-- Module: Mathlib.Topology.Category.TopCat.Limits.Products
Goal:
X Y : TopCat
c : BinaryCofan X Y
h₁ :
  IsOpenEmbedding (X := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u)) (Y :=
    (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
          { as := WalkingPair.left : Discrete WalkingPair }) :
      Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) c.inl) :
      (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
        (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u))
h₂ :
  IsOpenEmbedding (X := (↑(Functor.obj (pair X Y) { as := WalkingPair.right : Discrete WalkingPair }) : Type u)) (Y :=
    (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
          { as := WalkingPair.right : Discrete WalkingPair }) :
      Type u))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) c.inr) :
      (a : (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) →
        (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.right }) : Type u))
h₃ :
  IsCompl (α :=
    Set
      (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
            { as := WalkingPair.left : Discrete WalkingPair }) :
        Type u))
    (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) c.inl) :
        (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
          (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)))
    (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.right : Discrete WalkingPair }) : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) c.inr) :
        (a : (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) →
          (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.right }) : Type u)))
this_1 :
  ∀
    (x :
      (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
            { as := WalkingPair.left : Discrete WalkingPair }) :
        Type u)),
    Membership.mem (γ :=
        Set
          (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                { as := WalkingPair.left : Discrete WalkingPair }) :
            Type u))
        (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                c.inl) :
            (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
              (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)))
        x ∨
      Membership.mem (γ :=
        Set
          (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                { as := WalkingPair.right : Discrete WalkingPair }) :
            Type u))
        (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.right : Discrete WalkingPair }) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                c.inr) :
            (a : (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) →
              (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.right }) : Type u)))
        x
T : TopCat
f : X ⟶ T
g : Y ⟶ T
this :
  ∀
    a ∉
      range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
              c.inl) :
          (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
            (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)),
    Membership.mem (γ :=
      Set
        (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
              { as := WalkingPair.right : Discrete WalkingPair }) :
          Type u))
      (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.right : Discrete WalkingPair }) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
              c.inr) :
          (a : (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) →
            (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.right }) : Type u)))
      a
e_1 :
  (↑(range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
            (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                  c.inl) :
              (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
                (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)))ᶜ :
      Type u) =
    { a :
      (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
            { as := WalkingPair.left : Discrete WalkingPair }) :
        Type u) //
      a ∉
        range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                c.inl) :
            (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
              (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)) }
x : (↑c.1 : Type u)
hx :
  Membership.mem (γ := Set (↑c.1 : Type u))
    (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
              c.inl) :
          (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
            (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)))ᶜ
    x
⊢ Eq (α := (↑T : Type u))
    (restrict (π := fun (a : (↑c.1 : Type u)) ↦ (↑T : Type u))
        (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
            (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                  c.inl) :
              (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
                (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)))ᶜ
        (fun (x : (↑c.1 : Type u)) ↦
          if h :
              Membership.mem (γ :=
                Set
                  (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                        { as := WalkingPair.left : Discrete WalkingPair }) :
                    Type u))
                (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
                  (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                        carrier) c.inl) :
                    (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
                      (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)))
                x then
            (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) f :
                (a : (↑X : Type u)) → (↑T : Type u))
              (((Equiv.ofInjective
                        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                              carrier) c.inl) :
                          (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
                            (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) :
                              Type u))
                        ⋯).symm :
                    (a :
                        (↑(range
                              (⇑(ConcreteCategory.hom c.inl) :
                                (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
                                  (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) :
                                    Type u))) :
                          Type u)) →
                      (↑((pair X Y).obj { as := WalkingPair.left }) : Type u))
                  (Subtype.mk (p :=
                    fun
                      (x :
                        (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                              { as := WalkingPair.left : Discrete WalkingPair }) :
                          Type u)) ↦
                    Membership.mem (γ :=
                      Set
                        (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                              { as := WalkingPair.left : Discrete WalkingPair }) :
                          Type u))
                      (range (ι :=
                        (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
                        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                              carrier) c.inl) :
                          (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
                            (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) :
                              Type u)))
                      x)
                    x ⋯) :
                (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
          else
            (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) g :
                (a : (↑Y : Type u)) → (↑T : Type u))
              (((Equiv.ofInjective
                        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                              carrier) c.inr) :
                          (a : (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) →
                            (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.right }) :
                              Type u))
                        ⋯).symm :
                    (a :
                        (↑(range
                              (⇑(ConcreteCategory.hom c.inr) :
                                (a : (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) →
                                  (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj
                                        { as := WalkingPair.right }) :
                                    Type u))) :
                          Type u)) →
                      (↑((pair X Y).obj { as := WalkingPair.right }) : Type u))
                  (Subtype.mk (p :=
                    fun
                      (x :
                        (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                              { as := WalkingPair.right : Discrete WalkingPair }) :
                          Type u)) ↦
                    Membership.mem (γ :=
                      Set
                        (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                              { as := WalkingPair.right : Discrete WalkingPair }) :
                          Type u))
                      (range (ι :=
                        (↑(Functor.obj (pair X Y) { as := WalkingPair.right : Discrete WalkingPair }) : Type u))
                        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                              carrier) c.inr) :
                          (a : (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) →
                            (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.right }) :
                              Type u)))
                      x)
                    x ⋯) :
                (↑(Functor.obj (pair X Y) { as := WalkingPair.right : Discrete WalkingPair }) : Type u)))
        (Subtype.mk (p := fun (x : (↑c.1 : Type u)) ↦
          Membership.mem (γ := Set (↑c.1 : Type u))
            (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
                (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                      c.inl) :
                  (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
                    (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)))ᶜ
            x)
          x hx) :
      (↑T : Type u))
    (Function.comp (β := (↑Y : Type u))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) g) :
          (a : (↑Y : Type u)) → (↑T : Type u))
        ((⇑⋯.toHomeomorph.symm :
            (a :
                (↑(range
                      (⇑(ConcreteCategory.hom c.inr) :
                        (a : (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) →
                          (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.right }) :
                            Type u))) :
                  Type u)) →
              (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) ∘
          Subtype.map (p :=
            fun
              (a :
                (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                      { as := WalkingPair.left : Discrete WalkingPair }) :
                  Type u)) ↦
            a ∉
              range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
                (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                      c.inl) :
                  (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
                    (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)))
            (q :=
            fun
              (x :
                (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                      { as := WalkingPair.right : Discrete WalkingPair }) :
                  Type u)) ↦
            Membership.mem (γ :=
              Set
                (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                      { as := WalkingPair.right : Discrete WalkingPair }) :
                  Type u))
              (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.right : Discrete WalkingPair }) : Type u))
                (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                      c.inr) :
                  (a : (↑((pair X Y).obj { as := WalkingPair.right }) : Type u)) →
                    (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.right }) : Type u)))
              x)
            (fun
                (a :
                  (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
                        { as := WalkingPair.left : Discrete WalkingPair }) :
                    Type u)) ↦
              a)
            this)
        (Subtype.mk (p := fun (x : (↑c.1 : Type u)) ↦
          Membership.mem (γ := Set (↑c.1 : Type u))
            (range (ι := (↑(Functor.obj (pair X Y) { as := WalkingPair.left : Discrete WalkingPair }) : Type u))
                (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                      c.inl) :
                  (a : (↑((pair X Y).obj { as := WalkingPair.left }) : Type u)) →
                    (↑(((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }) : Type u)))ᶜ
            x)
          x hx) :
      (↑T : Type u))

Tactic:
exact dif_neg hx

Premises used: dif_neg