-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Topology.Basic (goal)
Goal:
R : Type u
inst : Ring R
inst_1 : TopologicalSpace R
X : TopModuleCat R
⊢ LE.le (α :=
    TopologicalSpace
      (↑(toModuleCat
            (Functor.obj
              (forget₂ (TopModuleCat R) (FC := fun (x1 x2 : TopModuleCat R) ↦
                  (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
                  fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v))) TopCat (FD := fun (X Y : TopCat) ↦
                  C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CD := TopCat.carrier) ⋙
                free R)
              X)) :
        Type (max u v)))
    (topologicalSpace
      (Functor.obj
        (forget₂ (TopModuleCat R) (FC := fun (x1 x2 : TopModuleCat R) ↦
            (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
            fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v))) TopCat (FD := fun (X Y : TopCat) ↦
            C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CD := TopCat.carrier) ⋙
          free R)
        X))
    (TopologicalSpace.induced
      (AddHom.toFun
        (LinearMap.toAddHom (R := R) (S := R) (σ := RingHom.id R)
          ((Finsupp.lift (↑(Functor.obj (𝟭 (TopModuleCat R)) X).toModuleCat : Type (max u v)) R
                (↑X.toModuleCat : Type (max u v)) :
              (a : (↑X.toModuleCat : Type (max u v)) → (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat : Type (max u v))) →
                ((↑X.toModuleCat : Type (max u v)) →₀ R) →ₗ[R]
                  (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat : Type (max u v)))
            id)))
      (Functor.obj (𝟭 (TopModuleCat R)) X).topologicalSpace)
have refine : LE.le (α :=
  TopologicalSpace
    ((↑(Functor.obj
            (forget₂ (TopModuleCat R) (FC := fun (x1 x2 : TopModuleCat R) ↦
              (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
              fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v))) TopCat (FD := fun (X Y : TopCat) ↦
              C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CD := TopCat.carrier))
            X) :
        Type (max v u)) →₀
      R))
  (TopologicalSpace.coinduced
    (fun
        (x :
          (↑(Functor.obj
                (forget₂ (TopModuleCat R) (FC := fun (x1 x2 : TopModuleCat R) ↦
                  (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC :=
                  fun (X : TopModuleCat R) ↦ (↑X.toModuleCat : Type (max u v))) TopCat (FD := fun (X Y : TopCat) ↦
                  C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CD := TopCat.carrier))
                X) :
            Type (max v u))) ↦
      Finsupp.single x 1)
    (Functor.obj
        (forget₂ (TopModuleCat R) (FC := fun (x1 x2 : TopModuleCat R) ↦
          (↑x1.toModuleCat : Type (max u v)) →L[R] (↑x2.toModuleCat : Type (max u v))) (CC := fun (X : TopModuleCat R) ↦
          (↑X.toModuleCat : Type (max u v))) TopCat (FD := fun (X Y : TopCat) ↦
          C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CD := TopCat.carrier))
        X).str)
  (TopologicalSpace.induced
    (AddHom.toFun
      (LinearMap.toAddHom (R := R) (S := R) (σ := RingHom.id R)
        ((Finsupp.lift (↑(Functor.obj (𝟭 (TopModuleCat R)) X).toModuleCat : Type (max u v)) R
              (↑X.toModuleCat : Type (max u v)) :
            (a : (↑X.toModuleCat : Type (max u v)) → (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat : Type (max u v))) →
              ((↑X.toModuleCat : Type (max u v)) →₀ R) →ₗ[R]
                (↑((𝟭 (TopModuleCat R)).obj X).toModuleCat : Type (max u v)))
          id)))
    (Functor.obj (𝟭 (TopModuleCat R)) X).topologicalSpace)
Tactic: refine sInf_le ⟨continuousSMul_induced (Finsupp.lift _ R X id),
        continuousAdd_induced (Finsupp.lift _ R X id), ?_⟩