An exact proof term for the current goal is (andEL (topology_on X Tx) (∀x : set, x ∈ X → ∃N : set, N ∈ Tx ∧ x ∈ N ∧ ∃S : set, finite S ∧ S ⊆ F ∧ ∀A : set, A ∈ F → A ∩ N ≠ Empty → A ∈ S) HLF).