-- Module: Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace
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)
a b :
  (↑((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)
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)) ⋯)))))
              (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))
        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)) ⋯)))))
                (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))
        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)) ⋯)))))
                (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))
refine_1 :
  ∀ (y : (↑(↑(ChartedSpace.locallyRingedSpace IM ↥U).toPresheafedSpace : TopCat) : Type u))
    (hy : y ∈ (↑V : Set (↑(↑(ChartedSpace.locallyRingedSpace IM ↥U).toPresheafedSpace : TopCat) : Type u))),
    (↑a :
          (x : ↥(Opposite.unop (Opposite.op (⋯.functor.obj V)))) →
            (fun (x : (↑(TopCat.of M) : Type u)) ↦ 𝕜) (↑x : (↑(TopCat.of M) : Type u)))
        (Subtype.mk (p := fun (x : (↑(TopCat.of M) : Type u)) ↦
          Membership.mem (γ := Opens (↑(TopCat.of M) : Type u))
            (Opposite.unop
              (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)))
            x)
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                (X := (↑(ChartedSpace.locallyRingedSpace IM ↥U).toPresheafedSpace : TopCat))
                (PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))) :
              (a : (↑(↑(ChartedSpace.locallyRingedSpace IM ↥U).toPresheafedSpace : TopCat) : Type u)) →
                (↑(↑(ChartedSpace.locallyRingedSpace IM M).toPresheafedSpace : TopCat) : Type u))
            y)
          ⋯) =
      (↑b :
          (x : ↥(Opposite.unop (Opposite.op (⋯.functor.obj V)))) →
            (fun (x : (↑(TopCat.of M) : Type u)) ↦ 𝕜) (↑x : (↑(TopCat.of M) : Type u)))
        (Subtype.mk (p := fun (x : (↑(TopCat.of M) : Type u)) ↦
          Membership.mem (γ := Opens (↑(TopCat.of M) : Type u))
            (Opposite.unop
              (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)))
            x)
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                (X := (↑(ChartedSpace.locallyRingedSpace IM ↥U).toPresheafedSpace : TopCat))
                (PresheafedSpace.Hom.base
                  (InducedCategory.Hom.hom
                    (LocallyRingedSpace.Hom.toShHom
                      (ChartedSpace.locallyRingedSpaceMap (Subtype.val (p := fun (x : M) ↦ x ∈ U)) ⋯)))) :
              (a : (↑(↑(ChartedSpace.locallyRingedSpace IM ↥U).toPresheafedSpace : TopCat) : Type u)) →
                (↑(↑(ChartedSpace.locallyRingedSpace IM M).toPresheafedSpace : TopCat) : Type u))
            y)
          ⋯)
⊢ (↑a :
      (x : ↥(Opposite.unop (Opposite.op (⋯.functor.obj V)))) →
        (fun (x : (↑(TopCat.of M) : Type u)) ↦ 𝕜) (↑x : (↑(TopCat.of M) : Type u))) =
    (↑b :
      (x : ↥(Opposite.unop (Opposite.op (⋯.functor.obj V)))) →
        (fun (x : (↑(TopCat.of M) : Type u)) ↦ 𝕜) (↑x : (↑(TopCat.of M) : Type u)))

Tactic:
ext ⟨x, y, hy, rfl⟩

Premises used: rfl