-- 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
a :
  (↑(Functor.obj (Functor.obj (Functor.const (Discrete WalkingPair)) c.pt)
        { as := WalkingPair.left : Discrete WalkingPair }) :
    Type u)
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)))ᶜ
    a
rwa rwa₁ rwa₂ rwa₃ :
  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.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
    a
⊢ 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

Tactic:
rwa [eq_compl_iff_isCompl.mpr h₃.symm]

Premises used: Iff.mpr, eq_compl_iff_isCompl, IsCompl.symm