-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Products (goal)
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
⊢ Continuous (Y := (↑T : 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))
have mpr : ∀ (x : (↑c.1 : Type u)),
  ContinuousAt (Y := (↑T : 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))
                    sorry).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))
                    sorry).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 sorry) :
            (↑(Functor.obj (pair X Y) { as := WalkingPair.right : Discrete WalkingPair }) : Type u)))
    x
Tactic: rw [continuous_iff_continuousAt]