-- Lean 4 proof step from Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace (goal)
Goal:
𝕜 : Type u
inst : NontriviallyNormedField 𝕜
EM : Type u_1
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_2
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
M : Type u
inst_4 : TopologicalSpace M
inst_5 : ChartedSpace HM M
EN : Type u_3
inst_6 : NormedAddCommGroup EN
inst_7 : NormedSpace 𝕜 EN
HN : Type u_4
inst_8 : TopologicalSpace HN
IN : ModelWithCorners 𝕜 EN HN
N : Type u
inst_9 : TopologicalSpace N
inst_10 : ChartedSpace HN N
EP : Type u_5
inst_11 : NormedAddCommGroup EP
inst_12 : NormedSpace 𝕜 EP
HP : Type u_6
inst_13 : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst_14 : TopologicalSpace P
inst_15 : ChartedSpace HP P
U : Opens M
V : Opens (↑(↑(ChartedSpace.locallyRingedSpace IM ↥U).toPresheafedSpace : TopCat) : Type u)
⊢ Function.Bijective (α :=
    (↑((ChartedSpace.locallyRingedSpace IM M).presheaf.obj
          (Opposite.op
            (Functor.obj
              (⋯.functor (f :=
                PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
              V))) :
      Type u))
    (β :=
    (↑((Functor.obj
              (TopCat.Presheaf.pushforward CommRingCat
                (PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
              (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
          (Opposite.op
            (Functor.obj
              (⋯.functor (f :=
                PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
              V))) :
      Type u))
    (⇑(ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
          CommRingCat.carrier) (X :=
          (ChartedSpace.locallyRingedSpace IM M).presheaf.obj
            (Opposite.op
              (Functor.obj
                (⋯.functor (f :=
                  PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
                V)))
          (Y :=
          (Functor.obj
                (TopCat.Presheaf.pushforward CommRingCat
                  (PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
                (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
            (Opposite.op
              (Functor.obj
                (⋯.functor (f :=
                  PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
                V)))
          (NatTrans.app
            (PresheafedSpace.Hom.c
              (InducedCategory.Hom.hom
                (LocallyRingedSpace.Hom.toShHom
                  (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯))))
            (Opposite.op
              (Functor.obj
                (⋯.functor (f :=
                  PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))))
                V)))) :
      (a : (↑((ChartedSpace.locallyRingedSpace IM M).presheaf.obj (Opposite.op (⋯.functor.obj V))) : Type u)) →
        (↑(((TopCat.Presheaf.pushforward CommRingCat
                      (LocallyRingedSpace.Hom.toShHom (ChartedSpace.locallyRingedSpaceMap Subtype.val ⋯)).hom.base).obj
                  (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
              (Opposite.op (⋯.functor.obj V))) :
          Type u))
have refine_1 : ∀
  (a b :
    (↑((ChartedSpace.locallyRingedSpace IM M).presheaf.obj
          (Opposite.op
            (Functor.obj
              (sorry.functor (f :=
                PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
              V))) :
      Type u))
  (hab :
    Eq (α :=
      (↑((Functor.obj
                (TopCat.Presheaf.pushforward CommRingCat
                  (PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
            (Opposite.op
              (Functor.obj
                (sorry.functor (f :=
                  PresheafedSpace.Hom.base
                    (InducedCategory.Hom.hom
                      (LocallyRingedSpace.Hom.toShHom
                        (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                V))) :
        Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (X :=
              (ChartedSpace.locallyRingedSpace IM M).presheaf.obj
                (Opposite.op
                  (Functor.obj
                    (sorry.functor (f :=
                      PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom
                          (LocallyRingedSpace.Hom.toShHom
                            (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                    V)))
              (Y :=
              (Functor.obj
                    (TopCat.Presheaf.pushforward CommRingCat
                      (PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom
                          (LocallyRingedSpace.Hom.toShHom
                            (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                    (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
                (Opposite.op
                  (Functor.obj
                    (sorry.functor (f :=
                      PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom
                          (LocallyRingedSpace.Hom.toShHom
                            (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                    V)))
              (NatTrans.app
                (PresheafedSpace.Hom.c
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry))))
                (Opposite.op
                  (Functor.obj
                    (sorry.functor (f :=
                      PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom
                          (LocallyRingedSpace.Hom.toShHom
                            (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                    V))) :
            (a : (↑((ChartedSpace.locallyRingedSpace IM M).presheaf.obj (Opposite.op (sorry.functor.obj V))) : Type u)) →
              (↑(((TopCat.Presheaf.pushforward CommRingCat
                            (LocallyRingedSpace.Hom.toShHom
                                  (ChartedSpace.locallyRingedSpaceMap Subtype.val sorry)).hom.base).obj
                        (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
                    (Opposite.op (sorry.functor.obj V))) :
                Type u))
          a :
        (↑((Functor.obj
                  (TopCat.Presheaf.pushforward CommRingCat
                    (PresheafedSpace.Hom.base
                      (InducedCategory.Hom.hom
                        (LocallyRingedSpace.Hom.toShHom
                          (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                  (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
              (Opposite.op
                (Functor.obj
                  (sorry.functor (f :=
                    PresheafedSpace.Hom.base
                      (InducedCategory.Hom.hom
                        (LocallyRingedSpace.Hom.toShHom
                          (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                  V))) :
          Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (X :=
              (ChartedSpace.locallyRingedSpace IM M).presheaf.obj
                (Opposite.op
                  (Functor.obj
                    (sorry.functor (f :=
                      PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom
                          (LocallyRingedSpace.Hom.toShHom
                            (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                    V)))
              (Y :=
              (Functor.obj
                    (TopCat.Presheaf.pushforward CommRingCat
                      (PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom
                          (LocallyRingedSpace.Hom.toShHom
                            (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                    (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
                (Opposite.op
                  (Functor.obj
                    (sorry.functor (f :=
                      PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom
                          (LocallyRingedSpace.Hom.toShHom
                            (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                    V)))
              (NatTrans.app
                (PresheafedSpace.Hom.c
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry))))
                (Opposite.op
                  (Functor.obj
                    (sorry.functor (f :=
                      PresheafedSpace.Hom.base
                        (InducedCategory.Hom.hom
                          (LocallyRingedSpace.Hom.toShHom
                            (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                    V))) :
            (a : (↑((ChartedSpace.locallyRingedSpace IM M).presheaf.obj (Opposite.op (sorry.functor.obj V))) : Type u)) →
              (↑(((TopCat.Presheaf.pushforward CommRingCat
                            (LocallyRingedSpace.Hom.toShHom
                                  (ChartedSpace.locallyRingedSpaceMap Subtype.val sorry)).hom.base).obj
                        (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
                    (Opposite.op (sorry.functor.obj V))) :
                Type u))
          b :
        (↑((Functor.obj
                  (TopCat.Presheaf.pushforward CommRingCat
                    (PresheafedSpace.Hom.base
                      (InducedCategory.Hom.hom
                        (LocallyRingedSpace.Hom.toShHom
                          (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                  (ChartedSpace.locallyRingedSpace IM ↥U).presheaf).obj
              (Opposite.op
                (Functor.obj
                  (sorry.functor (f :=
                    PresheafedSpace.Hom.base
                      (InducedCategory.Hom.hom
                        (LocallyRingedSpace.Hom.toShHom
                          (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) sorry)))))
                  V))) :
          Type u))),
  (↑a :
      (x : ↥(Opposite.unop (Opposite.op (sorry.functor.obj V)))) →
        (fun (x : (↑(TopCat.of M) : Type u)) ↦ 𝕜) (↑x : (↑(TopCat.of M) : Type u))) =
    (↑b :
      (x : ↥(Opposite.unop (Opposite.op (sorry.functor.obj V)))) →
        (fun (x : (↑(TopCat.of M) : Type u)) ↦ 𝕜) (↑x : (↑(TopCat.of M) : Type u)))
Tactic: refine ⟨fun a b hab ↦ Subtype.ext ?_, fun ⟨g, hg⟩ ↦ ?_⟩