We will prove closure_of X Tx Empty ⊆ Empty.

Let x be given.

Assume Hx: x ∈ closure_of X Tx Empty.

We will prove x ∈ Empty.

Apply (SepE X (λx0 ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ Empty ≠ Empty) x Hx) to the current goal.

Assume HxX: x ∈ X.

Assume Hcond: ∀U : set, U ∈ Tx → x ∈ U → U ∩ Empty ≠ Empty.

We prove the intermediate claim HXopen: X ∈ Tx.

An exact proof term for the current goal is (topology_has_X X Tx Htop).

We prove the intermediate claim HXne: X ∩ Empty ≠ Empty.

An exact proof term for the current goal is (Hcond X HXopen HxX).

We prove the intermediate claim HXempty: X ∩ Empty = Empty.

Apply set_ext to the current goal.

Let y be given.

Assume Hy: y ∈ X ∩ Empty.

An exact proof term for the current goal is (binintersectE2 X Empty y Hy).

An exact proof term for the current goal is (Subq_Empty (X ∩ Empty)).

Apply HXne to the current goal.

An exact proof term for the current goal is HXempty.