Eq.{u_2 + 1} (α := S)
    ((AlgHom.comp (B := MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)
            (↑((fun (q : { q : MonicDegreeEq S m × MonicDegreeEq S k // (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑p : S[X]) }) ↦
                    (Subtype.mk (p := fun (f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S) ↦
                        f.comp (universalFactorizationMap R n m k hn) =
                          (Equiv.symm (mapEquivMonic R S n) : (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S)
                            p)
                        (Algebra.TensorProduct.lift
                          ((Equiv.symm (mapEquivMonic R S m) : (a : MonicDegreeEq S m) → MvPolynomial (Fin m) R →ₐ[R] S)
                            (↑q : MonicDegreeEq S m × MonicDegreeEq S k).1)
                          ((Equiv.symm (mapEquivMonic R S k) : (a : MonicDegreeEq S k) → MvPolynomial (Fin k) R →ₐ[R] S)
                            (↑q : MonicDegreeEq S m × MonicDegreeEq S k).2)
                          sorry)
                        sorry :
                      { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
                        f.comp (universalFactorizationMap R n m k hn) =
                          (Equiv.symm (mapEquivMonic R S n) : (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S)
                            p }))
                  ((fun
                        (f :
                          { f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S //
                            f.comp (universalFactorizationMap R n m k hn) =
                              (Equiv.symm (mapEquivMonic R S n) :
                                  (a : MonicDegreeEq S n) → MvPolynomial (Fin n) R →ₐ[R] S)
                                p }) ↦
                      (Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
                          (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑p : S[X]))
                          ((mapEquivMonic R S m : (a : MvPolynomial (Fin m) R →ₐ[R] S) → MonicDegreeEq S m)
                              ((↑f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S).comp
                                Algebra.TensorProduct.includeLeft),
                            (mapEquivMonic R S k : (a : MvPolynomial (Fin k) R →ₐ[R] S) → MonicDegreeEq S k)
                              ((↑f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S).comp
                                Algebra.TensorProduct.includeRight))
                          sorry :
                        { q : MonicDegreeEq S m × MonicDegreeEq S k // (↑q.1 : S[X]) * (↑q.2 : S[X]) = (↑p : S[X]) }))
                    f)) :
              MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S)
            Algebra.TensorProduct.includeLeft :
          (a : MvPolynomial (Fin m) R) → S)
        (X i) :
      S)
    (((↑f : MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R →ₐ[R] S).comp Algebra.TensorProduct.includeLeft :
          (a : MvPolynomial (Fin m) R) → S)
        (X i) :
      S)
Tactic: ext