-- 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 :
  ∀
    (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
h :
  ∀ (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)))
        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)))
          (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 h) :
                  (↑(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)))
              x)
            x hx) :
        (↑T : Type u))
      (Function.comp (β := (↑X : Type u))
          (⇑(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))
          (⇑⋯.toHomeomorph.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 : (↑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)))
              x)
            x hx) :
        (↑T : Type u))
⊢ Eq (α :=
    (↑(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))) :
        Type u) →
      (↑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)))
        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 h) :
              (↑(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)) :
      (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))) :
            Type u)) →
        (↑T : Type u))
    (Function.comp (β := (↑X : Type u))
        (⇑(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))
        (⇑⋯.toHomeomorph.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)) :
      (↑(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))) :
          Type u) →
        (↑T : Type u))

Tactic:
ext ⟨x, hx⟩