Assume Hxcl: x ∈ closure_of X Tx A.

We will prove ∃seq : set, sequence_in seq A ∧ converges_to X Tx seq x.

We prove the intermediate claim HTx: topology_on X Tx.

An exact proof term for the current goal is (andEL (topology_on X Tx) (∀x0 : set, x0 ∈ X → countable_basis_at X Tx x0) Hfc).

We prove the intermediate claim HxX: x ∈ X.

An exact proof term for the current goal is (SepE1 X (λx0 : set ⇒ ∀U : set, U ∈ Tx → x0 ∈ U → U ∩ A ≠ Empty) x Hxcl).

We prove the intermediate claim Hbas_all: ∀x0 : set, x0 ∈ X → countable_basis_at X Tx x0.

An exact proof term for the current goal is (andER (topology_on X Tx) (∀x0 : set, x0 ∈ X → countable_basis_at X Tx x0) Hfc).

We prove the intermediate claim Hbas: countable_basis_at X Tx x.

An exact proof term for the current goal is (Hbas_all x HxX).

We prove the intermediate claim HBex: ∃B : set, B ⊆ Tx ∧ countable_set B ∧ (∀b : set, b ∈ B → x ∈ b) ∧ (∀U : set, U ∈ Tx → x ∈ U → ∃b : set, b ∈ B ∧ b ⊆ U).

We prove the intermediate claim Hpack: (topology_on X Tx ∧ x ∈ X) ∧ ∃B : set, B ⊆ Tx ∧ countable_set B ∧ (∀b : set, b ∈ B → x ∈ b) ∧ (∀U : set, U ∈ Tx → x ∈ U → ∃b : set, b ∈ B ∧ b ⊆ U).

An exact proof term for the current goal is Hbas.

An exact proof term for the current goal is (andER (topology_on X Tx ∧ x ∈ X) (∃B : set, B ⊆ Tx ∧ countable_set B ∧ (∀b : set, b ∈ B → x ∈ b) ∧ (∀U : set, U ∈ Tx → x ∈ U → ∃b : set, b ∈ B ∧ b ⊆ U)) Hpack).

We prove the intermediate claim Hclprop: ∀U : set, U ∈ Tx → x ∈ U → U ∩ A ≠ Empty.

An exact proof term for the current goal is (SepE2 X (λx0 : set ⇒ ∀U0 : set, U0 ∈ Tx → x0 ∈ U0 → U0 ∩ A ≠ Empty) x Hxcl).

Apply HBex to the current goal.

Let B be given.

Assume HB: B ⊆ Tx ∧ countable_set B ∧ (∀b : set, b ∈ B → x ∈ b) ∧ (∀U : set, U ∈ Tx → x ∈ U → ∃b : set, b ∈ B ∧ b ⊆ U).

We prove the intermediate claim HBCD: (B ⊆ Tx ∧ countable_set B) ∧ (∀b : set, b ∈ B → x ∈ b).

An exact proof term for the current goal is (andEL ((B ⊆ Tx ∧ countable_set B) ∧ (∀b : set, b ∈ B → x ∈ b)) (∀U : set, U ∈ Tx → x ∈ U → ∃b : set, b ∈ B ∧ b ⊆ U) HB).

We prove the intermediate claim HAB: B ⊆ Tx ∧ countable_set B.

An exact proof term for the current goal is (andEL (B ⊆ Tx ∧ countable_set B) (∀b : set, b ∈ B → x ∈ b) HBCD).

We prove the intermediate claim HBsubTx: B ⊆ Tx.

An exact proof term for the current goal is (andEL (B ⊆ Tx) (countable_set B) HAB).

We prove the intermediate claim HcountB: countable_set B.

An exact proof term for the current goal is (andER (B ⊆ Tx) (countable_set B) HAB).

We prove the intermediate claim Hx_in_B: ∀b : set, b ∈ B → x ∈ b.

An exact proof term for the current goal is (andER (B ⊆ Tx ∧ countable_set B) (∀b : set, b ∈ B → x ∈ b) HBCD).

We prove the intermediate claim HrefB: ∀U : set, U ∈ Tx → x ∈ U → ∃b : set, b ∈ B ∧ b ⊆ U.

An exact proof term for the current goal is (andER ((B ⊆ Tx ∧ countable_set B) ∧ (∀b : set, b ∈ B → x ∈ b)) (∀U : set, U ∈ Tx → x ∈ U → ∃b : set, b ∈ B ∧ b ⊆ U) HB).

Apply HcountB to the current goal.

Let code be given.

Assume Hcodeinj: inj B ω code.

We prove the intermediate claim Hcode_cod: ∀b ∈ B, code b ∈ ω.

An exact proof term for the current goal is (andEL (∀b ∈ B, code b ∈ ω) (∀b1 b2 ∈ B, code b1 = code b2 → b1 = b2) Hcodeinj).

We prove the intermediate claim Hcode_inj: ∀b1 b2 ∈ B, code b1 = code b2 → b1 = b2.

An exact proof term for the current goal is (andER (∀b ∈ B, code b ∈ ω) (∀b1 b2 ∈ B, code b1 = code b2 → b1 = b2) Hcodeinj).

Set F to be the term (λn : set ⇒ {b ∈ B|code b ∈ ordsucc n}).

Set Ubase to be the term (λn : set ⇒ intersection_of_family X (F n)).

We prove the intermediate claim HUinfo: ∀n : set, n ∈ ω → (Ubase n ∈ Tx ∧ x ∈ Ubase n) ∧ Ubase n ∩ A ≠ Empty.

Let n be given.

Assume HnO: n ∈ ω.

We will prove (Ubase n ∈ Tx ∧ x ∈ Ubase n) ∧ Ubase n ∩ A ≠ Empty.

We prove the intermediate claim HnNat: nat_p n.

An exact proof term for the current goal is (omega_nat_p n HnO).

We prove the intermediate claim HsuccNat: nat_p (ordsucc n).

An exact proof term for the current goal is (nat_ordsucc n HnNat).

We prove the intermediate claim HFsubB: F n ⊆ B.

Let b be given.

Assume Hb: b ∈ F n.

An exact proof term for the current goal is (SepE1 B (λb0 : set ⇒ code b0 ∈ ordsucc n) b Hb).

We prove the intermediate claim HFsubTx: F n ⊆ Tx.

Let b be given.

Assume Hb: b ∈ F n.

We prove the intermediate claim HbB: b ∈ B.

An exact proof term for the current goal is (HFsubB b Hb).

An exact proof term for the current goal is (HBsubTx b HbB).

We prove the intermediate claim HFpow: F n ∈ 𝒫 Tx.

An exact proof term for the current goal is (PowerI Tx (F n) HFsubTx).

We prove the intermediate claim HinjFn: inj (F n) (ordsucc n) code.

Apply injI to the current goal.

Let b be given.

Assume Hb: b ∈ F n.

An exact proof term for the current goal is (SepE2 B (λb0 : set ⇒ code b0 ∈ ordsucc n) b Hb).

Let b1 be given.

Assume Hb1: b1 ∈ F n.

Let b2 be given.

Assume Hb2: b2 ∈ F n.

Assume Heq: code b1 = code b2.

We prove the intermediate claim Hb1B: b1 ∈ B.

An exact proof term for the current goal is (HFsubB b1 Hb1).

We prove the intermediate claim Hb2B: b2 ∈ B.

An exact proof term for the current goal is (HFsubB b2 Hb2).

An exact proof term for the current goal is (Hcode_inj b1 Hb1B b2 Hb2B Heq).

We prove the intermediate claim HFin: finite (F n).

An exact proof term for the current goal is (inj_into_finite_nat (F n) (ordsucc n) code HsuccNat HinjFn).

We prove the intermediate claim HUinTx: Ubase n ∈ Tx.

An exact proof term for the current goal is (finite_intersection_in_topology X Tx (F n) HTx HFpow HFin).

We prove the intermediate claim HxU: x ∈ Ubase n.

We will prove x ∈ intersection_of_family X (F n).

We prove the intermediate claim HxAll: ∀U : set, U ∈ F n → x ∈ U.

Let T be given.

Assume HT: T ∈ F n.

We prove the intermediate claim HTB: T ∈ B.

An exact proof term for the current goal is (HFsubB T HT).

An exact proof term for the current goal is (Hx_in_B T HTB).

An exact proof term for the current goal is (SepI X (λz : set ⇒ ∀U : set, U ∈ F n → z ∈ U) x HxX HxAll).

We prove the intermediate claim Hne: Ubase n ∩ A ≠ Empty.

An exact proof term for the current goal is (Hclprop (Ubase n) HUinTx HxU).

Apply andI to the current goal.

An exact proof term for the current goal is HUinTx.

An exact proof term for the current goal is HxU.

An exact proof term for the current goal is Hne.

Set pick to be the term (λn : set ⇒ Eps_i (λy : set ⇒ y ∈ (Ubase n ∩ A))).

Set seq to be the term graph ω pick.

We prove the intermediate claim Hseqdef: seq = graph ω pick.

Use reflexivity.

We use seq to witness the existential quantifier.

Apply andI to the current goal.

We will prove sequence_in seq A.

We will prove function_on seq ω A.

Let n be given.

Assume HnO: n ∈ ω.

We will prove apply_fun seq n ∈ A.

We prove the intermediate claim Happ: apply_fun seq n = pick n.

rewrite the current goal using Hseqdef (from left to right).

An exact proof term for the current goal is (apply_fun_graph ω pick n HnO).

rewrite the current goal using Happ (from left to right).

We prove the intermediate claim HUA: (Ubase n ∈ Tx ∧ x ∈ Ubase n) ∧ Ubase n ∩ A ≠ Empty.

An exact proof term for the current goal is (HUinfo n HnO).

We prove the intermediate claim Hne: Ubase n ∩ A ≠ Empty.

An exact proof term for the current goal is (andER (Ubase n ∈ Tx ∧ x ∈ Ubase n) (Ubase n ∩ A ≠ Empty) HUA).

We prove the intermediate claim Hex: ∃y : set, y ∈ (Ubase n ∩ A).

An exact proof term for the current goal is (nonempty_has_element (Ubase n ∩ A) Hne).

We prove the intermediate claim Hpickin: pick n ∈ (Ubase n ∩ A).

An exact proof term for the current goal is (Eps_i_ex (λy : set ⇒ y ∈ (Ubase n ∩ A)) Hex).

An exact proof term for the current goal is (binintersectE2 (Ubase n) A (pick n) Hpickin).

We will prove converges_to X Tx seq x.

We will prove (topology_on X Tx ∧ sequence_on seq X) ∧ x ∈ X ∧ ∀V : set, V ∈ Tx → x ∈ V → ∃N : set, N ∈ ω ∧ ∀n : set, n ∈ ω → N ⊆ n → apply_fun seq n ∈ V.

Apply andI to the current goal.

An exact proof term for the current goal is HTx.

We will prove sequence_on seq X.

Let n be given.

Assume HnO: n ∈ ω.

We will prove apply_fun seq n ∈ X.

We prove the intermediate claim Happ: apply_fun seq n = pick n.

rewrite the current goal using Hseqdef (from left to right).

An exact proof term for the current goal is (apply_fun_graph ω pick n HnO).

rewrite the current goal using Happ (from left to right).

We prove the intermediate claim HUA: (Ubase n ∈ Tx ∧ x ∈ Ubase n) ∧ Ubase n ∩ A ≠ Empty.

An exact proof term for the current goal is (HUinfo n HnO).

We prove the intermediate claim Hne: Ubase n ∩ A ≠ Empty.

An exact proof term for the current goal is (andER (Ubase n ∈ Tx ∧ x ∈ Ubase n) (Ubase n ∩ A ≠ Empty) HUA).

We prove the intermediate claim Hex: ∃y : set, y ∈ (Ubase n ∩ A).

An exact proof term for the current goal is (nonempty_has_element (Ubase n ∩ A) Hne).

We prove the intermediate claim Hpickin: pick n ∈ (Ubase n ∩ A).

An exact proof term for the current goal is (Eps_i_ex (λy : set ⇒ y ∈ (Ubase n ∩ A)) Hex).

We prove the intermediate claim HpickU: pick n ∈ Ubase n.

An exact proof term for the current goal is (binintersectE1 (Ubase n) A (pick n) Hpickin).

An exact proof term for the current goal is (SepE1 X (λy : set ⇒ ∀T : set, T ∈ (F n) → y ∈ T) (pick n) HpickU).

An exact proof term for the current goal is HxX.

Let V be given.

Assume HV: V ∈ Tx.

Assume HxV: x ∈ V.

We prove the intermediate claim Hexb: ∃b : set, b ∈ B ∧ b ⊆ V.

An exact proof term for the current goal is (HrefB V HV HxV).

Apply Hexb to the current goal.

Let b0 be given.

Assume Hb0pair.

We prove the intermediate claim Hb0B: b0 ∈ B.

An exact proof term for the current goal is (andEL (b0 ∈ B) (b0 ⊆ V) Hb0pair).

We prove the intermediate claim Hb0subV: b0 ⊆ V.

An exact proof term for the current goal is (andER (b0 ∈ B) (b0 ⊆ V) Hb0pair).

Set N to be the term ordsucc (code b0).

We use N to witness the existential quantifier.

Apply andI to the current goal.

We prove the intermediate claim Hcodeb0: code b0 ∈ ω.

An exact proof term for the current goal is (Hcode_cod b0 Hb0B).

An exact proof term for the current goal is (omega_ordsucc (code b0) Hcodeb0).

Let n be given.

Assume HnO: n ∈ ω.

Assume HNsub: N ⊆ n.

We will prove apply_fun seq n ∈ V.

We prove the intermediate claim Happ: apply_fun seq n = pick n.

rewrite the current goal using Hseqdef (from left to right).

An exact proof term for the current goal is (apply_fun_graph ω pick n HnO).

rewrite the current goal using Happ (from left to right).

We prove the intermediate claim HUA: (Ubase n ∈ Tx ∧ x ∈ Ubase n) ∧ Ubase n ∩ A ≠ Empty.

An exact proof term for the current goal is (HUinfo n HnO).

We prove the intermediate claim Hne: Ubase n ∩ A ≠ Empty.

An exact proof term for the current goal is (andER (Ubase n ∈ Tx ∧ x ∈ Ubase n) (Ubase n ∩ A ≠ Empty) HUA).

We prove the intermediate claim Hex: ∃y : set, y ∈ (Ubase n ∩ A).

An exact proof term for the current goal is (nonempty_has_element (Ubase n ∩ A) Hne).

We prove the intermediate claim Hpickin: pick n ∈ (Ubase n ∩ A).

An exact proof term for the current goal is (Eps_i_ex (λy : set ⇒ y ∈ (Ubase n ∩ A)) Hex).

We prove the intermediate claim HpickU: pick n ∈ Ubase n.

An exact proof term for the current goal is (binintersectE1 (Ubase n) A (pick n) Hpickin).

We prove the intermediate claim Hcode_in_N: code b0 ∈ N.

An exact proof term for the current goal is (ordsuccI2 (code b0)).

We prove the intermediate claim Hcode_in_n: code b0 ∈ n.

An exact proof term for the current goal is (HNsub (code b0) Hcode_in_N).

We prove the intermediate claim Hcode_in_succn: code b0 ∈ ordsucc n.

An exact proof term for the current goal is (ordsuccI1 n (code b0) Hcode_in_n).

We prove the intermediate claim Hb0Fn: b0 ∈ F n.

An exact proof term for the current goal is (SepI B (λb : set ⇒ code b ∈ ordsucc n) b0 Hb0B Hcode_in_succn).

We prove the intermediate claim HUnsubb0: Ubase n ⊆ b0.

Let z be given.

Assume Hz: z ∈ Ubase n.

We will prove z ∈ b0.

We prove the intermediate claim Hall: ∀T : set, T ∈ F n → z ∈ T.

An exact proof term for the current goal is (SepE2 X (λz0 : set ⇒ ∀T : set, T ∈ F n → z0 ∈ T) z Hz).

An exact proof term for the current goal is (Hall b0 Hb0Fn).

We prove the intermediate claim Hzb0: pick n ∈ b0.

An exact proof term for the current goal is (HUnsubb0 (pick n) HpickU).

An exact proof term for the current goal is (Hb0subV (pick n) Hzb0).