-- 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 R n
q :
  { q : MonicDegreeEq S m × MonicDegreeEq S k //
    (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) ∧ IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]) }
⊢ (fun (f : 𝓡' →ₐ[R] S) ↦
        (Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
            (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) ∧ IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]))
            (↑((UniversalFactorizationRing.homEquiv S m k hn p :
                    (a : 𝓡 →ₐ[R] S) →
                      { q : MonicDegreeEq S m × MonicDegreeEq S k //
                        (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) })
                  (f.comp (IsScalarTower.toAlgHom R 𝓡 𝓡'))) :
              MonicDegreeEq S m × MonicDegreeEq S k)
            ⋯ :
          { q : MonicDegreeEq S m × MonicDegreeEq S k //
            (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) ∧ IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]) }))
      ((fun
            (q :
              { q : MonicDegreeEq S m × MonicDegreeEq S k //
                (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) ∧
                  IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]) }) ↦
          (IsLocalization.liftAlgHom (R := 𝓡) (M :=
              Submonoid.powers (UniversalFactorizationRing.presentation m k hn p).jacobian) (f :=
              (Equiv.symm (UniversalFactorizationRing.homEquiv S m k hn p) :
                  (a :
                      { q : MonicDegreeEq S m × MonicDegreeEq S k //
                        (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) }) →
                    𝓡 →ₐ[R] S)
                (Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
                  (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]))
                  (↑q : MonicDegreeEq S m × MonicDegreeEq S k) ⋯))
              ⋯ :
            𝓡' →ₐ[R] S))
        q) =
    q
have a : (↑((fun (f : 𝓡' →ₐ[R] S) ↦
          (Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
              (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) ∧ IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]))
              (↑((UniversalFactorizationRing.homEquiv S m k hn p :
                      (a : 𝓡 →ₐ[R] S) →
                        { q : MonicDegreeEq S m × MonicDegreeEq S k //
                          (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) })
                    (f.comp (IsScalarTower.toAlgHom R 𝓡 𝓡'))) :
                MonicDegreeEq S m × MonicDegreeEq S k)
              sorry :
            { q : MonicDegreeEq S m × MonicDegreeEq S k //
              (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) ∧
                IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]) }))
        ((fun
              (q :
                { q : MonicDegreeEq S m × MonicDegreeEq S k //
                  (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) ∧
                    IsCoprime (↑q.1 : S[X]) (↑q.2 : S[X]) }) ↦
            (IsLocalization.liftAlgHom (R := 𝓡) (M :=
                Submonoid.powers (UniversalFactorizationRing.presentation m k hn p).jacobian) (f :=
                (Equiv.symm (UniversalFactorizationRing.homEquiv S m k hn p) :
                    (a :
                        { q : MonicDegreeEq S m × MonicDegreeEq S k //
                          (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]) }) →
                      𝓡 →ₐ[R] S)
                  (Subtype.mk (p := fun (q : MonicDegreeEq S m × MonicDegreeEq S k) ↦
                    (↑q.1 : S[X]) * (↑q.2 : S[X]) = map (algebraMap R S) (↑p : R[X]))
                    (↑q : MonicDegreeEq S m × MonicDegreeEq S k) sorry))
                sorry :
              𝓡' →ₐ[R] S))
          q)) :
    MonicDegreeEq S m × MonicDegreeEq S k) =
  (↑q : MonicDegreeEq S m × MonicDegreeEq S k)
Tactic: apply Subtype.ext