%------------------------------------------------------------------------------
% File : LEO-II---2.3.1
% Problem : TOP004-1 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n018.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Oct 7 11:09:35 AM UTC 2026
% Result : Unsatisfiable 5.14s 1.07s
% Output : CNFRefutation 5.14s
% Verified :
% SZS Type : Refutation
% Derivation depth : 20
% Number of leaves : 114
% Syntax : Number of formulae : 1182 ( 588 unt; 0 typ; 0 def)
% Number of atoms : 8132 (2061 equ; 0 cnn)
% Maximal formula atoms : 8 ( 6 avg)
% Number of connectives : 17821 (2008 ~;2828 |; 0 &;12985 @)
% ( 0 <=>; 0 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 2 ( 0 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 62 ( 59 usr; 5 con; 0-5 aty)
% Number of variables : 3979 ( 0 ^;3979 !; 0 ?;3979 :)
% Comments :
%------------------------------------------------------------------------------
thf(tp_basis,type,
basis: $i > $i > $o ).
thf(tp_boundary,type,
boundary: $i > $i > $i > $i ).
thf(tp_closed,type,
closed: $i > $i > $i > $o ).
thf(tp_closure,type,
closure: $i > $i > $i > $i ).
thf(tp_compact_set,type,
compact_set: $i > $i > $i > $o ).
thf(tp_compact_space,type,
compact_space: $i > $i > $o ).
thf(tp_connected_set,type,
connected_set: $i > $i > $i > $o ).
thf(tp_connected_space,type,
connected_space: $i > $i > $o ).
thf(tp_cx,type,
cx: $i ).
thf(tp_disjoint_s,type,
disjoint_s: $i > $i > $o ).
thf(tp_element_of_collection,type,
element_of_collection: $i > $i > $o ).
thf(tp_element_of_set,type,
element_of_set: $i > $i > $o ).
thf(tp_empty_set,type,
empty_set: $i ).
thf(tp_eq_p,type,
eq_p: $i > $i > $o ).
thf(tp_equal_sets,type,
equal_sets: $i > $i > $o ).
thf(tp_f,type,
f: $i ).
thf(tp_f1,type,
f1: $i > $i > $i ).
thf(tp_f10,type,
f10: $i > $i > $i > $i ).
thf(tp_f11,type,
f11: $i > $i > $i ).
thf(tp_f12,type,
f12: $i > $i > $i > $i > $i ).
thf(tp_f13,type,
f13: $i > $i > $i > $i > $i ).
thf(tp_f14,type,
f14: $i > $i > $i > $i > $i ).
thf(tp_f15,type,
f15: $i > $i > $i > $i > $i > $i ).
thf(tp_f16,type,
f16: $i > $i > $i > $i > $i ).
thf(tp_f17,type,
f17: $i > $i > $i > $i > $i ).
thf(tp_f18,type,
f18: $i > $i > $i > $i > $i ).
thf(tp_f19,type,
f19: $i > $i > $i ).
thf(tp_f2,type,
f2: $i > $i > $i ).
thf(tp_f20,type,
f20: $i > $i > $i ).
thf(tp_f21,type,
f21: $i > $i > $i ).
thf(tp_f22,type,
f22: $i > $i > $i ).
thf(tp_f23,type,
f23: $i > $i > $i > $i ).
thf(tp_f24,type,
f24: $i > $i > $i ).
thf(tp_f3,type,
f3: $i > $i > $i ).
thf(tp_f4,type,
f4: $i > $i > $i ).
thf(tp_f5,type,
f5: $i > $i > $i ).
thf(tp_f6,type,
f6: $i > $i > $i > $i > $i > $i ).
thf(tp_f7,type,
f7: $i > $i > $i ).
thf(tp_f8,type,
f8: $i > $i > $i ).
thf(tp_f9,type,
f9: $i > $i > $i ).
thf(tp_finer,type,
finer: $i > $i > $i > $o ).
thf(tp_finite,type,
finite: $i > $o ).
thf(tp_hausdorff,type,
hausdorff: $i > $i > $o ).
thf(tp_interior,type,
interior: $i > $i > $i > $i ).
thf(tp_intersection_of_members,type,
intersection_of_members: $i > $i ).
thf(tp_intersection_of_sets,type,
intersection_of_sets: $i > $i > $i ).
thf(tp_limit_point,type,
limit_point: $i > $i > $i > $i > $o ).
thf(tp_neighborhood,type,
neighborhood: $i > $i > $i > $i > $o ).
thf(tp_open,type,
open: $i > $i > $i > $o ).
thf(tp_open_covering,type,
open_covering: $i > $i > $i > $o ).
thf(tp_relative_complement_sets,type,
relative_complement_sets: $i > $i > $i ).
thf(tp_separation,type,
separation: $i > $i > $i > $i > $o ).
thf(tp_subset_collections,type,
subset_collections: $i > $i > $o ).
thf(tp_subset_sets,type,
subset_sets: $i > $i > $o ).
thf(tp_subspace_topology,type,
subspace_topology: $i > $i > $i > $i ).
thf(tp_top_of_basis,type,
top_of_basis: $i > $i ).
thf(tp_topological_space,type,
topological_space: $i > $i > $o ).
thf(tp_union_of_members,type,
union_of_members: $i > $i ).
thf(tp_union_of_sets,type,
union_of_sets: $i > $i > $i ).
thf(1,axiom,
! [Va: $i,X: $i,Vt: $i] :
( ( compact_set @ Va @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Va @ X )
| ~ ( compact_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_set_109) ).
thf(2,axiom,
! [Va: $i,X: $i,Vt: $i] :
( ~ ( compact_set @ Va @ X @ Vt )
| ( compact_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_set_108) ).
thf(3,axiom,
! [Va: $i,X: $i,Vt: $i] :
( ~ ( compact_set @ Va @ X @ Vt )
| ( subset_sets @ Va @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_set_107) ).
thf(4,axiom,
! [Va: $i,X: $i,Vt: $i] :
( ~ ( compact_set @ Va @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_set_106) ).
thf(5,axiom,
! [X: $i,Vt: $i,Uu24: $i] :
( ( compact_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( finite @ Uu24 )
| ~ ( subset_collections @ Uu24 @ ( f24 @ X @ Vt ) )
| ~ ( open_covering @ Uu24 @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_space_105) ).
thf(6,axiom,
! [X: $i,Vt: $i] :
( ( compact_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( open_covering @ ( f24 @ X @ Vt ) @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_space_104) ).
thf(7,axiom,
! [X: $i,Vt: $i,Vf1: $i] :
( ~ ( compact_space @ X @ Vt )
| ~ ( open_covering @ Vf1 @ X @ Vt )
| ( open_covering @ ( f23 @ X @ Vt @ Vf1 ) @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_space_103) ).
thf(8,axiom,
! [X: $i,Vt: $i,Vf1: $i] :
( ~ ( compact_space @ X @ Vt )
| ~ ( open_covering @ Vf1 @ X @ Vt )
| ( subset_collections @ ( f23 @ X @ Vt @ Vf1 ) @ Vf1 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_space_102) ).
thf(9,axiom,
! [X: $i,Vt: $i,Vf1: $i] :
( ~ ( compact_space @ X @ Vt )
| ~ ( open_covering @ Vf1 @ X @ Vt )
| ( finite @ ( f23 @ X @ Vt @ Vf1 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_space_101) ).
thf(10,axiom,
! [X: $i,Vt: $i] :
( ~ ( compact_space @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',compact_space_100) ).
thf(11,axiom,
! [Vf: $i,X: $i,Vt: $i] :
( ( open_covering @ Vf @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_collections @ Vf @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',open_covering_99) ).
thf(12,axiom,
! [Vf: $i,X: $i,Vt: $i] :
( ~ ( open_covering @ Vf @ X @ Vt )
| ( equal_sets @ ( union_of_members @ Vf ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',open_covering_98) ).
thf(13,axiom,
! [Vf: $i,X: $i,Vt: $i] :
( ~ ( open_covering @ Vf @ X @ Vt )
| ( subset_collections @ Vf @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',open_covering_97) ).
thf(14,axiom,
! [Vf: $i,X: $i,Vt: $i] :
( ~ ( open_covering @ Vf @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',open_covering_96) ).
thf(15,axiom,
! [Va: $i,X: $i,Vt: $i] :
( ( connected_set @ Va @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Va @ X )
| ~ ( connected_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connected_set_95) ).
thf(16,axiom,
! [Va: $i,X: $i,Vt: $i] :
( ~ ( connected_set @ Va @ X @ Vt )
| ( connected_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connected_set_94) ).
thf(17,axiom,
! [Va: $i,X: $i,Vt: $i] :
( ~ ( connected_set @ Va @ X @ Vt )
| ( subset_sets @ Va @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connected_set_93) ).
thf(18,axiom,
! [Va: $i,X: $i,Vt: $i] :
( ~ ( connected_set @ Va @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connected_set_92) ).
thf(19,axiom,
! [X: $i,Vt: $i] :
( ( connected_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( separation @ ( f21 @ X @ Vt ) @ ( f22 @ X @ Vt ) @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connected_space_91) ).
thf(20,axiom,
! [X: $i,Vt: $i,Va1: $i,Va2: $i] :
( ~ ( connected_space @ X @ Vt )
| ~ ( separation @ Va1 @ Va2 @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connected_space_90) ).
thf(21,axiom,
! [X: $i,Vt: $i] :
( ~ ( connected_space @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',connected_space_89) ).
thf(22,axiom,
! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( equal_sets @ Va1 @ empty_set )
| ( equal_sets @ Va2 @ empty_set )
| ~ ( element_of_collection @ Va1 @ Vt )
| ~ ( element_of_collection @ Va2 @ Vt )
| ~ ( equal_sets @ ( union_of_sets @ Va1 @ Va2 ) @ X )
| ~ ( disjoint_s @ Va1 @ Va2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',separation_88) ).
thf(23,axiom,
! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( disjoint_s @ Va1 @ Va2 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',separation_87) ).
thf(24,axiom,
! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( equal_sets @ ( union_of_sets @ Va1 @ Va2 ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',separation_86) ).
thf(25,axiom,
! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( element_of_collection @ Va2 @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',separation_85) ).
thf(26,axiom,
! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( element_of_collection @ Va1 @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',separation_84) ).
thf(27,axiom,
! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( equal_sets @ Va2 @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',separation_83) ).
thf(28,axiom,
! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( equal_sets @ Va1 @ empty_set ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',separation_82) ).
thf(29,axiom,
! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',separation_81) ).
thf(30,axiom,
! [X: $i,Vt: $i,Uu19: $i,Uu20: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( neighborhood @ Uu19 @ ( f19 @ X @ Vt ) @ X @ Vt )
| ~ ( neighborhood @ Uu20 @ ( f20 @ X @ Vt ) @ X @ Vt )
| ~ ( disjoint_s @ Uu19 @ Uu20 ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hausdorff_80) ).
thf(31,axiom,
! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( eq_p @ ( f19 @ X @ Vt ) @ ( f20 @ X @ Vt ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hausdorff_79) ).
thf(32,axiom,
! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( element_of_set @ ( f20 @ X @ Vt ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hausdorff_78) ).
thf(33,axiom,
! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( element_of_set @ ( f19 @ X @ Vt ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hausdorff_77) ).
thf(34,axiom,
! [X: $i,Vt: $i,X_1: $i,X_2: $i] :
( ~ ( hausdorff @ X @ Vt )
| ~ ( element_of_set @ X_1 @ X )
| ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( disjoint_s @ ( f17 @ X @ Vt @ X_1 @ X_2 ) @ ( f18 @ X @ Vt @ X_1 @ X_2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hausdorff_76) ).
thf(35,axiom,
! [X: $i,Vt: $i,X_1: $i,X_2: $i] :
( ~ ( hausdorff @ X @ Vt )
| ~ ( element_of_set @ X_1 @ X )
| ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( neighborhood @ ( f18 @ X @ Vt @ X_1 @ X_2 ) @ X_2 @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hausdorff_75) ).
thf(36,axiom,
! [X: $i,Vt: $i,X_1: $i,X_2: $i] :
( ~ ( hausdorff @ X @ Vt )
| ~ ( element_of_set @ X_1 @ X )
| ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( neighborhood @ ( f17 @ X @ Vt @ X_1 @ X_2 ) @ X_1 @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hausdorff_74) ).
thf(37,axiom,
! [X: $i,Vt: $i] :
( ~ ( hausdorff @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',hausdorff_73) ).
thf(38,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( element_of_set @ U @ ( closure @ ( relative_complement_sets @ Y @ X ) @ X @ Vt ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',boundary_72) ).
thf(39,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ( element_of_set @ U @ ( closure @ ( relative_complement_sets @ Y @ X ) @ X @ Vt ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',boundary_71) ).
thf(40,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',boundary_70) ).
thf(41,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',boundary_69) ).
thf(42,axiom,
! [Z: $i,Y: $i,X: $i,Vt: $i,Uu16: $i] :
( ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ~ ( element_of_set @ Uu16 @ ( intersection_of_sets @ ( f16 @ Z @ Y @ X @ Vt ) @ Y ) )
| ( eq_p @ Uu16 @ Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',limit_point_68) ).
thf(43,axiom,
! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ( neighborhood @ ( f16 @ Z @ Y @ X @ Vt ) @ Z @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',limit_point_67) ).
thf(44,axiom,
! [Z: $i,Y: $i,X: $i,Vt: $i,U: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( neighborhood @ U @ Z @ X @ Vt )
| ~ ( eq_p @ ( f15 @ Z @ Y @ X @ Vt @ U ) @ Z ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',limit_point_66) ).
thf(45,axiom,
! [Z: $i,Y: $i,X: $i,Vt: $i,U: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( neighborhood @ U @ Z @ X @ Vt )
| ( element_of_set @ ( f15 @ Z @ Y @ X @ Vt @ U ) @ ( intersection_of_sets @ U @ Y ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',limit_point_65) ).
thf(46,axiom,
! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ( subset_sets @ Y @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',limit_point_64) ).
thf(47,axiom,
! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',limit_point_63) ).
thf(48,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( neighborhood @ U @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( open @ U @ X @ Vt )
| ~ ( element_of_set @ Y @ U ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',neighborhood_62) ).
thf(49,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( neighborhood @ U @ Y @ X @ Vt )
| ( element_of_set @ Y @ U ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',neighborhood_61) ).
thf(50,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( neighborhood @ U @ Y @ X @ Vt )
| ( open @ U @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',neighborhood_60) ).
thf(51,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( neighborhood @ U @ Y @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',neighborhood_59) ).
thf(52,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ~ ( element_of_set @ U @ ( f14 @ Y @ X @ Vt @ U ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closure_58) ).
thf(53,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ( closed @ ( f14 @ Y @ X @ Vt @ U ) @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closure_57) ).
thf(54,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ( subset_sets @ Y @ ( f14 @ Y @ X @ Vt @ U ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closure_56) ).
thf(55,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i,V: $i] :
( ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( subset_sets @ Y @ V )
| ~ ( closed @ V @ X @ Vt )
| ( element_of_set @ U @ V ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closure_55) ).
thf(56,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ( subset_sets @ Y @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closure_54) ).
thf(57,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closure_53) ).
thf(58,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i,Uu13: $i] :
( ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ~ ( element_of_set @ U @ Uu13 )
| ~ ( subset_sets @ Uu13 @ Y )
| ~ ( open @ Uu13 @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',interior_52) ).
thf(59,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( open @ ( f13 @ Y @ X @ Vt @ U ) @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',interior_51) ).
thf(60,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( subset_sets @ ( f13 @ Y @ X @ Vt @ U ) @ Y ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',interior_50) ).
thf(61,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( element_of_set @ U @ ( f13 @ Y @ X @ Vt @ U ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',interior_49) ).
thf(62,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( subset_sets @ Y @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',interior_48) ).
thf(63,axiom,
! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',interior_47) ).
thf(64,axiom,
! [U: $i,X: $i,Vt: $i,Y: $i,Uu12: $i] :
( ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ~ ( element_of_collection @ Uu12 @ Vt )
| ~ ( equal_sets @ U @ ( intersection_of_sets @ Y @ Uu12 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subspace_topology_46) ).
thf(65,axiom,
! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( equal_sets @ U @ ( intersection_of_sets @ Y @ ( f12 @ X @ Vt @ Y @ U ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subspace_topology_45) ).
thf(66,axiom,
! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( element_of_collection @ ( f12 @ X @ Vt @ Y @ U ) @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subspace_topology_44) ).
thf(67,axiom,
! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( subset_sets @ Y @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subspace_topology_43) ).
thf(68,axiom,
! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',subspace_topology_42) ).
thf(69,axiom,
! [U: $i,Vf: $i,Uu11: $i] :
( ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ~ ( element_of_set @ ( f11 @ Vf @ U ) @ Uu11 )
| ~ ( element_of_collection @ Uu11 @ Vf )
| ~ ( subset_sets @ Uu11 @ U ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topology_generated_41) ).
thf(70,axiom,
! [U: $i,Vf: $i] :
( ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ( element_of_set @ ( f11 @ Vf @ U ) @ U ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topology_generated_40) ).
thf(71,axiom,
! [U: $i,Vf: $i,X: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ~ ( element_of_set @ X @ U )
| ( subset_sets @ ( f10 @ Vf @ U @ X ) @ U ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topology_generated_39) ).
thf(72,axiom,
! [U: $i,Vf: $i,X: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ~ ( element_of_set @ X @ U )
| ( element_of_collection @ ( f10 @ Vf @ U @ X ) @ Vf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topology_generated_38) ).
thf(73,axiom,
! [U: $i,Vf: $i,X: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ~ ( element_of_set @ X @ U )
| ( element_of_set @ X @ ( f10 @ Vf @ U @ X ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topology_generated_37) ).
thf(74,axiom,
! [X: $i,Vf: $i,Uu9: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ~ ( element_of_set @ ( f7 @ X @ Vf ) @ Uu9 )
| ~ ( element_of_collection @ Uu9 @ Vf )
| ~ ( subset_sets @ Uu9 @ ( intersection_of_sets @ ( f8 @ X @ Vf ) @ ( f9 @ X @ Vf ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_36) ).
thf(75,axiom,
! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_set @ ( f7 @ X @ Vf ) @ ( intersection_of_sets @ ( f8 @ X @ Vf ) @ ( f9 @ X @ Vf ) ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_35) ).
thf(76,axiom,
! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_collection @ ( f9 @ X @ Vf ) @ Vf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_34) ).
thf(77,axiom,
! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_collection @ ( f8 @ X @ Vf ) @ Vf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_33) ).
thf(78,axiom,
! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_set @ ( f7 @ X @ Vf ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_32) ).
thf(79,axiom,
! [X: $i,Vf: $i,Y: $i,Vb1: $i,Vb2: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ~ ( element_of_collection @ Vb1 @ Vf )
| ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( subset_sets @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) @ ( intersection_of_sets @ Vb1 @ Vb2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_31) ).
thf(80,axiom,
! [X: $i,Vf: $i,Y: $i,Vb1: $i,Vb2: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ~ ( element_of_collection @ Vb1 @ Vf )
| ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( element_of_collection @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) @ Vf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_30) ).
thf(81,axiom,
! [X: $i,Vf: $i,Y: $i,Vb1: $i,Vb2: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ~ ( element_of_collection @ Vb1 @ Vf )
| ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( element_of_set @ Y @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_29) ).
thf(82,axiom,
! [X: $i,Vf: $i] :
( ~ ( basis @ X @ Vf )
| ( equal_sets @ ( union_of_members @ Vf ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',basis_for_topology_28) ).
thf(83,axiom,
! [Vt: $i,Vs: $i,X: $i] :
( ( finer @ Vt @ Vs @ X )
| ~ ( topological_space @ X @ Vt )
| ~ ( topological_space @ X @ Vs )
| ~ ( subset_collections @ Vs @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',finer_topology_27) ).
thf(84,axiom,
! [Vt: $i,Vs: $i,X: $i] :
( ~ ( finer @ Vt @ Vs @ X )
| ( subset_collections @ Vs @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',finer_topology_26) ).
thf(85,axiom,
! [Vt: $i,Vs: $i,X: $i] :
( ~ ( finer @ Vt @ Vs @ X )
| ( topological_space @ X @ Vs ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',finer_topology_25) ).
thf(86,axiom,
! [Vt: $i,Vs: $i,X: $i] :
( ~ ( finer @ Vt @ Vs @ X )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',finer_topology_24) ).
thf(87,axiom,
! [U: $i,X: $i,Vt: $i] :
( ( closed @ U @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( open @ ( relative_complement_sets @ U @ X ) @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed_set_23) ).
thf(88,axiom,
! [U: $i,X: $i,Vt: $i] :
( ~ ( closed @ U @ X @ Vt )
| ( open @ ( relative_complement_sets @ U @ X ) @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed_set_22) ).
thf(89,axiom,
! [U: $i,X: $i,Vt: $i] :
( ~ ( closed @ U @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',closed_set_21) ).
thf(90,axiom,
! [U: $i,X: $i,Vt: $i] :
( ( open @ U @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( element_of_collection @ U @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',open_set_20) ).
thf(91,axiom,
! [U: $i,X: $i,Vt: $i] :
( ~ ( open @ U @ X @ Vt )
| ( element_of_collection @ U @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',open_set_19) ).
thf(92,axiom,
! [U: $i,X: $i,Vt: $i] :
( ~ ( open @ U @ X @ Vt )
| ( topological_space @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',open_set_18) ).
thf(93,axiom,
! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ X @ Vt ) @ ( f4 @ X @ Vt ) ) @ Vt )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ X @ Vt ) ) @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_17) ).
thf(94,axiom,
! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ X @ Vt ) @ ( f4 @ X @ Vt ) ) @ Vt )
| ( subset_collections @ ( f5 @ X @ Vt ) @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_16) ).
thf(95,axiom,
! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f4 @ X @ Vt ) @ Vt )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ X @ Vt ) ) @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_15) ).
thf(96,axiom,
! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f4 @ X @ Vt ) @ Vt )
| ( subset_collections @ ( f5 @ X @ Vt ) @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_14) ).
thf(97,axiom,
! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f3 @ X @ Vt ) @ Vt )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ X @ Vt ) ) @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_13) ).
thf(98,axiom,
! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f3 @ X @ Vt ) @ Vt )
| ( subset_collections @ ( f5 @ X @ Vt ) @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_12) ).
thf(99,axiom,
! [X: $i,Vt: $i,Vf: $i] :
( ~ ( topological_space @ X @ Vt )
| ~ ( subset_collections @ Vf @ Vt )
| ( element_of_collection @ ( union_of_members @ Vf ) @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_11) ).
thf(100,axiom,
! [X: $i,Vt: $i,Y: $i,Z: $i] :
( ~ ( topological_space @ X @ Vt )
| ~ ( element_of_collection @ Y @ Vt )
| ~ ( element_of_collection @ Z @ Vt )
| ( element_of_collection @ ( intersection_of_sets @ Y @ Z ) @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_10) ).
thf(101,axiom,
! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ( element_of_collection @ X @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_9) ).
thf(102,axiom,
! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ( element_of_collection @ empty_set @ Vt ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_8) ).
thf(103,axiom,
! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ( equal_sets @ ( union_of_members @ Vt ) @ X ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',topological_space_7) ).
thf(104,axiom,
! [U: $i,Vf: $i] :
( ( element_of_set @ U @ ( intersection_of_members @ Vf ) )
| ~ ( element_of_set @ U @ ( f2 @ Vf @ U ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection_of_members_6) ).
thf(105,axiom,
! [U: $i,Vf: $i] :
( ( element_of_set @ U @ ( intersection_of_members @ Vf ) )
| ( element_of_collection @ ( f2 @ Vf @ U ) @ Vf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection_of_members_5) ).
thf(106,axiom,
! [U: $i,Vf: $i,Va: $i] :
( ~ ( element_of_set @ U @ ( intersection_of_members @ Vf ) )
| ~ ( element_of_collection @ Va @ Vf )
| ( element_of_set @ U @ Va ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',intersection_of_members_4) ).
thf(107,axiom,
! [U: $i,Vf: $i,Uu1: $i] :
( ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ~ ( element_of_set @ U @ Uu1 )
| ~ ( element_of_collection @ Uu1 @ Vf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_of_members_3) ).
thf(108,axiom,
! [U: $i,Vf: $i] :
( ~ ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ( element_of_collection @ ( f1 @ Vf @ U ) @ Vf ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_of_members_2) ).
thf(109,axiom,
! [U: $i,Vf: $i] :
( ~ ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ( element_of_set @ U @ ( f1 @ Vf @ U ) ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',union_of_members_1) ).
thf(110,conjecture,
$false,
file('no conjecture given, we try to refute the axioms',dummy_conjecture) ).
thf(111,negated_conjecture,
$false = $false,
inference(negate_conjecture,[status(cth)],[110]) ).
thf(112,negated_conjecture,
! [U: $i,V: $i] :
~ ( element_of_collection @ ( intersection_of_sets @ U @ V ) @ ( top_of_basis @ f ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',lemma_1d_4) ).
thf(113,negated_conjecture,
! [V: $i] : ( element_of_collection @ V @ ( top_of_basis @ f ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',lemma_1d_3) ).
thf(114,negated_conjecture,
! [U: $i] : ( element_of_collection @ U @ ( top_of_basis @ f ) ),
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',lemma_1d_2) ).
thf(115,negated_conjecture,
basis @ cx @ f,
file('/export/starexec/sandbox2/benchmark/theBenchmark.p',lemma_1d_1) ).
thf(116,plain,
$false = $false,
inference(unfold_def,[status(thm)],[111]) ).
thf(117,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ( compact_set @ Va @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Va @ X )
| ~ ( compact_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[1]) ).
thf(118,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( compact_set @ Va @ X @ Vt )
| ( compact_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[2]) ).
thf(119,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( compact_set @ Va @ X @ Vt )
| ( subset_sets @ Va @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[3]) ).
thf(120,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( compact_set @ Va @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[4]) ).
thf(121,plain,
( ( ! [X: $i,Vt: $i,Uu24: $i] :
( ( compact_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( finite @ Uu24 )
| ~ ( subset_collections @ Uu24 @ ( f24 @ X @ Vt ) )
| ~ ( open_covering @ Uu24 @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[5]) ).
thf(122,plain,
( ( ! [X: $i,Vt: $i] :
( ( compact_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( open_covering @ ( f24 @ X @ Vt ) @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[6]) ).
thf(123,plain,
( ( ! [X: $i,Vt: $i,Vf1: $i] :
( ~ ( compact_space @ X @ Vt )
| ~ ( open_covering @ Vf1 @ X @ Vt )
| ( open_covering @ ( f23 @ X @ Vt @ Vf1 ) @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[7]) ).
thf(124,plain,
( ( ! [X: $i,Vt: $i,Vf1: $i] :
( ~ ( compact_space @ X @ Vt )
| ~ ( open_covering @ Vf1 @ X @ Vt )
| ( subset_collections @ ( f23 @ X @ Vt @ Vf1 ) @ Vf1 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[8]) ).
thf(125,plain,
( ( ! [X: $i,Vt: $i,Vf1: $i] :
( ~ ( compact_space @ X @ Vt )
| ~ ( open_covering @ Vf1 @ X @ Vt )
| ( finite @ ( f23 @ X @ Vt @ Vf1 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[9]) ).
thf(126,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( compact_space @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[10]) ).
thf(127,plain,
( ( ! [Vf: $i,X: $i,Vt: $i] :
( ( open_covering @ Vf @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_collections @ Vf @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[11]) ).
thf(128,plain,
( ( ! [Vf: $i,X: $i,Vt: $i] :
( ~ ( open_covering @ Vf @ X @ Vt )
| ( equal_sets @ ( union_of_members @ Vf ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[12]) ).
thf(129,plain,
( ( ! [Vf: $i,X: $i,Vt: $i] :
( ~ ( open_covering @ Vf @ X @ Vt )
| ( subset_collections @ Vf @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[13]) ).
thf(130,plain,
( ( ! [Vf: $i,X: $i,Vt: $i] :
( ~ ( open_covering @ Vf @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[14]) ).
thf(131,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ( connected_set @ Va @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Va @ X )
| ~ ( connected_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[15]) ).
thf(132,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( connected_set @ Va @ X @ Vt )
| ( connected_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[16]) ).
thf(133,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( connected_set @ Va @ X @ Vt )
| ( subset_sets @ Va @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[17]) ).
thf(134,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( connected_set @ Va @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[18]) ).
thf(135,plain,
( ( ! [X: $i,Vt: $i] :
( ( connected_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( separation @ ( f21 @ X @ Vt ) @ ( f22 @ X @ Vt ) @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[19]) ).
thf(136,plain,
( ( ! [X: $i,Vt: $i,Va1: $i,Va2: $i] :
( ~ ( connected_space @ X @ Vt )
| ~ ( separation @ Va1 @ Va2 @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[20]) ).
thf(137,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( connected_space @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[21]) ).
thf(138,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( equal_sets @ Va1 @ empty_set )
| ( equal_sets @ Va2 @ empty_set )
| ~ ( element_of_collection @ Va1 @ Vt )
| ~ ( element_of_collection @ Va2 @ Vt )
| ~ ( equal_sets @ ( union_of_sets @ Va1 @ Va2 ) @ X )
| ~ ( disjoint_s @ Va1 @ Va2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[22]) ).
thf(139,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( disjoint_s @ Va1 @ Va2 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[23]) ).
thf(140,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( equal_sets @ ( union_of_sets @ Va1 @ Va2 ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[24]) ).
thf(141,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( element_of_collection @ Va2 @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[25]) ).
thf(142,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( element_of_collection @ Va1 @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[26]) ).
thf(143,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( equal_sets @ Va2 @ empty_set ) ) )
= $true ),
inference(unfold_def,[status(thm)],[27]) ).
thf(144,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( equal_sets @ Va1 @ empty_set ) ) )
= $true ),
inference(unfold_def,[status(thm)],[28]) ).
thf(145,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[29]) ).
thf(146,plain,
( ( ! [X: $i,Vt: $i,Uu19: $i,Uu20: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( neighborhood @ Uu19 @ ( f19 @ X @ Vt ) @ X @ Vt )
| ~ ( neighborhood @ Uu20 @ ( f20 @ X @ Vt ) @ X @ Vt )
| ~ ( disjoint_s @ Uu19 @ Uu20 ) ) )
= $true ),
inference(unfold_def,[status(thm)],[30]) ).
thf(147,plain,
( ( ! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( eq_p @ ( f19 @ X @ Vt ) @ ( f20 @ X @ Vt ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[31]) ).
thf(148,plain,
( ( ! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( element_of_set @ ( f20 @ X @ Vt ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[32]) ).
thf(149,plain,
( ( ! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( element_of_set @ ( f19 @ X @ Vt ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[33]) ).
thf(150,plain,
( ( ! [X: $i,Vt: $i,X_1: $i,X_2: $i] :
( ~ ( hausdorff @ X @ Vt )
| ~ ( element_of_set @ X_1 @ X )
| ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( disjoint_s @ ( f17 @ X @ Vt @ X_1 @ X_2 ) @ ( f18 @ X @ Vt @ X_1 @ X_2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[34]) ).
thf(151,plain,
( ( ! [X: $i,Vt: $i,X_1: $i,X_2: $i] :
( ~ ( hausdorff @ X @ Vt )
| ~ ( element_of_set @ X_1 @ X )
| ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( neighborhood @ ( f18 @ X @ Vt @ X_1 @ X_2 ) @ X_2 @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[35]) ).
thf(152,plain,
( ( ! [X: $i,Vt: $i,X_1: $i,X_2: $i] :
( ~ ( hausdorff @ X @ Vt )
| ~ ( element_of_set @ X_1 @ X )
| ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( neighborhood @ ( f17 @ X @ Vt @ X_1 @ X_2 ) @ X_1 @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[36]) ).
thf(153,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( hausdorff @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[37]) ).
thf(154,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( element_of_set @ U @ ( closure @ ( relative_complement_sets @ Y @ X ) @ X @ Vt ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[38]) ).
thf(155,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ( element_of_set @ U @ ( closure @ ( relative_complement_sets @ Y @ X ) @ X @ Vt ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[39]) ).
thf(156,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[40]) ).
thf(157,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[41]) ).
thf(158,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i,Uu16: $i] :
( ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ~ ( element_of_set @ Uu16 @ ( intersection_of_sets @ ( f16 @ Z @ Y @ X @ Vt ) @ Y ) )
| ( eq_p @ Uu16 @ Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[42]) ).
thf(159,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ( neighborhood @ ( f16 @ Z @ Y @ X @ Vt ) @ Z @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[43]) ).
thf(160,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i,U: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( neighborhood @ U @ Z @ X @ Vt )
| ~ ( eq_p @ ( f15 @ Z @ Y @ X @ Vt @ U ) @ Z ) ) )
= $true ),
inference(unfold_def,[status(thm)],[44]) ).
thf(161,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i,U: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( neighborhood @ U @ Z @ X @ Vt )
| ( element_of_set @ ( f15 @ Z @ Y @ X @ Vt @ U ) @ ( intersection_of_sets @ U @ Y ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[45]) ).
thf(162,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[46]) ).
thf(163,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[47]) ).
thf(164,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( neighborhood @ U @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( open @ U @ X @ Vt )
| ~ ( element_of_set @ Y @ U ) ) )
= $true ),
inference(unfold_def,[status(thm)],[48]) ).
thf(165,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( neighborhood @ U @ Y @ X @ Vt )
| ( element_of_set @ Y @ U ) ) )
= $true ),
inference(unfold_def,[status(thm)],[49]) ).
thf(166,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( neighborhood @ U @ Y @ X @ Vt )
| ( open @ U @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[50]) ).
thf(167,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( neighborhood @ U @ Y @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[51]) ).
thf(168,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ~ ( element_of_set @ U @ ( f14 @ Y @ X @ Vt @ U ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[52]) ).
thf(169,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ( closed @ ( f14 @ Y @ X @ Vt @ U ) @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[53]) ).
thf(170,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ( subset_sets @ Y @ ( f14 @ Y @ X @ Vt @ U ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[54]) ).
thf(171,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i,V: $i] :
( ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( subset_sets @ Y @ V )
| ~ ( closed @ V @ X @ Vt )
| ( element_of_set @ U @ V ) ) )
= $true ),
inference(unfold_def,[status(thm)],[55]) ).
thf(172,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[56]) ).
thf(173,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[57]) ).
thf(174,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i,Uu13: $i] :
( ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ~ ( element_of_set @ U @ Uu13 )
| ~ ( subset_sets @ Uu13 @ Y )
| ~ ( open @ Uu13 @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[58]) ).
thf(175,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( open @ ( f13 @ Y @ X @ Vt @ U ) @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[59]) ).
thf(176,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( subset_sets @ ( f13 @ Y @ X @ Vt @ U ) @ Y ) ) )
= $true ),
inference(unfold_def,[status(thm)],[60]) ).
thf(177,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( element_of_set @ U @ ( f13 @ Y @ X @ Vt @ U ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[61]) ).
thf(178,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[62]) ).
thf(179,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[63]) ).
thf(180,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i,Uu12: $i] :
( ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ~ ( element_of_collection @ Uu12 @ Vt )
| ~ ( equal_sets @ U @ ( intersection_of_sets @ Y @ Uu12 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[64]) ).
thf(181,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( equal_sets @ U @ ( intersection_of_sets @ Y @ ( f12 @ X @ Vt @ Y @ U ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[65]) ).
thf(182,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( element_of_collection @ ( f12 @ X @ Vt @ Y @ U ) @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[66]) ).
thf(183,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[67]) ).
thf(184,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[68]) ).
thf(185,plain,
( ( ! [U: $i,Vf: $i,Uu11: $i] :
( ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ~ ( element_of_set @ ( f11 @ Vf @ U ) @ Uu11 )
| ~ ( element_of_collection @ Uu11 @ Vf )
| ~ ( subset_sets @ Uu11 @ U ) ) )
= $true ),
inference(unfold_def,[status(thm)],[69]) ).
thf(186,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ( element_of_set @ ( f11 @ Vf @ U ) @ U ) ) )
= $true ),
inference(unfold_def,[status(thm)],[70]) ).
thf(187,plain,
( ( ! [U: $i,Vf: $i,X: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ~ ( element_of_set @ X @ U )
| ( subset_sets @ ( f10 @ Vf @ U @ X ) @ U ) ) )
= $true ),
inference(unfold_def,[status(thm)],[71]) ).
thf(188,plain,
( ( ! [U: $i,Vf: $i,X: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ~ ( element_of_set @ X @ U )
| ( element_of_collection @ ( f10 @ Vf @ U @ X ) @ Vf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[72]) ).
thf(189,plain,
( ( ! [U: $i,Vf: $i,X: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ~ ( element_of_set @ X @ U )
| ( element_of_set @ X @ ( f10 @ Vf @ U @ X ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[73]) ).
thf(190,plain,
( ( ! [X: $i,Vf: $i,Uu9: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ~ ( element_of_set @ ( f7 @ X @ Vf ) @ Uu9 )
| ~ ( element_of_collection @ Uu9 @ Vf )
| ~ ( subset_sets @ Uu9 @ ( intersection_of_sets @ ( f8 @ X @ Vf ) @ ( f9 @ X @ Vf ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[74]) ).
thf(191,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_set @ ( f7 @ X @ Vf ) @ ( intersection_of_sets @ ( f8 @ X @ Vf ) @ ( f9 @ X @ Vf ) ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[75]) ).
thf(192,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_collection @ ( f9 @ X @ Vf ) @ Vf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[76]) ).
thf(193,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_collection @ ( f8 @ X @ Vf ) @ Vf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[77]) ).
thf(194,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_set @ ( f7 @ X @ Vf ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[78]) ).
thf(195,plain,
( ( ! [X: $i,Vf: $i,Y: $i,Vb1: $i,Vb2: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ~ ( element_of_collection @ Vb1 @ Vf )
| ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( subset_sets @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) @ ( intersection_of_sets @ Vb1 @ Vb2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[79]) ).
thf(196,plain,
( ( ! [X: $i,Vf: $i,Y: $i,Vb1: $i,Vb2: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ~ ( element_of_collection @ Vb1 @ Vf )
| ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( element_of_collection @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) @ Vf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[80]) ).
thf(197,plain,
( ( ! [X: $i,Vf: $i,Y: $i,Vb1: $i,Vb2: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ~ ( element_of_collection @ Vb1 @ Vf )
| ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( element_of_set @ Y @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[81]) ).
thf(198,plain,
( ( ! [X: $i,Vf: $i] :
( ~ ( basis @ X @ Vf )
| ( equal_sets @ ( union_of_members @ Vf ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[82]) ).
thf(199,plain,
( ( ! [Vt: $i,Vs: $i,X: $i] :
( ( finer @ Vt @ Vs @ X )
| ~ ( topological_space @ X @ Vt )
| ~ ( topological_space @ X @ Vs )
| ~ ( subset_collections @ Vs @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[83]) ).
thf(200,plain,
( ( ! [Vt: $i,Vs: $i,X: $i] :
( ~ ( finer @ Vt @ Vs @ X )
| ( subset_collections @ Vs @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[84]) ).
thf(201,plain,
( ( ! [Vt: $i,Vs: $i,X: $i] :
( ~ ( finer @ Vt @ Vs @ X )
| ( topological_space @ X @ Vs ) ) )
= $true ),
inference(unfold_def,[status(thm)],[85]) ).
thf(202,plain,
( ( ! [Vt: $i,Vs: $i,X: $i] :
( ~ ( finer @ Vt @ Vs @ X )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[86]) ).
thf(203,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ( closed @ U @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( open @ ( relative_complement_sets @ U @ X ) @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[87]) ).
thf(204,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ~ ( closed @ U @ X @ Vt )
| ( open @ ( relative_complement_sets @ U @ X ) @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[88]) ).
thf(205,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ~ ( closed @ U @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[89]) ).
thf(206,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ( open @ U @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( element_of_collection @ U @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[90]) ).
thf(207,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ~ ( open @ U @ X @ Vt )
| ( element_of_collection @ U @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[91]) ).
thf(208,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ~ ( open @ U @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[92]) ).
thf(209,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ X @ Vt ) @ ( f4 @ X @ Vt ) ) @ Vt )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ X @ Vt ) ) @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[93]) ).
thf(210,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ X @ Vt ) @ ( f4 @ X @ Vt ) ) @ Vt )
| ( subset_collections @ ( f5 @ X @ Vt ) @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[94]) ).
thf(211,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f4 @ X @ Vt ) @ Vt )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ X @ Vt ) ) @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[95]) ).
thf(212,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f4 @ X @ Vt ) @ Vt )
| ( subset_collections @ ( f5 @ X @ Vt ) @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[96]) ).
thf(213,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f3 @ X @ Vt ) @ Vt )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ X @ Vt ) ) @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[97]) ).
thf(214,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f3 @ X @ Vt ) @ Vt )
| ( subset_collections @ ( f5 @ X @ Vt ) @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[98]) ).
thf(215,plain,
( ( ! [X: $i,Vt: $i,Vf: $i] :
( ~ ( topological_space @ X @ Vt )
| ~ ( subset_collections @ Vf @ Vt )
| ( element_of_collection @ ( union_of_members @ Vf ) @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[99]) ).
thf(216,plain,
( ( ! [X: $i,Vt: $i,Y: $i,Z: $i] :
( ~ ( topological_space @ X @ Vt )
| ~ ( element_of_collection @ Y @ Vt )
| ~ ( element_of_collection @ Z @ Vt )
| ( element_of_collection @ ( intersection_of_sets @ Y @ Z ) @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[100]) ).
thf(217,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ( element_of_collection @ X @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[101]) ).
thf(218,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ( element_of_collection @ empty_set @ Vt ) ) )
= $true ),
inference(unfold_def,[status(thm)],[102]) ).
thf(219,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ( equal_sets @ ( union_of_members @ Vt ) @ X ) ) )
= $true ),
inference(unfold_def,[status(thm)],[103]) ).
thf(220,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_set @ U @ ( intersection_of_members @ Vf ) )
| ~ ( element_of_set @ U @ ( f2 @ Vf @ U ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[104]) ).
thf(221,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_set @ U @ ( intersection_of_members @ Vf ) )
| ( element_of_collection @ ( f2 @ Vf @ U ) @ Vf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[105]) ).
thf(222,plain,
( ( ! [U: $i,Vf: $i,Va: $i] :
( ~ ( element_of_set @ U @ ( intersection_of_members @ Vf ) )
| ~ ( element_of_collection @ Va @ Vf )
| ( element_of_set @ U @ Va ) ) )
= $true ),
inference(unfold_def,[status(thm)],[106]) ).
thf(223,plain,
( ( ! [U: $i,Vf: $i,Uu1: $i] :
( ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ~ ( element_of_set @ U @ Uu1 )
| ~ ( element_of_collection @ Uu1 @ Vf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[107]) ).
thf(224,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ( element_of_collection @ ( f1 @ Vf @ U ) @ Vf ) ) )
= $true ),
inference(unfold_def,[status(thm)],[108]) ).
thf(225,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ( element_of_set @ U @ ( f1 @ Vf @ U ) ) ) )
= $true ),
inference(unfold_def,[status(thm)],[109]) ).
thf(226,plain,
( ( ! [U: $i,V: $i] :
~ ( element_of_collection @ ( intersection_of_sets @ U @ V ) @ ( top_of_basis @ f ) ) )
= $true ),
inference(unfold_def,[status(thm)],[112]) ).
thf(227,plain,
( ( ! [V: $i] : ( element_of_collection @ V @ ( top_of_basis @ f ) ) )
= $true ),
inference(unfold_def,[status(thm)],[113]) ).
thf(228,plain,
( ( ! [U: $i] : ( element_of_collection @ U @ ( top_of_basis @ f ) ) )
= $true ),
inference(unfold_def,[status(thm)],[114]) ).
thf(229,plain,
( ( basis @ cx @ f )
= $true ),
inference(unfold_def,[status(thm)],[115]) ).
thf(230,plain,
( ( ~ $false ) = $true ),
inference(polarity_switch,[status(thm)],[116]) ).
thf(231,plain,
( ( ! [Va: $i,X: $i] :
( ! [Vt: $i] :
~ ( compact_set @ Va @ X @ Vt )
| ( subset_sets @ Va @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[119]) ).
thf(232,plain,
( ( ! [X: $i,Vt: $i] :
( ( compact_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ! [Uu24: $i] :
( ~ ( finite @ Uu24 )
| ~ ( subset_collections @ Uu24 @ ( f24 @ X @ Vt ) )
| ~ ( open_covering @ Uu24 @ X @ Vt ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[121]) ).
thf(233,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( compact_space @ X @ Vt )
| ! [Vf1: $i] :
( ~ ( open_covering @ Vf1 @ X @ Vt )
| ( open_covering @ ( f23 @ X @ Vt @ Vf1 ) @ X @ Vt ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[123]) ).
thf(234,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( compact_space @ X @ Vt )
| ! [Vf1: $i] :
( ~ ( open_covering @ Vf1 @ X @ Vt )
| ( subset_collections @ ( f23 @ X @ Vt @ Vf1 ) @ Vf1 ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[124]) ).
thf(235,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( compact_space @ X @ Vt )
| ! [Vf1: $i] :
( ~ ( open_covering @ Vf1 @ X @ Vt )
| ( finite @ ( f23 @ X @ Vt @ Vf1 ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[125]) ).
thf(236,plain,
( ( ! [Vf: $i,X: $i] :
( ! [Vt: $i] :
~ ( open_covering @ Vf @ X @ Vt )
| ( equal_sets @ ( union_of_members @ Vf ) @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[128]) ).
thf(237,plain,
( ( ! [Va: $i,X: $i] :
( ! [Vt: $i] :
~ ( connected_set @ Va @ X @ Vt )
| ( subset_sets @ Va @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[133]) ).
thf(238,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( connected_space @ X @ Vt )
| ! [Va1: $i,Va2: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[136]) ).
thf(239,plain,
( ( ! [Va1: $i,Va2: $i] :
( ! [X: $i,Vt: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( disjoint_s @ Va1 @ Va2 ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[139]) ).
thf(240,plain,
( ( ! [Va1: $i,Va2: $i,X: $i] :
( ! [Vt: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( equal_sets @ ( union_of_sets @ Va1 @ Va2 ) @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[140]) ).
thf(241,plain,
( ( ! [Va1: $i,Va2: $i] :
( ! [X: $i,Vt: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( equal_sets @ Va2 @ empty_set ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[143]) ).
thf(242,plain,
( ( ! [Va1: $i,Va2: $i] :
( ! [X: $i,Vt: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( equal_sets @ Va1 @ empty_set ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[144]) ).
thf(243,plain,
( ( ! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ! [Uu19: $i] :
( ~ ( neighborhood @ Uu19 @ ( f19 @ X @ Vt ) @ X @ Vt )
| ! [Uu20: $i] :
( ~ ( neighborhood @ Uu20 @ ( f20 @ X @ Vt ) @ X @ Vt )
| ~ ( disjoint_s @ Uu19 @ Uu20 ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[146]) ).
thf(244,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( hausdorff @ X @ Vt )
| ! [X_1: $i] :
( ~ ( element_of_set @ X_1 @ X )
| ! [X_2: $i] :
( ~ ( element_of_set @ X_2 @ X )
| ( disjoint_s @ ( f17 @ X @ Vt @ X_1 @ X_2 ) @ ( f18 @ X @ Vt @ X_1 @ X_2 ) )
| ( eq_p @ X_1 @ X_2 ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[150]) ).
thf(245,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( hausdorff @ X @ Vt )
| ! [X_1: $i] :
( ~ ( element_of_set @ X_1 @ X )
| ! [X_2: $i] :
( ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( neighborhood @ ( f18 @ X @ Vt @ X_1 @ X_2 ) @ X_2 @ X @ Vt ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[151]) ).
thf(246,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( hausdorff @ X @ Vt )
| ! [X_1: $i] :
( ~ ( element_of_set @ X_1 @ X )
| ! [X_2: $i] :
( ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( neighborhood @ ( f17 @ X @ Vt @ X_1 @ X_2 ) @ X_1 @ X @ Vt ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[152]) ).
thf(247,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ! [Uu16: $i] :
( ~ ( element_of_set @ Uu16 @ ( intersection_of_sets @ ( f16 @ Z @ Y @ X @ Vt ) @ Y ) )
| ( eq_p @ Uu16 @ Z ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[158]) ).
thf(248,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ! [U: $i] :
( ~ ( neighborhood @ U @ Z @ X @ Vt )
| ~ ( eq_p @ ( f15 @ Z @ Y @ X @ Vt @ U ) @ Z ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[160]) ).
thf(249,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ! [U: $i] :
( ~ ( neighborhood @ U @ Z @ X @ Vt )
| ( element_of_set @ ( f15 @ Z @ Y @ X @ Vt @ U ) @ ( intersection_of_sets @ U @ Y ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[161]) ).
thf(250,plain,
( ( ! [Z: $i,Y: $i,X: $i] :
( ! [Vt: $i] :
~ ( limit_point @ Z @ Y @ X @ Vt )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[162]) ).
thf(251,plain,
( ( ! [U: $i,Y: $i] :
( ! [X: $i,Vt: $i] :
~ ( neighborhood @ U @ Y @ X @ Vt )
| ( element_of_set @ Y @ U ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[165]) ).
thf(252,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ! [V: $i] :
( ~ ( subset_sets @ Y @ V )
| ~ ( closed @ V @ X @ Vt )
| ( element_of_set @ U @ V ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[171]) ).
thf(253,plain,
( ( ! [U: $i,Y: $i,X: $i] :
( ! [Vt: $i] :
~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[172]) ).
thf(254,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ! [Uu13: $i] :
( ~ ( element_of_set @ U @ Uu13 )
| ~ ( subset_sets @ Uu13 @ Y )
| ~ ( open @ Uu13 @ X @ Vt ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[174]) ).
thf(255,plain,
( ( ! [U: $i,Y: $i,X: $i] :
( ! [Vt: $i] :
~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[178]) ).
thf(256,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i] :
( ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ! [Uu12: $i] :
( ~ ( element_of_collection @ Uu12 @ Vt )
| ~ ( equal_sets @ U @ ( intersection_of_sets @ Y @ Uu12 ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[180]) ).
thf(257,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ! [Y: $i] :
~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[184]) ).
thf(258,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ! [Uu11: $i] :
( ~ ( element_of_set @ ( f11 @ Vf @ U ) @ Uu11 )
| ~ ( element_of_collection @ Uu11 @ Vf )
| ~ ( subset_sets @ Uu11 @ U ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[185]) ).
thf(259,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ! [X: $i] :
( ~ ( element_of_set @ X @ U )
| ( subset_sets @ ( f10 @ Vf @ U @ X ) @ U ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[187]) ).
thf(260,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ! [X: $i] :
( ~ ( element_of_set @ X @ U )
| ( element_of_collection @ ( f10 @ Vf @ U @ X ) @ Vf ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[188]) ).
thf(261,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ! [X: $i] :
( ~ ( element_of_set @ X @ U )
| ( element_of_set @ X @ ( f10 @ Vf @ U @ X ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[189]) ).
thf(262,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ! [Uu9: $i] :
( ~ ( element_of_set @ ( f7 @ X @ Vf ) @ Uu9 )
| ~ ( element_of_collection @ Uu9 @ Vf )
| ~ ( subset_sets @ Uu9 @ ( intersection_of_sets @ ( f8 @ X @ Vf ) @ ( f9 @ X @ Vf ) ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[190]) ).
thf(263,plain,
( ( ! [X: $i,Vf: $i,Y: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ! [Vb1: $i] :
( ~ ( element_of_collection @ Vb1 @ Vf )
| ! [Vb2: $i] :
( ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( subset_sets @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) @ ( intersection_of_sets @ Vb1 @ Vb2 ) ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[195]) ).
thf(264,plain,
( ( ! [X: $i,Vf: $i,Y: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ! [Vb1: $i] :
( ~ ( element_of_collection @ Vb1 @ Vf )
| ! [Vb2: $i] :
( ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( element_of_collection @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) @ Vf ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[196]) ).
thf(265,plain,
( ( ! [X: $i,Vf: $i,Y: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ! [Vb1: $i] :
( ~ ( element_of_collection @ Vb1 @ Vf )
| ! [Vb2: $i] :
( ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( element_of_set @ Y @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[197]) ).
thf(266,plain,
( ( ! [Vt: $i,Vs: $i] :
( ! [X: $i] :
~ ( finer @ Vt @ Vs @ X )
| ( subset_collections @ Vs @ Vt ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[200]) ).
thf(267,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ! [Vf: $i] :
( ~ ( subset_collections @ Vf @ Vt )
| ( element_of_collection @ ( union_of_members @ Vf ) @ Vt ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[215]) ).
thf(268,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ! [Y: $i] :
( ~ ( element_of_collection @ Y @ Vt )
| ! [Z: $i] :
( ~ ( element_of_collection @ Z @ Vt )
| ( element_of_collection @ ( intersection_of_sets @ Y @ Z ) @ Vt ) ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[216]) ).
thf(269,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_collection @ ( f2 @ Vf @ U ) @ Vf )
| ( element_of_set @ U @ ( intersection_of_members @ Vf ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[221]) ).
thf(270,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_set @ U @ ( intersection_of_members @ Vf ) )
| ! [Va: $i] :
( ~ ( element_of_collection @ Va @ Vf )
| ( element_of_set @ U @ Va ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[222]) ).
thf(271,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ! [Uu1: $i] :
( ~ ( element_of_set @ U @ Uu1 )
| ~ ( element_of_collection @ Uu1 @ Vf ) ) ) )
= $true ),
inference(extcnf_combined,[status(esa)],[223]) ).
thf(272,plain,
( ( basis @ cx @ f )
= $true ),
inference(copy,[status(thm)],[229]) ).
thf(273,plain,
( ( ! [U: $i] : ( element_of_collection @ U @ ( top_of_basis @ f ) ) )
= $true ),
inference(copy,[status(thm)],[228]) ).
thf(274,plain,
( ( ! [V: $i] : ( element_of_collection @ V @ ( top_of_basis @ f ) ) )
= $true ),
inference(copy,[status(thm)],[227]) ).
thf(275,plain,
( ( ! [U: $i,V: $i] :
~ ( element_of_collection @ ( intersection_of_sets @ U @ V ) @ ( top_of_basis @ f ) ) )
= $true ),
inference(copy,[status(thm)],[226]) ).
thf(276,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ( element_of_set @ U @ ( f1 @ Vf @ U ) ) ) )
= $true ),
inference(copy,[status(thm)],[225]) ).
thf(277,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ( element_of_collection @ ( f1 @ Vf @ U ) @ Vf ) ) )
= $true ),
inference(copy,[status(thm)],[224]) ).
thf(278,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_set @ U @ ( union_of_members @ Vf ) )
| ! [Uu1: $i] :
( ~ ( element_of_set @ U @ Uu1 )
| ~ ( element_of_collection @ Uu1 @ Vf ) ) ) )
= $true ),
inference(copy,[status(thm)],[271]) ).
thf(279,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_set @ U @ ( intersection_of_members @ Vf ) )
| ! [Va: $i] :
( ~ ( element_of_collection @ Va @ Vf )
| ( element_of_set @ U @ Va ) ) ) )
= $true ),
inference(copy,[status(thm)],[270]) ).
thf(280,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_collection @ ( f2 @ Vf @ U ) @ Vf )
| ( element_of_set @ U @ ( intersection_of_members @ Vf ) ) ) )
= $true ),
inference(copy,[status(thm)],[269]) ).
thf(281,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_set @ U @ ( intersection_of_members @ Vf ) )
| ~ ( element_of_set @ U @ ( f2 @ Vf @ U ) ) ) )
= $true ),
inference(copy,[status(thm)],[220]) ).
thf(282,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ( equal_sets @ ( union_of_members @ Vt ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[219]) ).
thf(283,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ( element_of_collection @ empty_set @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[218]) ).
thf(284,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ( element_of_collection @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[217]) ).
thf(285,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ! [Y: $i] :
( ~ ( element_of_collection @ Y @ Vt )
| ! [Z: $i] :
( ~ ( element_of_collection @ Z @ Vt )
| ( element_of_collection @ ( intersection_of_sets @ Y @ Z ) @ Vt ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[268]) ).
thf(286,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( topological_space @ X @ Vt )
| ! [Vf: $i] :
( ~ ( subset_collections @ Vf @ Vt )
| ( element_of_collection @ ( union_of_members @ Vf ) @ Vt ) ) ) )
= $true ),
inference(copy,[status(thm)],[267]) ).
thf(287,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f3 @ X @ Vt ) @ Vt )
| ( subset_collections @ ( f5 @ X @ Vt ) @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[214]) ).
thf(288,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f3 @ X @ Vt ) @ Vt )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ X @ Vt ) ) @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[213]) ).
thf(289,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f4 @ X @ Vt ) @ Vt )
| ( subset_collections @ ( f5 @ X @ Vt ) @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[212]) ).
thf(290,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ( element_of_collection @ ( f4 @ X @ Vt ) @ Vt )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ X @ Vt ) ) @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[211]) ).
thf(291,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ X @ Vt ) @ ( f4 @ X @ Vt ) ) @ Vt )
| ( subset_collections @ ( f5 @ X @ Vt ) @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[210]) ).
thf(292,plain,
( ( ! [X: $i,Vt: $i] :
( ( topological_space @ X @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vt ) @ X )
| ~ ( element_of_collection @ empty_set @ Vt )
| ~ ( element_of_collection @ X @ Vt )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ X @ Vt ) @ ( f4 @ X @ Vt ) ) @ Vt )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ X @ Vt ) ) @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[209]) ).
thf(293,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ~ ( open @ U @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[208]) ).
thf(294,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ~ ( open @ U @ X @ Vt )
| ( element_of_collection @ U @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[207]) ).
thf(295,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ( open @ U @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( element_of_collection @ U @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[206]) ).
thf(296,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ~ ( closed @ U @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[205]) ).
thf(297,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ~ ( closed @ U @ X @ Vt )
| ( open @ ( relative_complement_sets @ U @ X ) @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[204]) ).
thf(298,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ( closed @ U @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( open @ ( relative_complement_sets @ U @ X ) @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[203]) ).
thf(299,plain,
( ( ! [Vt: $i,Vs: $i,X: $i] :
( ~ ( finer @ Vt @ Vs @ X )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[202]) ).
thf(300,plain,
( ( ! [Vt: $i,Vs: $i,X: $i] :
( ~ ( finer @ Vt @ Vs @ X )
| ( topological_space @ X @ Vs ) ) )
= $true ),
inference(copy,[status(thm)],[201]) ).
thf(301,plain,
( ( ! [Vt: $i,Vs: $i] :
( ! [X: $i] :
~ ( finer @ Vt @ Vs @ X )
| ( subset_collections @ Vs @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[266]) ).
thf(302,plain,
( ( ! [Vt: $i,Vs: $i,X: $i] :
( ( finer @ Vt @ Vs @ X )
| ~ ( topological_space @ X @ Vt )
| ~ ( topological_space @ X @ Vs )
| ~ ( subset_collections @ Vs @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[199]) ).
thf(303,plain,
( ( ! [X: $i,Vf: $i] :
( ~ ( basis @ X @ Vf )
| ( equal_sets @ ( union_of_members @ Vf ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[198]) ).
thf(304,plain,
( ( ! [X: $i,Vf: $i,Y: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ! [Vb1: $i] :
( ~ ( element_of_collection @ Vb1 @ Vf )
| ! [Vb2: $i] :
( ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( element_of_set @ Y @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[265]) ).
thf(305,plain,
( ( ! [X: $i,Vf: $i,Y: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ! [Vb1: $i] :
( ~ ( element_of_collection @ Vb1 @ Vf )
| ! [Vb2: $i] :
( ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( element_of_collection @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) @ Vf ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[264]) ).
thf(306,plain,
( ( ! [X: $i,Vf: $i,Y: $i] :
( ~ ( basis @ X @ Vf )
| ~ ( element_of_set @ Y @ X )
| ! [Vb1: $i] :
( ~ ( element_of_collection @ Vb1 @ Vf )
| ! [Vb2: $i] :
( ~ ( element_of_collection @ Vb2 @ Vf )
| ~ ( element_of_set @ Y @ ( intersection_of_sets @ Vb1 @ Vb2 ) )
| ( subset_sets @ ( f6 @ X @ Vf @ Y @ Vb1 @ Vb2 ) @ ( intersection_of_sets @ Vb1 @ Vb2 ) ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[263]) ).
thf(307,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_set @ ( f7 @ X @ Vf ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[194]) ).
thf(308,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_collection @ ( f8 @ X @ Vf ) @ Vf ) ) )
= $true ),
inference(copy,[status(thm)],[193]) ).
thf(309,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_collection @ ( f9 @ X @ Vf ) @ Vf ) ) )
= $true ),
inference(copy,[status(thm)],[192]) ).
thf(310,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ( element_of_set @ ( f7 @ X @ Vf ) @ ( intersection_of_sets @ ( f8 @ X @ Vf ) @ ( f9 @ X @ Vf ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[191]) ).
thf(311,plain,
( ( ! [X: $i,Vf: $i] :
( ( basis @ X @ Vf )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X )
| ! [Uu9: $i] :
( ~ ( element_of_set @ ( f7 @ X @ Vf ) @ Uu9 )
| ~ ( element_of_collection @ Uu9 @ Vf )
| ~ ( subset_sets @ Uu9 @ ( intersection_of_sets @ ( f8 @ X @ Vf ) @ ( f9 @ X @ Vf ) ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[262]) ).
thf(312,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ! [X: $i] :
( ~ ( element_of_set @ X @ U )
| ( element_of_set @ X @ ( f10 @ Vf @ U @ X ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[261]) ).
thf(313,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ! [X: $i] :
( ~ ( element_of_set @ X @ U )
| ( element_of_collection @ ( f10 @ Vf @ U @ X ) @ Vf ) ) ) )
= $true ),
inference(copy,[status(thm)],[260]) ).
thf(314,plain,
( ( ! [U: $i,Vf: $i] :
( ~ ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ! [X: $i] :
( ~ ( element_of_set @ X @ U )
| ( subset_sets @ ( f10 @ Vf @ U @ X ) @ U ) ) ) )
= $true ),
inference(copy,[status(thm)],[259]) ).
thf(315,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ( element_of_set @ ( f11 @ Vf @ U ) @ U ) ) )
= $true ),
inference(copy,[status(thm)],[186]) ).
thf(316,plain,
( ( ! [U: $i,Vf: $i] :
( ( element_of_collection @ U @ ( top_of_basis @ Vf ) )
| ! [Uu11: $i] :
( ~ ( element_of_set @ ( f11 @ Vf @ U ) @ Uu11 )
| ~ ( element_of_collection @ Uu11 @ Vf )
| ~ ( subset_sets @ Uu11 @ U ) ) ) )
= $true ),
inference(copy,[status(thm)],[258]) ).
thf(317,plain,
( ( ! [U: $i,X: $i,Vt: $i] :
( ! [Y: $i] :
~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[257]) ).
thf(318,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(copy,[status(thm)],[183]) ).
thf(319,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( element_of_collection @ ( f12 @ X @ Vt @ Y @ U ) @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[182]) ).
thf(320,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i] :
( ~ ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ( equal_sets @ U @ ( intersection_of_sets @ Y @ ( f12 @ X @ Vt @ Y @ U ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[181]) ).
thf(321,plain,
( ( ! [U: $i,X: $i,Vt: $i,Y: $i] :
( ( element_of_collection @ U @ ( subspace_topology @ X @ Vt @ Y ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ! [Uu12: $i] :
( ~ ( element_of_collection @ Uu12 @ Vt )
| ~ ( equal_sets @ U @ ( intersection_of_sets @ Y @ Uu12 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[256]) ).
thf(322,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[179]) ).
thf(323,plain,
( ( ! [U: $i,Y: $i,X: $i] :
( ! [Vt: $i] :
~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(copy,[status(thm)],[255]) ).
thf(324,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( element_of_set @ U @ ( f13 @ Y @ X @ Vt @ U ) ) ) )
= $true ),
inference(copy,[status(thm)],[177]) ).
thf(325,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( subset_sets @ ( f13 @ Y @ X @ Vt @ U ) @ Y ) ) )
= $true ),
inference(copy,[status(thm)],[176]) ).
thf(326,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ( open @ ( f13 @ Y @ X @ Vt @ U ) @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[175]) ).
thf(327,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( interior @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ! [Uu13: $i] :
( ~ ( element_of_set @ U @ Uu13 )
| ~ ( subset_sets @ Uu13 @ Y )
| ~ ( open @ Uu13 @ X @ Vt ) ) ) )
= $true ),
inference(copy,[status(thm)],[254]) ).
thf(328,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[173]) ).
thf(329,plain,
( ( ! [U: $i,Y: $i,X: $i] :
( ! [Vt: $i] :
~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(copy,[status(thm)],[253]) ).
thf(330,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ! [V: $i] :
( ~ ( subset_sets @ Y @ V )
| ~ ( closed @ V @ X @ Vt )
| ( element_of_set @ U @ V ) ) ) )
= $true ),
inference(copy,[status(thm)],[252]) ).
thf(331,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ( subset_sets @ Y @ ( f14 @ Y @ X @ Vt @ U ) ) ) )
= $true ),
inference(copy,[status(thm)],[170]) ).
thf(332,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ( closed @ ( f14 @ Y @ X @ Vt @ U ) @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[169]) ).
thf(333,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ~ ( element_of_set @ U @ ( f14 @ Y @ X @ Vt @ U ) ) ) )
= $true ),
inference(copy,[status(thm)],[168]) ).
thf(334,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( neighborhood @ U @ Y @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[167]) ).
thf(335,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( neighborhood @ U @ Y @ X @ Vt )
| ( open @ U @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[166]) ).
thf(336,plain,
( ( ! [U: $i,Y: $i] :
( ! [X: $i,Vt: $i] :
~ ( neighborhood @ U @ Y @ X @ Vt )
| ( element_of_set @ Y @ U ) ) )
= $true ),
inference(copy,[status(thm)],[251]) ).
thf(337,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( neighborhood @ U @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( open @ U @ X @ Vt )
| ~ ( element_of_set @ Y @ U ) ) )
= $true ),
inference(copy,[status(thm)],[164]) ).
thf(338,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[163]) ).
thf(339,plain,
( ( ! [Z: $i,Y: $i,X: $i] :
( ! [Vt: $i] :
~ ( limit_point @ Z @ Y @ X @ Vt )
| ( subset_sets @ Y @ X ) ) )
= $true ),
inference(copy,[status(thm)],[250]) ).
thf(340,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ! [U: $i] :
( ~ ( neighborhood @ U @ Z @ X @ Vt )
| ( element_of_set @ ( f15 @ Z @ Y @ X @ Vt @ U ) @ ( intersection_of_sets @ U @ Y ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[249]) ).
thf(341,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( limit_point @ Z @ Y @ X @ Vt )
| ! [U: $i] :
( ~ ( neighborhood @ U @ Z @ X @ Vt )
| ~ ( eq_p @ ( f15 @ Z @ Y @ X @ Vt @ U ) @ Z ) ) ) )
= $true ),
inference(copy,[status(thm)],[248]) ).
thf(342,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ( neighborhood @ ( f16 @ Z @ Y @ X @ Vt ) @ Z @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[159]) ).
thf(343,plain,
( ( ! [Z: $i,Y: $i,X: $i,Vt: $i] :
( ( limit_point @ Z @ Y @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Y @ X )
| ! [Uu16: $i] :
( ~ ( element_of_set @ Uu16 @ ( intersection_of_sets @ ( f16 @ Z @ Y @ X @ Vt ) @ Y ) )
| ( eq_p @ Uu16 @ Z ) ) ) )
= $true ),
inference(copy,[status(thm)],[247]) ).
thf(344,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[157]) ).
thf(345,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) ) ) )
= $true ),
inference(copy,[status(thm)],[156]) ).
thf(346,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ~ ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ( element_of_set @ U @ ( closure @ ( relative_complement_sets @ Y @ X ) @ X @ Vt ) ) ) )
= $true ),
inference(copy,[status(thm)],[155]) ).
thf(347,plain,
( ( ! [U: $i,Y: $i,X: $i,Vt: $i] :
( ( element_of_set @ U @ ( boundary @ Y @ X @ Vt ) )
| ~ ( topological_space @ X @ Vt )
| ~ ( element_of_set @ U @ ( closure @ Y @ X @ Vt ) )
| ~ ( element_of_set @ U @ ( closure @ ( relative_complement_sets @ Y @ X ) @ X @ Vt ) ) ) )
= $true ),
inference(copy,[status(thm)],[154]) ).
thf(348,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( hausdorff @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[153]) ).
thf(349,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( hausdorff @ X @ Vt )
| ! [X_1: $i] :
( ~ ( element_of_set @ X_1 @ X )
| ! [X_2: $i] :
( ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( neighborhood @ ( f17 @ X @ Vt @ X_1 @ X_2 ) @ X_1 @ X @ Vt ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[246]) ).
thf(350,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( hausdorff @ X @ Vt )
| ! [X_1: $i] :
( ~ ( element_of_set @ X_1 @ X )
| ! [X_2: $i] :
( ~ ( element_of_set @ X_2 @ X )
| ( eq_p @ X_1 @ X_2 )
| ( neighborhood @ ( f18 @ X @ Vt @ X_1 @ X_2 ) @ X_2 @ X @ Vt ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[245]) ).
thf(351,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( hausdorff @ X @ Vt )
| ! [X_1: $i] :
( ~ ( element_of_set @ X_1 @ X )
| ! [X_2: $i] :
( ~ ( element_of_set @ X_2 @ X )
| ( disjoint_s @ ( f17 @ X @ Vt @ X_1 @ X_2 ) @ ( f18 @ X @ Vt @ X_1 @ X_2 ) )
| ( eq_p @ X_1 @ X_2 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[244]) ).
thf(352,plain,
( ( ! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( element_of_set @ ( f19 @ X @ Vt ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[149]) ).
thf(353,plain,
( ( ! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( element_of_set @ ( f20 @ X @ Vt ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[148]) ).
thf(354,plain,
( ( ! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( eq_p @ ( f19 @ X @ Vt ) @ ( f20 @ X @ Vt ) ) ) )
= $true ),
inference(copy,[status(thm)],[147]) ).
thf(355,plain,
( ( ! [X: $i,Vt: $i] :
( ( hausdorff @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ! [Uu19: $i] :
( ~ ( neighborhood @ Uu19 @ ( f19 @ X @ Vt ) @ X @ Vt )
| ! [Uu20: $i] :
( ~ ( neighborhood @ Uu20 @ ( f20 @ X @ Vt ) @ X @ Vt )
| ~ ( disjoint_s @ Uu19 @ Uu20 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[243]) ).
thf(356,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[145]) ).
thf(357,plain,
( ( ! [Va1: $i,Va2: $i] :
( ! [X: $i,Vt: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( equal_sets @ Va1 @ empty_set ) ) )
= $true ),
inference(copy,[status(thm)],[242]) ).
thf(358,plain,
( ( ! [Va1: $i,Va2: $i] :
( ! [X: $i,Vt: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( equal_sets @ Va2 @ empty_set ) ) )
= $true ),
inference(copy,[status(thm)],[241]) ).
thf(359,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( element_of_collection @ Va1 @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[142]) ).
thf(360,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( element_of_collection @ Va2 @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[141]) ).
thf(361,plain,
( ( ! [Va1: $i,Va2: $i,X: $i] :
( ! [Vt: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( equal_sets @ ( union_of_sets @ Va1 @ Va2 ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[240]) ).
thf(362,plain,
( ( ! [Va1: $i,Va2: $i] :
( ! [X: $i,Vt: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt )
| ( disjoint_s @ Va1 @ Va2 ) ) )
= $true ),
inference(copy,[status(thm)],[239]) ).
thf(363,plain,
( ( ! [Va1: $i,Va2: $i,X: $i,Vt: $i] :
( ( separation @ Va1 @ Va2 @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( equal_sets @ Va1 @ empty_set )
| ( equal_sets @ Va2 @ empty_set )
| ~ ( element_of_collection @ Va1 @ Vt )
| ~ ( element_of_collection @ Va2 @ Vt )
| ~ ( equal_sets @ ( union_of_sets @ Va1 @ Va2 ) @ X )
| ~ ( disjoint_s @ Va1 @ Va2 ) ) )
= $true ),
inference(copy,[status(thm)],[138]) ).
thf(364,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( connected_space @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[137]) ).
thf(365,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( connected_space @ X @ Vt )
| ! [Va1: $i,Va2: $i] :
~ ( separation @ Va1 @ Va2 @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[238]) ).
thf(366,plain,
( ( ! [X: $i,Vt: $i] :
( ( connected_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( separation @ ( f21 @ X @ Vt ) @ ( f22 @ X @ Vt ) @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[135]) ).
thf(367,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( connected_set @ Va @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[134]) ).
thf(368,plain,
( ( ! [Va: $i,X: $i] :
( ! [Vt: $i] :
~ ( connected_set @ Va @ X @ Vt )
| ( subset_sets @ Va @ X ) ) )
= $true ),
inference(copy,[status(thm)],[237]) ).
thf(369,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( connected_set @ Va @ X @ Vt )
| ( connected_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ) )
= $true ),
inference(copy,[status(thm)],[132]) ).
thf(370,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ( connected_set @ Va @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Va @ X )
| ~ ( connected_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ) )
= $true ),
inference(copy,[status(thm)],[131]) ).
thf(371,plain,
( ( ! [Vf: $i,X: $i,Vt: $i] :
( ~ ( open_covering @ Vf @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[130]) ).
thf(372,plain,
( ( ! [Vf: $i,X: $i,Vt: $i] :
( ~ ( open_covering @ Vf @ X @ Vt )
| ( subset_collections @ Vf @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[129]) ).
thf(373,plain,
( ( ! [Vf: $i,X: $i] :
( ! [Vt: $i] :
~ ( open_covering @ Vf @ X @ Vt )
| ( equal_sets @ ( union_of_members @ Vf ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[236]) ).
thf(374,plain,
( ( ! [Vf: $i,X: $i,Vt: $i] :
( ( open_covering @ Vf @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_collections @ Vf @ Vt )
| ~ ( equal_sets @ ( union_of_members @ Vf ) @ X ) ) )
= $true ),
inference(copy,[status(thm)],[127]) ).
thf(375,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( compact_space @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[126]) ).
thf(376,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( compact_space @ X @ Vt )
| ! [Vf1: $i] :
( ~ ( open_covering @ Vf1 @ X @ Vt )
| ( finite @ ( f23 @ X @ Vt @ Vf1 ) ) ) ) )
= $true ),
inference(copy,[status(thm)],[235]) ).
thf(377,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( compact_space @ X @ Vt )
| ! [Vf1: $i] :
( ~ ( open_covering @ Vf1 @ X @ Vt )
| ( subset_collections @ ( f23 @ X @ Vt @ Vf1 ) @ Vf1 ) ) ) )
= $true ),
inference(copy,[status(thm)],[234]) ).
thf(378,plain,
( ( ! [X: $i,Vt: $i] :
( ~ ( compact_space @ X @ Vt )
| ! [Vf1: $i] :
( ~ ( open_covering @ Vf1 @ X @ Vt )
| ( open_covering @ ( f23 @ X @ Vt @ Vf1 ) @ X @ Vt ) ) ) )
= $true ),
inference(copy,[status(thm)],[233]) ).
thf(379,plain,
( ( ! [X: $i,Vt: $i] :
( ( compact_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ( open_covering @ ( f24 @ X @ Vt ) @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[122]) ).
thf(380,plain,
( ( ! [X: $i,Vt: $i] :
( ( compact_space @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ! [Uu24: $i] :
( ~ ( finite @ Uu24 )
| ~ ( subset_collections @ Uu24 @ ( f24 @ X @ Vt ) )
| ~ ( open_covering @ Uu24 @ X @ Vt ) ) ) )
= $true ),
inference(copy,[status(thm)],[232]) ).
thf(381,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( compact_set @ Va @ X @ Vt )
| ( topological_space @ X @ Vt ) ) )
= $true ),
inference(copy,[status(thm)],[120]) ).
thf(382,plain,
( ( ! [Va: $i,X: $i] :
( ! [Vt: $i] :
~ ( compact_set @ Va @ X @ Vt )
| ( subset_sets @ Va @ X ) ) )
= $true ),
inference(copy,[status(thm)],[231]) ).
thf(383,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ~ ( compact_set @ Va @ X @ Vt )
| ( compact_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ) )
= $true ),
inference(copy,[status(thm)],[118]) ).
thf(384,plain,
( ( ! [Va: $i,X: $i,Vt: $i] :
( ( compact_set @ Va @ X @ Vt )
| ~ ( topological_space @ X @ Vt )
| ~ ( subset_sets @ Va @ X )
| ~ ( compact_space @ Va @ ( subspace_topology @ X @ Vt @ Va ) ) ) )
= $true ),
inference(copy,[status(thm)],[117]) ).
thf(385,plain,
( ( ~ $false ) = $true ),
inference(copy,[status(thm)],[230]) ).
thf(386,plain,
! [SV1: $i] :
( ( element_of_collection @ SV1 @ ( top_of_basis @ f ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[273]) ).
thf(387,plain,
! [SV2: $i] :
( ( element_of_collection @ SV2 @ ( top_of_basis @ f ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[274]) ).
thf(388,plain,
! [SV3: $i] :
( ( ! [SY361: $i] :
~ ( element_of_collection @ ( intersection_of_sets @ SV3 @ SY361 ) @ ( top_of_basis @ f ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[275]) ).
thf(389,plain,
! [SV4: $i] :
( ( ! [SY362: $i] :
( ~ ( element_of_set @ SV4 @ ( union_of_members @ SY362 ) )
| ( element_of_set @ SV4 @ ( f1 @ SY362 @ SV4 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[276]) ).
thf(390,plain,
! [SV5: $i] :
( ( ! [SY363: $i] :
( ~ ( element_of_set @ SV5 @ ( union_of_members @ SY363 ) )
| ( element_of_collection @ ( f1 @ SY363 @ SV5 ) @ SY363 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[277]) ).
thf(391,plain,
! [SV6: $i] :
( ( ! [SY364: $i] :
( ( element_of_set @ SV6 @ ( union_of_members @ SY364 ) )
| ! [SY365: $i] :
( ~ ( element_of_set @ SV6 @ SY365 )
| ~ ( element_of_collection @ SY365 @ SY364 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[278]) ).
thf(392,plain,
! [SV7: $i] :
( ( ! [SY366: $i] :
( ~ ( element_of_set @ SV7 @ ( intersection_of_members @ SY366 ) )
| ! [SY367: $i] :
( ~ ( element_of_collection @ SY367 @ SY366 )
| ( element_of_set @ SV7 @ SY367 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[279]) ).
thf(393,plain,
! [SV8: $i] :
( ( ! [SY368: $i] :
( ( element_of_collection @ ( f2 @ SY368 @ SV8 ) @ SY368 )
| ( element_of_set @ SV8 @ ( intersection_of_members @ SY368 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[280]) ).
thf(394,plain,
! [SV9: $i] :
( ( ! [SY369: $i] :
( ( element_of_set @ SV9 @ ( intersection_of_members @ SY369 ) )
| ~ ( element_of_set @ SV9 @ ( f2 @ SY369 @ SV9 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[281]) ).
thf(395,plain,
! [SV10: $i] :
( ( ! [SY370: $i] :
( ~ ( topological_space @ SV10 @ SY370 )
| ( equal_sets @ ( union_of_members @ SY370 ) @ SV10 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[282]) ).
thf(396,plain,
! [SV11: $i] :
( ( ! [SY371: $i] :
( ~ ( topological_space @ SV11 @ SY371 )
| ( element_of_collection @ empty_set @ SY371 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[283]) ).
thf(397,plain,
! [SV12: $i] :
( ( ! [SY372: $i] :
( ~ ( topological_space @ SV12 @ SY372 )
| ( element_of_collection @ SV12 @ SY372 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[284]) ).
thf(398,plain,
! [SV13: $i] :
( ( ! [SY373: $i] :
( ~ ( topological_space @ SV13 @ SY373 )
| ! [Y: $i] :
( ~ ( element_of_collection @ Y @ SY373 )
| ! [Z: $i] :
( ~ ( element_of_collection @ Z @ SY373 )
| ( element_of_collection @ ( intersection_of_sets @ Y @ Z ) @ SY373 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[285]) ).
thf(399,plain,
! [SV14: $i] :
( ( ! [SY376: $i] :
( ~ ( topological_space @ SV14 @ SY376 )
| ! [Vf: $i] :
( ~ ( subset_collections @ Vf @ SY376 )
| ( element_of_collection @ ( union_of_members @ Vf ) @ SY376 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[286]) ).
thf(400,plain,
! [SV15: $i] :
( ( ! [SY378: $i] :
( ( topological_space @ SV15 @ SY378 )
| ~ ( equal_sets @ ( union_of_members @ SY378 ) @ SV15 )
| ~ ( element_of_collection @ empty_set @ SY378 )
| ~ ( element_of_collection @ SV15 @ SY378 )
| ( element_of_collection @ ( f3 @ SV15 @ SY378 ) @ SY378 )
| ( subset_collections @ ( f5 @ SV15 @ SY378 ) @ SY378 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[287]) ).
thf(401,plain,
! [SV16: $i] :
( ( ! [SY379: $i] :
( ( topological_space @ SV16 @ SY379 )
| ~ ( equal_sets @ ( union_of_members @ SY379 ) @ SV16 )
| ~ ( element_of_collection @ empty_set @ SY379 )
| ~ ( element_of_collection @ SV16 @ SY379 )
| ( element_of_collection @ ( f3 @ SV16 @ SY379 ) @ SY379 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SY379 ) ) @ SY379 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[288]) ).
thf(402,plain,
! [SV17: $i] :
( ( ! [SY380: $i] :
( ( topological_space @ SV17 @ SY380 )
| ~ ( equal_sets @ ( union_of_members @ SY380 ) @ SV17 )
| ~ ( element_of_collection @ empty_set @ SY380 )
| ~ ( element_of_collection @ SV17 @ SY380 )
| ( element_of_collection @ ( f4 @ SV17 @ SY380 ) @ SY380 )
| ( subset_collections @ ( f5 @ SV17 @ SY380 ) @ SY380 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[289]) ).
thf(403,plain,
! [SV18: $i] :
( ( ! [SY381: $i] :
( ( topological_space @ SV18 @ SY381 )
| ~ ( equal_sets @ ( union_of_members @ SY381 ) @ SV18 )
| ~ ( element_of_collection @ empty_set @ SY381 )
| ~ ( element_of_collection @ SV18 @ SY381 )
| ( element_of_collection @ ( f4 @ SV18 @ SY381 ) @ SY381 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SY381 ) ) @ SY381 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[290]) ).
thf(404,plain,
! [SV19: $i] :
( ( ! [SY382: $i] :
( ( topological_space @ SV19 @ SY382 )
| ~ ( equal_sets @ ( union_of_members @ SY382 ) @ SV19 )
| ~ ( element_of_collection @ empty_set @ SY382 )
| ~ ( element_of_collection @ SV19 @ SY382 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SY382 ) @ ( f4 @ SV19 @ SY382 ) ) @ SY382 )
| ( subset_collections @ ( f5 @ SV19 @ SY382 ) @ SY382 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[291]) ).
thf(405,plain,
! [SV20: $i] :
( ( ! [SY383: $i] :
( ( topological_space @ SV20 @ SY383 )
| ~ ( equal_sets @ ( union_of_members @ SY383 ) @ SV20 )
| ~ ( element_of_collection @ empty_set @ SY383 )
| ~ ( element_of_collection @ SV20 @ SY383 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SY383 ) @ ( f4 @ SV20 @ SY383 ) ) @ SY383 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SY383 ) ) @ SY383 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[292]) ).
thf(406,plain,
! [SV21: $i] :
( ( ! [SY384: $i,SY385: $i] :
( ~ ( open @ SV21 @ SY384 @ SY385 )
| ( topological_space @ SY384 @ SY385 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[293]) ).
thf(407,plain,
! [SV22: $i] :
( ( ! [SY386: $i,SY387: $i] :
( ~ ( open @ SV22 @ SY386 @ SY387 )
| ( element_of_collection @ SV22 @ SY387 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[294]) ).
thf(408,plain,
! [SV23: $i] :
( ( ! [SY388: $i,SY389: $i] :
( ( open @ SV23 @ SY388 @ SY389 )
| ~ ( topological_space @ SY388 @ SY389 )
| ~ ( element_of_collection @ SV23 @ SY389 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[295]) ).
thf(409,plain,
! [SV24: $i] :
( ( ! [SY390: $i,SY391: $i] :
( ~ ( closed @ SV24 @ SY390 @ SY391 )
| ( topological_space @ SY390 @ SY391 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[296]) ).
thf(410,plain,
! [SV25: $i] :
( ( ! [SY392: $i,SY393: $i] :
( ~ ( closed @ SV25 @ SY392 @ SY393 )
| ( open @ ( relative_complement_sets @ SV25 @ SY392 ) @ SY392 @ SY393 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[297]) ).
thf(411,plain,
! [SV26: $i] :
( ( ! [SY394: $i,SY395: $i] :
( ( closed @ SV26 @ SY394 @ SY395 )
| ~ ( topological_space @ SY394 @ SY395 )
| ~ ( open @ ( relative_complement_sets @ SV26 @ SY394 ) @ SY394 @ SY395 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[298]) ).
thf(412,plain,
! [SV27: $i] :
( ( ! [SY396: $i,SY397: $i] :
( ~ ( finer @ SV27 @ SY396 @ SY397 )
| ( topological_space @ SY397 @ SV27 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[299]) ).
thf(413,plain,
! [SV28: $i] :
( ( ! [SY398: $i,SY399: $i] :
( ~ ( finer @ SV28 @ SY398 @ SY399 )
| ( topological_space @ SY399 @ SY398 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[300]) ).
thf(414,plain,
! [SV29: $i] :
( ( ! [SY400: $i] :
( ! [SY401: $i] :
~ ( finer @ SV29 @ SY400 @ SY401 )
| ( subset_collections @ SY400 @ SV29 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[301]) ).
thf(415,plain,
! [SV30: $i] :
( ( ! [SY402: $i,SY403: $i] :
( ( finer @ SV30 @ SY402 @ SY403 )
| ~ ( topological_space @ SY403 @ SV30 )
| ~ ( topological_space @ SY403 @ SY402 )
| ~ ( subset_collections @ SY402 @ SV30 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[302]) ).
thf(416,plain,
! [SV31: $i] :
( ( ! [SY404: $i] :
( ~ ( basis @ SV31 @ SY404 )
| ( equal_sets @ ( union_of_members @ SY404 ) @ SV31 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[303]) ).
thf(417,plain,
! [SV32: $i] :
( ( ! [SY405: $i,SY406: $i] :
( ~ ( basis @ SV32 @ SY405 )
| ~ ( element_of_set @ SY406 @ SV32 )
| ! [SY407: $i] :
( ~ ( element_of_collection @ SY407 @ SY405 )
| ! [SY408: $i] :
( ~ ( element_of_collection @ SY408 @ SY405 )
| ~ ( element_of_set @ SY406 @ ( intersection_of_sets @ SY407 @ SY408 ) )
| ( element_of_set @ SY406 @ ( f6 @ SV32 @ SY405 @ SY406 @ SY407 @ SY408 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[304]) ).
thf(418,plain,
! [SV33: $i] :
( ( ! [SY409: $i,SY410: $i] :
( ~ ( basis @ SV33 @ SY409 )
| ~ ( element_of_set @ SY410 @ SV33 )
| ! [SY411: $i] :
( ~ ( element_of_collection @ SY411 @ SY409 )
| ! [SY412: $i] :
( ~ ( element_of_collection @ SY412 @ SY409 )
| ~ ( element_of_set @ SY410 @ ( intersection_of_sets @ SY411 @ SY412 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SY409 @ SY410 @ SY411 @ SY412 ) @ SY409 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[305]) ).
thf(419,plain,
! [SV34: $i] :
( ( ! [SY413: $i,SY414: $i] :
( ~ ( basis @ SV34 @ SY413 )
| ~ ( element_of_set @ SY414 @ SV34 )
| ! [SY415: $i] :
( ~ ( element_of_collection @ SY415 @ SY413 )
| ! [SY416: $i] :
( ~ ( element_of_collection @ SY416 @ SY413 )
| ~ ( element_of_set @ SY414 @ ( intersection_of_sets @ SY415 @ SY416 ) )
| ( subset_sets @ ( f6 @ SV34 @ SY413 @ SY414 @ SY415 @ SY416 ) @ ( intersection_of_sets @ SY415 @ SY416 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[306]) ).
thf(420,plain,
! [SV35: $i] :
( ( ! [SY417: $i] :
( ( basis @ SV35 @ SY417 )
| ~ ( equal_sets @ ( union_of_members @ SY417 ) @ SV35 )
| ( element_of_set @ ( f7 @ SV35 @ SY417 ) @ SV35 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[307]) ).
thf(421,plain,
! [SV36: $i] :
( ( ! [SY418: $i] :
( ( basis @ SV36 @ SY418 )
| ~ ( equal_sets @ ( union_of_members @ SY418 ) @ SV36 )
| ( element_of_collection @ ( f8 @ SV36 @ SY418 ) @ SY418 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[308]) ).
thf(422,plain,
! [SV37: $i] :
( ( ! [SY419: $i] :
( ( basis @ SV37 @ SY419 )
| ~ ( equal_sets @ ( union_of_members @ SY419 ) @ SV37 )
| ( element_of_collection @ ( f9 @ SV37 @ SY419 ) @ SY419 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[309]) ).
thf(423,plain,
! [SV38: $i] :
( ( ! [SY420: $i] :
( ( basis @ SV38 @ SY420 )
| ~ ( equal_sets @ ( union_of_members @ SY420 ) @ SV38 )
| ( element_of_set @ ( f7 @ SV38 @ SY420 ) @ ( intersection_of_sets @ ( f8 @ SV38 @ SY420 ) @ ( f9 @ SV38 @ SY420 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[310]) ).
thf(424,plain,
! [SV39: $i] :
( ( ! [SY421: $i] :
( ( basis @ SV39 @ SY421 )
| ~ ( equal_sets @ ( union_of_members @ SY421 ) @ SV39 )
| ! [SY422: $i] :
( ~ ( element_of_set @ ( f7 @ SV39 @ SY421 ) @ SY422 )
| ~ ( element_of_collection @ SY422 @ SY421 )
| ~ ( subset_sets @ SY422 @ ( intersection_of_sets @ ( f8 @ SV39 @ SY421 ) @ ( f9 @ SV39 @ SY421 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[311]) ).
thf(425,plain,
! [SV40: $i] :
( ( ! [SY423: $i] :
( ~ ( element_of_collection @ SV40 @ ( top_of_basis @ SY423 ) )
| ! [SY424: $i] :
( ~ ( element_of_set @ SY424 @ SV40 )
| ( element_of_set @ SY424 @ ( f10 @ SY423 @ SV40 @ SY424 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[312]) ).
thf(426,plain,
! [SV41: $i] :
( ( ! [SY425: $i] :
( ~ ( element_of_collection @ SV41 @ ( top_of_basis @ SY425 ) )
| ! [SY426: $i] :
( ~ ( element_of_set @ SY426 @ SV41 )
| ( element_of_collection @ ( f10 @ SY425 @ SV41 @ SY426 ) @ SY425 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[313]) ).
thf(427,plain,
! [SV42: $i] :
( ( ! [SY427: $i] :
( ~ ( element_of_collection @ SV42 @ ( top_of_basis @ SY427 ) )
| ! [SY428: $i] :
( ~ ( element_of_set @ SY428 @ SV42 )
| ( subset_sets @ ( f10 @ SY427 @ SV42 @ SY428 ) @ SV42 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[314]) ).
thf(428,plain,
! [SV43: $i] :
( ( ! [SY429: $i] :
( ( element_of_collection @ SV43 @ ( top_of_basis @ SY429 ) )
| ( element_of_set @ ( f11 @ SY429 @ SV43 ) @ SV43 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[315]) ).
thf(429,plain,
! [SV44: $i] :
( ( ! [SY430: $i] :
( ( element_of_collection @ SV44 @ ( top_of_basis @ SY430 ) )
| ! [SY431: $i] :
( ~ ( element_of_set @ ( f11 @ SY430 @ SV44 ) @ SY431 )
| ~ ( element_of_collection @ SY431 @ SY430 )
| ~ ( subset_sets @ SY431 @ SV44 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[316]) ).
thf(430,plain,
! [SV45: $i] :
( ( ! [SY432: $i,SY433: $i] :
( ! [SY434: $i] :
~ ( element_of_collection @ SV45 @ ( subspace_topology @ SY432 @ SY433 @ SY434 ) )
| ( topological_space @ SY432 @ SY433 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[317]) ).
thf(431,plain,
! [SV46: $i] :
( ( ! [SY435: $i,SY436: $i,SY437: $i] :
( ~ ( element_of_collection @ SV46 @ ( subspace_topology @ SY435 @ SY436 @ SY437 ) )
| ( subset_sets @ SY437 @ SY435 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[318]) ).
thf(432,plain,
! [SV47: $i] :
( ( ! [SY438: $i,SY439: $i,SY440: $i] :
( ~ ( element_of_collection @ SV47 @ ( subspace_topology @ SY438 @ SY439 @ SY440 ) )
| ( element_of_collection @ ( f12 @ SY438 @ SY439 @ SY440 @ SV47 ) @ SY439 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[319]) ).
thf(433,plain,
! [SV48: $i] :
( ( ! [SY441: $i,SY442: $i,SY443: $i] :
( ~ ( element_of_collection @ SV48 @ ( subspace_topology @ SY441 @ SY442 @ SY443 ) )
| ( equal_sets @ SV48 @ ( intersection_of_sets @ SY443 @ ( f12 @ SY441 @ SY442 @ SY443 @ SV48 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[320]) ).
thf(434,plain,
! [SV49: $i] :
( ( ! [SY444: $i,SY445: $i,SY446: $i] :
( ( element_of_collection @ SV49 @ ( subspace_topology @ SY444 @ SY445 @ SY446 ) )
| ~ ( topological_space @ SY444 @ SY445 )
| ~ ( subset_sets @ SY446 @ SY444 )
| ! [SY447: $i] :
( ~ ( element_of_collection @ SY447 @ SY445 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SY446 @ SY447 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[321]) ).
thf(435,plain,
! [SV50: $i] :
( ( ! [SY448: $i,SY449: $i,SY450: $i] :
( ~ ( element_of_set @ SV50 @ ( interior @ SY448 @ SY449 @ SY450 ) )
| ( topological_space @ SY449 @ SY450 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[322]) ).
thf(436,plain,
! [SV51: $i] :
( ( ! [SY451: $i,SY452: $i] :
( ! [SY453: $i] :
~ ( element_of_set @ SV51 @ ( interior @ SY451 @ SY452 @ SY453 ) )
| ( subset_sets @ SY451 @ SY452 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[323]) ).
thf(437,plain,
! [SV52: $i] :
( ( ! [SY454: $i,SY455: $i,SY456: $i] :
( ~ ( element_of_set @ SV52 @ ( interior @ SY454 @ SY455 @ SY456 ) )
| ( element_of_set @ SV52 @ ( f13 @ SY454 @ SY455 @ SY456 @ SV52 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[324]) ).
thf(438,plain,
! [SV53: $i] :
( ( ! [SY457: $i,SY458: $i,SY459: $i] :
( ~ ( element_of_set @ SV53 @ ( interior @ SY457 @ SY458 @ SY459 ) )
| ( subset_sets @ ( f13 @ SY457 @ SY458 @ SY459 @ SV53 ) @ SY457 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[325]) ).
thf(439,plain,
! [SV54: $i] :
( ( ! [SY460: $i,SY461: $i,SY462: $i] :
( ~ ( element_of_set @ SV54 @ ( interior @ SY460 @ SY461 @ SY462 ) )
| ( open @ ( f13 @ SY460 @ SY461 @ SY462 @ SV54 ) @ SY461 @ SY462 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[326]) ).
thf(440,plain,
! [SV55: $i] :
( ( ! [SY463: $i,SY464: $i,SY465: $i] :
( ( element_of_set @ SV55 @ ( interior @ SY463 @ SY464 @ SY465 ) )
| ~ ( topological_space @ SY464 @ SY465 )
| ~ ( subset_sets @ SY463 @ SY464 )
| ! [SY466: $i] :
( ~ ( element_of_set @ SV55 @ SY466 )
| ~ ( subset_sets @ SY466 @ SY463 )
| ~ ( open @ SY466 @ SY464 @ SY465 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[327]) ).
thf(441,plain,
! [SV56: $i] :
( ( ! [SY467: $i,SY468: $i,SY469: $i] :
( ~ ( element_of_set @ SV56 @ ( closure @ SY467 @ SY468 @ SY469 ) )
| ( topological_space @ SY468 @ SY469 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[328]) ).
thf(442,plain,
! [SV57: $i] :
( ( ! [SY470: $i,SY471: $i] :
( ! [SY472: $i] :
~ ( element_of_set @ SV57 @ ( closure @ SY470 @ SY471 @ SY472 ) )
| ( subset_sets @ SY470 @ SY471 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[329]) ).
thf(443,plain,
! [SV58: $i] :
( ( ! [SY473: $i,SY474: $i,SY475: $i] :
( ~ ( element_of_set @ SV58 @ ( closure @ SY473 @ SY474 @ SY475 ) )
| ! [SY476: $i] :
( ~ ( subset_sets @ SY473 @ SY476 )
| ~ ( closed @ SY476 @ SY474 @ SY475 )
| ( element_of_set @ SV58 @ SY476 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[330]) ).
thf(444,plain,
! [SV59: $i] :
( ( ! [SY477: $i,SY478: $i,SY479: $i] :
( ( element_of_set @ SV59 @ ( closure @ SY477 @ SY478 @ SY479 ) )
| ~ ( topological_space @ SY478 @ SY479 )
| ~ ( subset_sets @ SY477 @ SY478 )
| ( subset_sets @ SY477 @ ( f14 @ SY477 @ SY478 @ SY479 @ SV59 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[331]) ).
thf(445,plain,
! [SV60: $i] :
( ( ! [SY480: $i,SY481: $i,SY482: $i] :
( ( element_of_set @ SV60 @ ( closure @ SY480 @ SY481 @ SY482 ) )
| ~ ( topological_space @ SY481 @ SY482 )
| ~ ( subset_sets @ SY480 @ SY481 )
| ( closed @ ( f14 @ SY480 @ SY481 @ SY482 @ SV60 ) @ SY481 @ SY482 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[332]) ).
thf(446,plain,
! [SV61: $i] :
( ( ! [SY483: $i,SY484: $i,SY485: $i] :
( ( element_of_set @ SV61 @ ( closure @ SY483 @ SY484 @ SY485 ) )
| ~ ( topological_space @ SY484 @ SY485 )
| ~ ( subset_sets @ SY483 @ SY484 )
| ~ ( element_of_set @ SV61 @ ( f14 @ SY483 @ SY484 @ SY485 @ SV61 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[333]) ).
thf(447,plain,
! [SV62: $i] :
( ( ! [SY486: $i,SY487: $i,SY488: $i] :
( ~ ( neighborhood @ SV62 @ SY486 @ SY487 @ SY488 )
| ( topological_space @ SY487 @ SY488 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[334]) ).
thf(448,plain,
! [SV63: $i] :
( ( ! [SY489: $i,SY490: $i,SY491: $i] :
( ~ ( neighborhood @ SV63 @ SY489 @ SY490 @ SY491 )
| ( open @ SV63 @ SY490 @ SY491 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[335]) ).
thf(449,plain,
! [SV64: $i] :
( ( ! [SY492: $i] :
( ! [SY493: $i,SY494: $i] :
~ ( neighborhood @ SV64 @ SY492 @ SY493 @ SY494 )
| ( element_of_set @ SY492 @ SV64 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[336]) ).
thf(450,plain,
! [SV65: $i] :
( ( ! [SY495: $i,SY496: $i,SY497: $i] :
( ( neighborhood @ SV65 @ SY495 @ SY496 @ SY497 )
| ~ ( topological_space @ SY496 @ SY497 )
| ~ ( open @ SV65 @ SY496 @ SY497 )
| ~ ( element_of_set @ SY495 @ SV65 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[337]) ).
thf(451,plain,
! [SV66: $i] :
( ( ! [SY498: $i,SY499: $i,SY500: $i] :
( ~ ( limit_point @ SV66 @ SY498 @ SY499 @ SY500 )
| ( topological_space @ SY499 @ SY500 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[338]) ).
thf(452,plain,
! [SV67: $i] :
( ( ! [SY501: $i,SY502: $i] :
( ! [SY503: $i] :
~ ( limit_point @ SV67 @ SY501 @ SY502 @ SY503 )
| ( subset_sets @ SY501 @ SY502 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[339]) ).
thf(453,plain,
! [SV68: $i] :
( ( ! [SY504: $i,SY505: $i,SY506: $i] :
( ~ ( limit_point @ SV68 @ SY504 @ SY505 @ SY506 )
| ! [SY507: $i] :
( ~ ( neighborhood @ SY507 @ SV68 @ SY505 @ SY506 )
| ( element_of_set @ ( f15 @ SV68 @ SY504 @ SY505 @ SY506 @ SY507 ) @ ( intersection_of_sets @ SY507 @ SY504 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[340]) ).
thf(454,plain,
! [SV69: $i] :
( ( ! [SY508: $i,SY509: $i,SY510: $i] :
( ~ ( limit_point @ SV69 @ SY508 @ SY509 @ SY510 )
| ! [SY511: $i] :
( ~ ( neighborhood @ SY511 @ SV69 @ SY509 @ SY510 )
| ~ ( eq_p @ ( f15 @ SV69 @ SY508 @ SY509 @ SY510 @ SY511 ) @ SV69 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[341]) ).
thf(455,plain,
! [SV70: $i] :
( ( ! [SY512: $i,SY513: $i,SY514: $i] :
( ( limit_point @ SV70 @ SY512 @ SY513 @ SY514 )
| ~ ( topological_space @ SY513 @ SY514 )
| ~ ( subset_sets @ SY512 @ SY513 )
| ( neighborhood @ ( f16 @ SV70 @ SY512 @ SY513 @ SY514 ) @ SV70 @ SY513 @ SY514 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[342]) ).
thf(456,plain,
! [SV71: $i] :
( ( ! [SY515: $i,SY516: $i,SY517: $i] :
( ( limit_point @ SV71 @ SY515 @ SY516 @ SY517 )
| ~ ( topological_space @ SY516 @ SY517 )
| ~ ( subset_sets @ SY515 @ SY516 )
| ! [SY518: $i] :
( ~ ( element_of_set @ SY518 @ ( intersection_of_sets @ ( f16 @ SV71 @ SY515 @ SY516 @ SY517 ) @ SY515 ) )
| ( eq_p @ SY518 @ SV71 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[343]) ).
thf(457,plain,
! [SV72: $i] :
( ( ! [SY519: $i,SY520: $i,SY521: $i] :
( ~ ( element_of_set @ SV72 @ ( boundary @ SY519 @ SY520 @ SY521 ) )
| ( topological_space @ SY520 @ SY521 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[344]) ).
thf(458,plain,
! [SV73: $i] :
( ( ! [SY522: $i,SY523: $i,SY524: $i] :
( ~ ( element_of_set @ SV73 @ ( boundary @ SY522 @ SY523 @ SY524 ) )
| ( element_of_set @ SV73 @ ( closure @ SY522 @ SY523 @ SY524 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[345]) ).
thf(459,plain,
! [SV74: $i] :
( ( ! [SY525: $i,SY526: $i,SY527: $i] :
( ~ ( element_of_set @ SV74 @ ( boundary @ SY525 @ SY526 @ SY527 ) )
| ( element_of_set @ SV74 @ ( closure @ ( relative_complement_sets @ SY525 @ SY526 ) @ SY526 @ SY527 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[346]) ).
thf(460,plain,
! [SV75: $i] :
( ( ! [SY528: $i,SY529: $i,SY530: $i] :
( ( element_of_set @ SV75 @ ( boundary @ SY528 @ SY529 @ SY530 ) )
| ~ ( topological_space @ SY529 @ SY530 )
| ~ ( element_of_set @ SV75 @ ( closure @ SY528 @ SY529 @ SY530 ) )
| ~ ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SY528 @ SY529 ) @ SY529 @ SY530 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[347]) ).
thf(461,plain,
! [SV76: $i] :
( ( ! [SY531: $i] :
( ~ ( hausdorff @ SV76 @ SY531 )
| ( topological_space @ SV76 @ SY531 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[348]) ).
thf(462,plain,
! [SV77: $i] :
( ( ! [SY532: $i] :
( ~ ( hausdorff @ SV77 @ SY532 )
| ! [SY533: $i] :
( ~ ( element_of_set @ SY533 @ SV77 )
| ! [SY534: $i] :
( ~ ( element_of_set @ SY534 @ SV77 )
| ( eq_p @ SY533 @ SY534 )
| ( neighborhood @ ( f17 @ SV77 @ SY532 @ SY533 @ SY534 ) @ SY533 @ SV77 @ SY532 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[349]) ).
thf(463,plain,
! [SV78: $i] :
( ( ! [SY535: $i] :
( ~ ( hausdorff @ SV78 @ SY535 )
| ! [SY536: $i] :
( ~ ( element_of_set @ SY536 @ SV78 )
| ! [SY537: $i] :
( ~ ( element_of_set @ SY537 @ SV78 )
| ( eq_p @ SY536 @ SY537 )
| ( neighborhood @ ( f18 @ SV78 @ SY535 @ SY536 @ SY537 ) @ SY537 @ SV78 @ SY535 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[350]) ).
thf(464,plain,
! [SV79: $i] :
( ( ! [SY538: $i] :
( ~ ( hausdorff @ SV79 @ SY538 )
| ! [SY539: $i] :
( ~ ( element_of_set @ SY539 @ SV79 )
| ! [SY540: $i] :
( ~ ( element_of_set @ SY540 @ SV79 )
| ( disjoint_s @ ( f17 @ SV79 @ SY538 @ SY539 @ SY540 ) @ ( f18 @ SV79 @ SY538 @ SY539 @ SY540 ) )
| ( eq_p @ SY539 @ SY540 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[351]) ).
thf(465,plain,
! [SV80: $i] :
( ( ! [SY541: $i] :
( ( hausdorff @ SV80 @ SY541 )
| ~ ( topological_space @ SV80 @ SY541 )
| ( element_of_set @ ( f19 @ SV80 @ SY541 ) @ SV80 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[352]) ).
thf(466,plain,
! [SV81: $i] :
( ( ! [SY542: $i] :
( ( hausdorff @ SV81 @ SY542 )
| ~ ( topological_space @ SV81 @ SY542 )
| ( element_of_set @ ( f20 @ SV81 @ SY542 ) @ SV81 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[353]) ).
thf(467,plain,
! [SV82: $i] :
( ( ! [SY543: $i] :
( ( hausdorff @ SV82 @ SY543 )
| ~ ( topological_space @ SV82 @ SY543 )
| ~ ( eq_p @ ( f19 @ SV82 @ SY543 ) @ ( f20 @ SV82 @ SY543 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[354]) ).
thf(468,plain,
! [SV83: $i] :
( ( ! [SY544: $i] :
( ( hausdorff @ SV83 @ SY544 )
| ~ ( topological_space @ SV83 @ SY544 )
| ! [SY545: $i] :
( ~ ( neighborhood @ SY545 @ ( f19 @ SV83 @ SY544 ) @ SV83 @ SY544 )
| ! [SY546: $i] :
( ~ ( neighborhood @ SY546 @ ( f20 @ SV83 @ SY544 ) @ SV83 @ SY544 )
| ~ ( disjoint_s @ SY545 @ SY546 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[355]) ).
thf(469,plain,
! [SV84: $i] :
( ( ! [SY547: $i,SY548: $i,SY549: $i] :
( ~ ( separation @ SV84 @ SY547 @ SY548 @ SY549 )
| ( topological_space @ SY548 @ SY549 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[356]) ).
thf(470,plain,
! [SV85: $i] :
( ( ! [SY550: $i] :
( ! [SY551: $i,SY552: $i] :
~ ( separation @ SV85 @ SY550 @ SY551 @ SY552 )
| ~ ( equal_sets @ SV85 @ empty_set ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[357]) ).
thf(471,plain,
! [SV86: $i] :
( ( ! [SY553: $i] :
( ! [SY554: $i,SY555: $i] :
~ ( separation @ SV86 @ SY553 @ SY554 @ SY555 )
| ~ ( equal_sets @ SY553 @ empty_set ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[358]) ).
thf(472,plain,
! [SV87: $i] :
( ( ! [SY556: $i,SY557: $i,SY558: $i] :
( ~ ( separation @ SV87 @ SY556 @ SY557 @ SY558 )
| ( element_of_collection @ SV87 @ SY558 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[359]) ).
thf(473,plain,
! [SV88: $i] :
( ( ! [SY559: $i,SY560: $i,SY561: $i] :
( ~ ( separation @ SV88 @ SY559 @ SY560 @ SY561 )
| ( element_of_collection @ SY559 @ SY561 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[360]) ).
thf(474,plain,
! [SV89: $i] :
( ( ! [SY562: $i,SY563: $i] :
( ! [SY564: $i] :
~ ( separation @ SV89 @ SY562 @ SY563 @ SY564 )
| ( equal_sets @ ( union_of_sets @ SV89 @ SY562 ) @ SY563 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[361]) ).
thf(475,plain,
! [SV90: $i] :
( ( ! [SY565: $i] :
( ! [SY566: $i,SY567: $i] :
~ ( separation @ SV90 @ SY565 @ SY566 @ SY567 )
| ( disjoint_s @ SV90 @ SY565 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[362]) ).
thf(476,plain,
! [SV91: $i] :
( ( ! [SY568: $i,SY569: $i,SY570: $i] :
( ( separation @ SV91 @ SY568 @ SY569 @ SY570 )
| ~ ( topological_space @ SY569 @ SY570 )
| ( equal_sets @ SV91 @ empty_set )
| ( equal_sets @ SY568 @ empty_set )
| ~ ( element_of_collection @ SV91 @ SY570 )
| ~ ( element_of_collection @ SY568 @ SY570 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SY568 ) @ SY569 )
| ~ ( disjoint_s @ SV91 @ SY568 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[363]) ).
thf(477,plain,
! [SV92: $i] :
( ( ! [SY571: $i] :
( ~ ( connected_space @ SV92 @ SY571 )
| ( topological_space @ SV92 @ SY571 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[364]) ).
thf(478,plain,
! [SV93: $i] :
( ( ! [SY572: $i] :
( ~ ( connected_space @ SV93 @ SY572 )
| ! [SY573: $i,SY574: $i] :
~ ( separation @ SY573 @ SY574 @ SV93 @ SY572 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[365]) ).
thf(479,plain,
! [SV94: $i] :
( ( ! [SY575: $i] :
( ( connected_space @ SV94 @ SY575 )
| ~ ( topological_space @ SV94 @ SY575 )
| ( separation @ ( f21 @ SV94 @ SY575 ) @ ( f22 @ SV94 @ SY575 ) @ SV94 @ SY575 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[366]) ).
thf(480,plain,
! [SV95: $i] :
( ( ! [SY576: $i,SY577: $i] :
( ~ ( connected_set @ SV95 @ SY576 @ SY577 )
| ( topological_space @ SY576 @ SY577 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[367]) ).
thf(481,plain,
! [SV96: $i] :
( ( ! [SY578: $i] :
( ! [SY579: $i] :
~ ( connected_set @ SV96 @ SY578 @ SY579 )
| ( subset_sets @ SV96 @ SY578 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[368]) ).
thf(482,plain,
! [SV97: $i] :
( ( ! [SY580: $i,SY581: $i] :
( ~ ( connected_set @ SV97 @ SY580 @ SY581 )
| ( connected_space @ SV97 @ ( subspace_topology @ SY580 @ SY581 @ SV97 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[369]) ).
thf(483,plain,
! [SV98: $i] :
( ( ! [SY582: $i,SY583: $i] :
( ( connected_set @ SV98 @ SY582 @ SY583 )
| ~ ( topological_space @ SY582 @ SY583 )
| ~ ( subset_sets @ SV98 @ SY582 )
| ~ ( connected_space @ SV98 @ ( subspace_topology @ SY582 @ SY583 @ SV98 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[370]) ).
thf(484,plain,
! [SV99: $i] :
( ( ! [SY584: $i,SY585: $i] :
( ~ ( open_covering @ SV99 @ SY584 @ SY585 )
| ( topological_space @ SY584 @ SY585 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[371]) ).
thf(485,plain,
! [SV100: $i] :
( ( ! [SY586: $i,SY587: $i] :
( ~ ( open_covering @ SV100 @ SY586 @ SY587 )
| ( subset_collections @ SV100 @ SY587 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[372]) ).
thf(486,plain,
! [SV101: $i] :
( ( ! [SY588: $i] :
( ! [SY589: $i] :
~ ( open_covering @ SV101 @ SY588 @ SY589 )
| ( equal_sets @ ( union_of_members @ SV101 ) @ SY588 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[373]) ).
thf(487,plain,
! [SV102: $i] :
( ( ! [SY590: $i,SY591: $i] :
( ( open_covering @ SV102 @ SY590 @ SY591 )
| ~ ( topological_space @ SY590 @ SY591 )
| ~ ( subset_collections @ SV102 @ SY591 )
| ~ ( equal_sets @ ( union_of_members @ SV102 ) @ SY590 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[374]) ).
thf(488,plain,
! [SV103: $i] :
( ( ! [SY592: $i] :
( ~ ( compact_space @ SV103 @ SY592 )
| ( topological_space @ SV103 @ SY592 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[375]) ).
thf(489,plain,
! [SV104: $i] :
( ( ! [SY593: $i] :
( ~ ( compact_space @ SV104 @ SY593 )
| ! [SY594: $i] :
( ~ ( open_covering @ SY594 @ SV104 @ SY593 )
| ( finite @ ( f23 @ SV104 @ SY593 @ SY594 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[376]) ).
thf(490,plain,
! [SV105: $i] :
( ( ! [SY595: $i] :
( ~ ( compact_space @ SV105 @ SY595 )
| ! [SY596: $i] :
( ~ ( open_covering @ SY596 @ SV105 @ SY595 )
| ( subset_collections @ ( f23 @ SV105 @ SY595 @ SY596 ) @ SY596 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[377]) ).
thf(491,plain,
! [SV106: $i] :
( ( ! [SY597: $i] :
( ~ ( compact_space @ SV106 @ SY597 )
| ! [SY598: $i] :
( ~ ( open_covering @ SY598 @ SV106 @ SY597 )
| ( open_covering @ ( f23 @ SV106 @ SY597 @ SY598 ) @ SV106 @ SY597 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[378]) ).
thf(492,plain,
! [SV107: $i] :
( ( ! [SY599: $i] :
( ( compact_space @ SV107 @ SY599 )
| ~ ( topological_space @ SV107 @ SY599 )
| ( open_covering @ ( f24 @ SV107 @ SY599 ) @ SV107 @ SY599 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[379]) ).
thf(493,plain,
! [SV108: $i] :
( ( ! [SY600: $i] :
( ( compact_space @ SV108 @ SY600 )
| ~ ( topological_space @ SV108 @ SY600 )
| ! [SY601: $i] :
( ~ ( finite @ SY601 )
| ~ ( subset_collections @ SY601 @ ( f24 @ SV108 @ SY600 ) )
| ~ ( open_covering @ SY601 @ SV108 @ SY600 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[380]) ).
thf(494,plain,
! [SV109: $i] :
( ( ! [SY602: $i,SY603: $i] :
( ~ ( compact_set @ SV109 @ SY602 @ SY603 )
| ( topological_space @ SY602 @ SY603 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[381]) ).
thf(495,plain,
! [SV110: $i] :
( ( ! [SY604: $i] :
( ! [SY605: $i] :
~ ( compact_set @ SV110 @ SY604 @ SY605 )
| ( subset_sets @ SV110 @ SY604 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[382]) ).
thf(496,plain,
! [SV111: $i] :
( ( ! [SY606: $i,SY607: $i] :
( ~ ( compact_set @ SV111 @ SY606 @ SY607 )
| ( compact_space @ SV111 @ ( subspace_topology @ SY606 @ SY607 @ SV111 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[383]) ).
thf(497,plain,
! [SV112: $i] :
( ( ! [SY608: $i,SY609: $i] :
( ( compact_set @ SV112 @ SY608 @ SY609 )
| ~ ( topological_space @ SY608 @ SY609 )
| ~ ( subset_sets @ SV112 @ SY608 )
| ~ ( compact_space @ SV112 @ ( subspace_topology @ SY608 @ SY609 @ SV112 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[384]) ).
thf(498,plain,
$false = $false,
inference(extcnf_not_pos,[status(thm)],[385]) ).
thf(499,plain,
! [SV113: $i,SV3: $i] :
( ( ~ ( element_of_collection @ ( intersection_of_sets @ SV3 @ SV113 ) @ ( top_of_basis @ f ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[388]) ).
thf(500,plain,
! [SV114: $i,SV4: $i] :
( ( ~ ( element_of_set @ SV4 @ ( union_of_members @ SV114 ) )
| ( element_of_set @ SV4 @ ( f1 @ SV114 @ SV4 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[389]) ).
thf(501,plain,
! [SV115: $i,SV5: $i] :
( ( ~ ( element_of_set @ SV5 @ ( union_of_members @ SV115 ) )
| ( element_of_collection @ ( f1 @ SV115 @ SV5 ) @ SV115 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[390]) ).
thf(502,plain,
! [SV116: $i,SV6: $i] :
( ( ( element_of_set @ SV6 @ ( union_of_members @ SV116 ) )
| ! [SY610: $i] :
( ~ ( element_of_set @ SV6 @ SY610 )
| ~ ( element_of_collection @ SY610 @ SV116 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[391]) ).
thf(503,plain,
! [SV117: $i,SV7: $i] :
( ( ~ ( element_of_set @ SV7 @ ( intersection_of_members @ SV117 ) )
| ! [SY611: $i] :
( ~ ( element_of_collection @ SY611 @ SV117 )
| ( element_of_set @ SV7 @ SY611 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[392]) ).
thf(504,plain,
! [SV8: $i,SV118: $i] :
( ( ( element_of_collection @ ( f2 @ SV118 @ SV8 ) @ SV118 )
| ( element_of_set @ SV8 @ ( intersection_of_members @ SV118 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[393]) ).
thf(505,plain,
! [SV119: $i,SV9: $i] :
( ( ( element_of_set @ SV9 @ ( intersection_of_members @ SV119 ) )
| ~ ( element_of_set @ SV9 @ ( f2 @ SV119 @ SV9 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[394]) ).
thf(506,plain,
! [SV120: $i,SV10: $i] :
( ( ~ ( topological_space @ SV10 @ SV120 )
| ( equal_sets @ ( union_of_members @ SV120 ) @ SV10 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[395]) ).
thf(507,plain,
! [SV121: $i,SV11: $i] :
( ( ~ ( topological_space @ SV11 @ SV121 )
| ( element_of_collection @ empty_set @ SV121 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[396]) ).
thf(508,plain,
! [SV122: $i,SV12: $i] :
( ( ~ ( topological_space @ SV12 @ SV122 )
| ( element_of_collection @ SV12 @ SV122 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[397]) ).
thf(509,plain,
! [SV123: $i,SV13: $i] :
( ( ~ ( topological_space @ SV13 @ SV123 )
| ! [SY612: $i] :
( ~ ( element_of_collection @ SY612 @ SV123 )
| ! [SY613: $i] :
( ~ ( element_of_collection @ SY613 @ SV123 )
| ( element_of_collection @ ( intersection_of_sets @ SY612 @ SY613 ) @ SV123 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[398]) ).
thf(510,plain,
! [SV124: $i,SV14: $i] :
( ( ~ ( topological_space @ SV14 @ SV124 )
| ! [SY614: $i] :
( ~ ( subset_collections @ SY614 @ SV124 )
| ( element_of_collection @ ( union_of_members @ SY614 ) @ SV124 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[399]) ).
thf(511,plain,
! [SV125: $i,SV15: $i] :
( ( ( topological_space @ SV15 @ SV125 )
| ~ ( equal_sets @ ( union_of_members @ SV125 ) @ SV15 )
| ~ ( element_of_collection @ empty_set @ SV125 )
| ~ ( element_of_collection @ SV15 @ SV125 )
| ( element_of_collection @ ( f3 @ SV15 @ SV125 ) @ SV125 )
| ( subset_collections @ ( f5 @ SV15 @ SV125 ) @ SV125 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[400]) ).
thf(512,plain,
! [SV126: $i,SV16: $i] :
( ( ( topological_space @ SV16 @ SV126 )
| ~ ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 )
| ~ ( element_of_collection @ empty_set @ SV126 )
| ~ ( element_of_collection @ SV16 @ SV126 )
| ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[401]) ).
thf(513,plain,
! [SV127: $i,SV17: $i] :
( ( ( topological_space @ SV17 @ SV127 )
| ~ ( equal_sets @ ( union_of_members @ SV127 ) @ SV17 )
| ~ ( element_of_collection @ empty_set @ SV127 )
| ~ ( element_of_collection @ SV17 @ SV127 )
| ( element_of_collection @ ( f4 @ SV17 @ SV127 ) @ SV127 )
| ( subset_collections @ ( f5 @ SV17 @ SV127 ) @ SV127 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[402]) ).
thf(514,plain,
! [SV128: $i,SV18: $i] :
( ( ( topological_space @ SV18 @ SV128 )
| ~ ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 )
| ~ ( element_of_collection @ empty_set @ SV128 )
| ~ ( element_of_collection @ SV18 @ SV128 )
| ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[403]) ).
thf(515,plain,
! [SV129: $i,SV19: $i] :
( ( ( topological_space @ SV19 @ SV129 )
| ~ ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 )
| ~ ( element_of_collection @ empty_set @ SV129 )
| ~ ( element_of_collection @ SV19 @ SV129 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 )
| ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[404]) ).
thf(516,plain,
! [SV130: $i,SV20: $i] :
( ( ( topological_space @ SV20 @ SV130 )
| ~ ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
| ~ ( element_of_collection @ empty_set @ SV130 )
| ~ ( element_of_collection @ SV20 @ SV130 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[405]) ).
thf(517,plain,
! [SV131: $i,SV21: $i] :
( ( ! [SY615: $i] :
( ~ ( open @ SV21 @ SV131 @ SY615 )
| ( topological_space @ SV131 @ SY615 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[406]) ).
thf(518,plain,
! [SV132: $i,SV22: $i] :
( ( ! [SY616: $i] :
( ~ ( open @ SV22 @ SV132 @ SY616 )
| ( element_of_collection @ SV22 @ SY616 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[407]) ).
thf(519,plain,
! [SV133: $i,SV23: $i] :
( ( ! [SY617: $i] :
( ( open @ SV23 @ SV133 @ SY617 )
| ~ ( topological_space @ SV133 @ SY617 )
| ~ ( element_of_collection @ SV23 @ SY617 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[408]) ).
thf(520,plain,
! [SV134: $i,SV24: $i] :
( ( ! [SY618: $i] :
( ~ ( closed @ SV24 @ SV134 @ SY618 )
| ( topological_space @ SV134 @ SY618 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[409]) ).
thf(521,plain,
! [SV135: $i,SV25: $i] :
( ( ! [SY619: $i] :
( ~ ( closed @ SV25 @ SV135 @ SY619 )
| ( open @ ( relative_complement_sets @ SV25 @ SV135 ) @ SV135 @ SY619 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[410]) ).
thf(522,plain,
! [SV136: $i,SV26: $i] :
( ( ! [SY620: $i] :
( ( closed @ SV26 @ SV136 @ SY620 )
| ~ ( topological_space @ SV136 @ SY620 )
| ~ ( open @ ( relative_complement_sets @ SV26 @ SV136 ) @ SV136 @ SY620 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[411]) ).
thf(523,plain,
! [SV137: $i,SV27: $i] :
( ( ! [SY621: $i] :
( ~ ( finer @ SV27 @ SV137 @ SY621 )
| ( topological_space @ SY621 @ SV27 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[412]) ).
thf(524,plain,
! [SV138: $i,SV28: $i] :
( ( ! [SY622: $i] :
( ~ ( finer @ SV28 @ SV138 @ SY622 )
| ( topological_space @ SY622 @ SV138 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[413]) ).
thf(525,plain,
! [SV139: $i,SV29: $i] :
( ( ! [SY623: $i] :
~ ( finer @ SV29 @ SV139 @ SY623 )
| ( subset_collections @ SV139 @ SV29 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[414]) ).
thf(526,plain,
! [SV140: $i,SV30: $i] :
( ( ! [SY624: $i] :
( ( finer @ SV30 @ SV140 @ SY624 )
| ~ ( topological_space @ SY624 @ SV30 )
| ~ ( topological_space @ SY624 @ SV140 )
| ~ ( subset_collections @ SV140 @ SV30 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[415]) ).
thf(527,plain,
! [SV141: $i,SV31: $i] :
( ( ~ ( basis @ SV31 @ SV141 )
| ( equal_sets @ ( union_of_members @ SV141 ) @ SV31 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[416]) ).
thf(528,plain,
! [SV142: $i,SV32: $i] :
( ( ! [SY625: $i] :
( ~ ( basis @ SV32 @ SV142 )
| ~ ( element_of_set @ SY625 @ SV32 )
| ! [SY626: $i] :
( ~ ( element_of_collection @ SY626 @ SV142 )
| ! [SY627: $i] :
( ~ ( element_of_collection @ SY627 @ SV142 )
| ~ ( element_of_set @ SY625 @ ( intersection_of_sets @ SY626 @ SY627 ) )
| ( element_of_set @ SY625 @ ( f6 @ SV32 @ SV142 @ SY625 @ SY626 @ SY627 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[417]) ).
thf(529,plain,
! [SV143: $i,SV33: $i] :
( ( ! [SY628: $i] :
( ~ ( basis @ SV33 @ SV143 )
| ~ ( element_of_set @ SY628 @ SV33 )
| ! [SY629: $i] :
( ~ ( element_of_collection @ SY629 @ SV143 )
| ! [SY630: $i] :
( ~ ( element_of_collection @ SY630 @ SV143 )
| ~ ( element_of_set @ SY628 @ ( intersection_of_sets @ SY629 @ SY630 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SY628 @ SY629 @ SY630 ) @ SV143 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[418]) ).
thf(530,plain,
! [SV144: $i,SV34: $i] :
( ( ! [SY631: $i] :
( ~ ( basis @ SV34 @ SV144 )
| ~ ( element_of_set @ SY631 @ SV34 )
| ! [SY632: $i] :
( ~ ( element_of_collection @ SY632 @ SV144 )
| ! [SY633: $i] :
( ~ ( element_of_collection @ SY633 @ SV144 )
| ~ ( element_of_set @ SY631 @ ( intersection_of_sets @ SY632 @ SY633 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SY631 @ SY632 @ SY633 ) @ ( intersection_of_sets @ SY632 @ SY633 ) ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[419]) ).
thf(531,plain,
! [SV145: $i,SV35: $i] :
( ( ( basis @ SV35 @ SV145 )
| ~ ( equal_sets @ ( union_of_members @ SV145 ) @ SV35 )
| ( element_of_set @ ( f7 @ SV35 @ SV145 ) @ SV35 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[420]) ).
thf(532,plain,
! [SV146: $i,SV36: $i] :
( ( ( basis @ SV36 @ SV146 )
| ~ ( equal_sets @ ( union_of_members @ SV146 ) @ SV36 )
| ( element_of_collection @ ( f8 @ SV36 @ SV146 ) @ SV146 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[421]) ).
thf(533,plain,
! [SV147: $i,SV37: $i] :
( ( ( basis @ SV37 @ SV147 )
| ~ ( equal_sets @ ( union_of_members @ SV147 ) @ SV37 )
| ( element_of_collection @ ( f9 @ SV37 @ SV147 ) @ SV147 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[422]) ).
thf(534,plain,
! [SV148: $i,SV38: $i] :
( ( ( basis @ SV38 @ SV148 )
| ~ ( equal_sets @ ( union_of_members @ SV148 ) @ SV38 )
| ( element_of_set @ ( f7 @ SV38 @ SV148 ) @ ( intersection_of_sets @ ( f8 @ SV38 @ SV148 ) @ ( f9 @ SV38 @ SV148 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[423]) ).
thf(535,plain,
! [SV149: $i,SV39: $i] :
( ( ( basis @ SV39 @ SV149 )
| ~ ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 )
| ! [SY634: $i] :
( ~ ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SY634 )
| ~ ( element_of_collection @ SY634 @ SV149 )
| ~ ( subset_sets @ SY634 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[424]) ).
thf(536,plain,
! [SV150: $i,SV40: $i] :
( ( ~ ( element_of_collection @ SV40 @ ( top_of_basis @ SV150 ) )
| ! [SY635: $i] :
( ~ ( element_of_set @ SY635 @ SV40 )
| ( element_of_set @ SY635 @ ( f10 @ SV150 @ SV40 @ SY635 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[425]) ).
thf(537,plain,
! [SV151: $i,SV41: $i] :
( ( ~ ( element_of_collection @ SV41 @ ( top_of_basis @ SV151 ) )
| ! [SY636: $i] :
( ~ ( element_of_set @ SY636 @ SV41 )
| ( element_of_collection @ ( f10 @ SV151 @ SV41 @ SY636 ) @ SV151 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[426]) ).
thf(538,plain,
! [SV152: $i,SV42: $i] :
( ( ~ ( element_of_collection @ SV42 @ ( top_of_basis @ SV152 ) )
| ! [SY637: $i] :
( ~ ( element_of_set @ SY637 @ SV42 )
| ( subset_sets @ ( f10 @ SV152 @ SV42 @ SY637 ) @ SV42 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[427]) ).
thf(539,plain,
! [SV153: $i,SV43: $i] :
( ( ( element_of_collection @ SV43 @ ( top_of_basis @ SV153 ) )
| ( element_of_set @ ( f11 @ SV153 @ SV43 ) @ SV43 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[428]) ).
thf(540,plain,
! [SV154: $i,SV44: $i] :
( ( ( element_of_collection @ SV44 @ ( top_of_basis @ SV154 ) )
| ! [SY638: $i] :
( ~ ( element_of_set @ ( f11 @ SV154 @ SV44 ) @ SY638 )
| ~ ( element_of_collection @ SY638 @ SV154 )
| ~ ( subset_sets @ SY638 @ SV44 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[429]) ).
thf(541,plain,
! [SV155: $i,SV45: $i] :
( ( ! [SY639: $i] :
( ! [SY640: $i] :
~ ( element_of_collection @ SV45 @ ( subspace_topology @ SV155 @ SY639 @ SY640 ) )
| ( topological_space @ SV155 @ SY639 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[430]) ).
thf(542,plain,
! [SV156: $i,SV46: $i] :
( ( ! [SY641: $i,SY642: $i] :
( ~ ( element_of_collection @ SV46 @ ( subspace_topology @ SV156 @ SY641 @ SY642 ) )
| ( subset_sets @ SY642 @ SV156 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[431]) ).
thf(543,plain,
! [SV157: $i,SV47: $i] :
( ( ! [SY643: $i,SY644: $i] :
( ~ ( element_of_collection @ SV47 @ ( subspace_topology @ SV157 @ SY643 @ SY644 ) )
| ( element_of_collection @ ( f12 @ SV157 @ SY643 @ SY644 @ SV47 ) @ SY643 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[432]) ).
thf(544,plain,
! [SV158: $i,SV48: $i] :
( ( ! [SY645: $i,SY646: $i] :
( ~ ( element_of_collection @ SV48 @ ( subspace_topology @ SV158 @ SY645 @ SY646 ) )
| ( equal_sets @ SV48 @ ( intersection_of_sets @ SY646 @ ( f12 @ SV158 @ SY645 @ SY646 @ SV48 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[433]) ).
thf(545,plain,
! [SV159: $i,SV49: $i] :
( ( ! [SY647: $i,SY648: $i] :
( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SY647 @ SY648 ) )
| ~ ( topological_space @ SV159 @ SY647 )
| ~ ( subset_sets @ SY648 @ SV159 )
| ! [SY447: $i] :
( ~ ( element_of_collection @ SY447 @ SY647 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SY648 @ SY447 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[434]) ).
thf(546,plain,
! [SV160: $i,SV50: $i] :
( ( ! [SY650: $i,SY651: $i] :
( ~ ( element_of_set @ SV50 @ ( interior @ SV160 @ SY650 @ SY651 ) )
| ( topological_space @ SY650 @ SY651 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[435]) ).
thf(547,plain,
! [SV161: $i,SV51: $i] :
( ( ! [SY652: $i] :
( ! [SY653: $i] :
~ ( element_of_set @ SV51 @ ( interior @ SV161 @ SY652 @ SY653 ) )
| ( subset_sets @ SV161 @ SY652 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[436]) ).
thf(548,plain,
! [SV162: $i,SV52: $i] :
( ( ! [SY654: $i,SY655: $i] :
( ~ ( element_of_set @ SV52 @ ( interior @ SV162 @ SY654 @ SY655 ) )
| ( element_of_set @ SV52 @ ( f13 @ SV162 @ SY654 @ SY655 @ SV52 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[437]) ).
thf(549,plain,
! [SV163: $i,SV53: $i] :
( ( ! [SY656: $i,SY657: $i] :
( ~ ( element_of_set @ SV53 @ ( interior @ SV163 @ SY656 @ SY657 ) )
| ( subset_sets @ ( f13 @ SV163 @ SY656 @ SY657 @ SV53 ) @ SV163 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[438]) ).
thf(550,plain,
! [SV164: $i,SV54: $i] :
( ( ! [SY658: $i,SY659: $i] :
( ~ ( element_of_set @ SV54 @ ( interior @ SV164 @ SY658 @ SY659 ) )
| ( open @ ( f13 @ SV164 @ SY658 @ SY659 @ SV54 ) @ SY658 @ SY659 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[439]) ).
thf(551,plain,
! [SV165: $i,SV55: $i] :
( ( ! [SY660: $i,SY661: $i] :
( ( element_of_set @ SV55 @ ( interior @ SV165 @ SY660 @ SY661 ) )
| ~ ( topological_space @ SY660 @ SY661 )
| ~ ( subset_sets @ SV165 @ SY660 )
| ! [SY662: $i] :
( ~ ( element_of_set @ SV55 @ SY662 )
| ~ ( subset_sets @ SY662 @ SV165 )
| ~ ( open @ SY662 @ SY660 @ SY661 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[440]) ).
thf(552,plain,
! [SV166: $i,SV56: $i] :
( ( ! [SY663: $i,SY664: $i] :
( ~ ( element_of_set @ SV56 @ ( closure @ SV166 @ SY663 @ SY664 ) )
| ( topological_space @ SY663 @ SY664 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[441]) ).
thf(553,plain,
! [SV167: $i,SV57: $i] :
( ( ! [SY665: $i] :
( ! [SY666: $i] :
~ ( element_of_set @ SV57 @ ( closure @ SV167 @ SY665 @ SY666 ) )
| ( subset_sets @ SV167 @ SY665 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[442]) ).
thf(554,plain,
! [SV168: $i,SV58: $i] :
( ( ! [SY667: $i,SY668: $i] :
( ~ ( element_of_set @ SV58 @ ( closure @ SV168 @ SY667 @ SY668 ) )
| ! [SY669: $i] :
( ~ ( subset_sets @ SV168 @ SY669 )
| ~ ( closed @ SY669 @ SY667 @ SY668 )
| ( element_of_set @ SV58 @ SY669 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[443]) ).
thf(555,plain,
! [SV169: $i,SV59: $i] :
( ( ! [SY670: $i,SY671: $i] :
( ( element_of_set @ SV59 @ ( closure @ SV169 @ SY670 @ SY671 ) )
| ~ ( topological_space @ SY670 @ SY671 )
| ~ ( subset_sets @ SV169 @ SY670 )
| ( subset_sets @ SV169 @ ( f14 @ SV169 @ SY670 @ SY671 @ SV59 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[444]) ).
thf(556,plain,
! [SV170: $i,SV60: $i] :
( ( ! [SY672: $i,SY673: $i] :
( ( element_of_set @ SV60 @ ( closure @ SV170 @ SY672 @ SY673 ) )
| ~ ( topological_space @ SY672 @ SY673 )
| ~ ( subset_sets @ SV170 @ SY672 )
| ( closed @ ( f14 @ SV170 @ SY672 @ SY673 @ SV60 ) @ SY672 @ SY673 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[445]) ).
thf(557,plain,
! [SV171: $i,SV61: $i] :
( ( ! [SY674: $i,SY675: $i] :
( ( element_of_set @ SV61 @ ( closure @ SV171 @ SY674 @ SY675 ) )
| ~ ( topological_space @ SY674 @ SY675 )
| ~ ( subset_sets @ SV171 @ SY674 )
| ~ ( element_of_set @ SV61 @ ( f14 @ SV171 @ SY674 @ SY675 @ SV61 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[446]) ).
thf(558,plain,
! [SV172: $i,SV62: $i] :
( ( ! [SY676: $i,SY677: $i] :
( ~ ( neighborhood @ SV62 @ SV172 @ SY676 @ SY677 )
| ( topological_space @ SY676 @ SY677 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[447]) ).
thf(559,plain,
! [SV173: $i,SV63: $i] :
( ( ! [SY678: $i,SY679: $i] :
( ~ ( neighborhood @ SV63 @ SV173 @ SY678 @ SY679 )
| ( open @ SV63 @ SY678 @ SY679 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[448]) ).
thf(560,plain,
! [SV174: $i,SV64: $i] :
( ( ! [SY680: $i,SY681: $i] :
~ ( neighborhood @ SV64 @ SV174 @ SY680 @ SY681 )
| ( element_of_set @ SV174 @ SV64 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[449]) ).
thf(561,plain,
! [SV175: $i,SV65: $i] :
( ( ! [SY682: $i,SY683: $i] :
( ( neighborhood @ SV65 @ SV175 @ SY682 @ SY683 )
| ~ ( topological_space @ SY682 @ SY683 )
| ~ ( open @ SV65 @ SY682 @ SY683 )
| ~ ( element_of_set @ SV175 @ SV65 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[450]) ).
thf(562,plain,
! [SV176: $i,SV66: $i] :
( ( ! [SY684: $i,SY685: $i] :
( ~ ( limit_point @ SV66 @ SV176 @ SY684 @ SY685 )
| ( topological_space @ SY684 @ SY685 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[451]) ).
thf(563,plain,
! [SV177: $i,SV67: $i] :
( ( ! [SY686: $i] :
( ! [SY687: $i] :
~ ( limit_point @ SV67 @ SV177 @ SY686 @ SY687 )
| ( subset_sets @ SV177 @ SY686 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[452]) ).
thf(564,plain,
! [SV178: $i,SV68: $i] :
( ( ! [SY688: $i,SY689: $i] :
( ~ ( limit_point @ SV68 @ SV178 @ SY688 @ SY689 )
| ! [SY690: $i] :
( ~ ( neighborhood @ SY690 @ SV68 @ SY688 @ SY689 )
| ( element_of_set @ ( f15 @ SV68 @ SV178 @ SY688 @ SY689 @ SY690 ) @ ( intersection_of_sets @ SY690 @ SV178 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[453]) ).
thf(565,plain,
! [SV179: $i,SV69: $i] :
( ( ! [SY691: $i,SY692: $i] :
( ~ ( limit_point @ SV69 @ SV179 @ SY691 @ SY692 )
| ! [SY693: $i] :
( ~ ( neighborhood @ SY693 @ SV69 @ SY691 @ SY692 )
| ~ ( eq_p @ ( f15 @ SV69 @ SV179 @ SY691 @ SY692 @ SY693 ) @ SV69 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[454]) ).
thf(566,plain,
! [SV180: $i,SV70: $i] :
( ( ! [SY694: $i,SY695: $i] :
( ( limit_point @ SV70 @ SV180 @ SY694 @ SY695 )
| ~ ( topological_space @ SY694 @ SY695 )
| ~ ( subset_sets @ SV180 @ SY694 )
| ( neighborhood @ ( f16 @ SV70 @ SV180 @ SY694 @ SY695 ) @ SV70 @ SY694 @ SY695 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[455]) ).
thf(567,plain,
! [SV181: $i,SV71: $i] :
( ( ! [SY696: $i,SY697: $i] :
( ( limit_point @ SV71 @ SV181 @ SY696 @ SY697 )
| ~ ( topological_space @ SY696 @ SY697 )
| ~ ( subset_sets @ SV181 @ SY696 )
| ! [SY698: $i] :
( ~ ( element_of_set @ SY698 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SY696 @ SY697 ) @ SV181 ) )
| ( eq_p @ SY698 @ SV71 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[456]) ).
thf(568,plain,
! [SV182: $i,SV72: $i] :
( ( ! [SY699: $i,SY700: $i] :
( ~ ( element_of_set @ SV72 @ ( boundary @ SV182 @ SY699 @ SY700 ) )
| ( topological_space @ SY699 @ SY700 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[457]) ).
thf(569,plain,
! [SV183: $i,SV73: $i] :
( ( ! [SY701: $i,SY702: $i] :
( ~ ( element_of_set @ SV73 @ ( boundary @ SV183 @ SY701 @ SY702 ) )
| ( element_of_set @ SV73 @ ( closure @ SV183 @ SY701 @ SY702 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[458]) ).
thf(570,plain,
! [SV184: $i,SV74: $i] :
( ( ! [SY703: $i,SY704: $i] :
( ~ ( element_of_set @ SV74 @ ( boundary @ SV184 @ SY703 @ SY704 ) )
| ( element_of_set @ SV74 @ ( closure @ ( relative_complement_sets @ SV184 @ SY703 ) @ SY703 @ SY704 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[459]) ).
thf(571,plain,
! [SV185: $i,SV75: $i] :
( ( ! [SY705: $i,SY706: $i] :
( ( element_of_set @ SV75 @ ( boundary @ SV185 @ SY705 @ SY706 ) )
| ~ ( topological_space @ SY705 @ SY706 )
| ~ ( element_of_set @ SV75 @ ( closure @ SV185 @ SY705 @ SY706 ) )
| ~ ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SV185 @ SY705 ) @ SY705 @ SY706 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[460]) ).
thf(572,plain,
! [SV186: $i,SV76: $i] :
( ( ~ ( hausdorff @ SV76 @ SV186 )
| ( topological_space @ SV76 @ SV186 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[461]) ).
thf(573,plain,
! [SV187: $i,SV77: $i] :
( ( ~ ( hausdorff @ SV77 @ SV187 )
| ! [SY707: $i] :
( ~ ( element_of_set @ SY707 @ SV77 )
| ! [SY708: $i] :
( ~ ( element_of_set @ SY708 @ SV77 )
| ( eq_p @ SY707 @ SY708 )
| ( neighborhood @ ( f17 @ SV77 @ SV187 @ SY707 @ SY708 ) @ SY707 @ SV77 @ SV187 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[462]) ).
thf(574,plain,
! [SV188: $i,SV78: $i] :
( ( ~ ( hausdorff @ SV78 @ SV188 )
| ! [SY709: $i] :
( ~ ( element_of_set @ SY709 @ SV78 )
| ! [SY710: $i] :
( ~ ( element_of_set @ SY710 @ SV78 )
| ( eq_p @ SY709 @ SY710 )
| ( neighborhood @ ( f18 @ SV78 @ SV188 @ SY709 @ SY710 ) @ SY710 @ SV78 @ SV188 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[463]) ).
thf(575,plain,
! [SV189: $i,SV79: $i] :
( ( ~ ( hausdorff @ SV79 @ SV189 )
| ! [SY711: $i] :
( ~ ( element_of_set @ SY711 @ SV79 )
| ! [SY712: $i] :
( ~ ( element_of_set @ SY712 @ SV79 )
| ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SY711 @ SY712 ) @ ( f18 @ SV79 @ SV189 @ SY711 @ SY712 ) )
| ( eq_p @ SY711 @ SY712 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[464]) ).
thf(576,plain,
! [SV190: $i,SV80: $i] :
( ( ( hausdorff @ SV80 @ SV190 )
| ~ ( topological_space @ SV80 @ SV190 )
| ( element_of_set @ ( f19 @ SV80 @ SV190 ) @ SV80 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[465]) ).
thf(577,plain,
! [SV191: $i,SV81: $i] :
( ( ( hausdorff @ SV81 @ SV191 )
| ~ ( topological_space @ SV81 @ SV191 )
| ( element_of_set @ ( f20 @ SV81 @ SV191 ) @ SV81 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[466]) ).
thf(578,plain,
! [SV192: $i,SV82: $i] :
( ( ( hausdorff @ SV82 @ SV192 )
| ~ ( topological_space @ SV82 @ SV192 )
| ~ ( eq_p @ ( f19 @ SV82 @ SV192 ) @ ( f20 @ SV82 @ SV192 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[467]) ).
thf(579,plain,
! [SV193: $i,SV83: $i] :
( ( ( hausdorff @ SV83 @ SV193 )
| ~ ( topological_space @ SV83 @ SV193 )
| ! [SY713: $i] :
( ~ ( neighborhood @ SY713 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ! [SY714: $i] :
( ~ ( neighborhood @ SY714 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ~ ( disjoint_s @ SY713 @ SY714 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[468]) ).
thf(580,plain,
! [SV194: $i,SV84: $i] :
( ( ! [SY715: $i,SY716: $i] :
( ~ ( separation @ SV84 @ SV194 @ SY715 @ SY716 )
| ( topological_space @ SY715 @ SY716 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[469]) ).
thf(581,plain,
! [SV195: $i,SV85: $i] :
( ( ! [SY717: $i,SY718: $i] :
~ ( separation @ SV85 @ SV195 @ SY717 @ SY718 )
| ~ ( equal_sets @ SV85 @ empty_set ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[470]) ).
thf(582,plain,
! [SV196: $i,SV86: $i] :
( ( ! [SY719: $i,SY720: $i] :
~ ( separation @ SV86 @ SV196 @ SY719 @ SY720 )
| ~ ( equal_sets @ SV196 @ empty_set ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[471]) ).
thf(583,plain,
! [SV197: $i,SV87: $i] :
( ( ! [SY721: $i,SY722: $i] :
( ~ ( separation @ SV87 @ SV197 @ SY721 @ SY722 )
| ( element_of_collection @ SV87 @ SY722 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[472]) ).
thf(584,plain,
! [SV198: $i,SV88: $i] :
( ( ! [SY723: $i,SY724: $i] :
( ~ ( separation @ SV88 @ SV198 @ SY723 @ SY724 )
| ( element_of_collection @ SV198 @ SY724 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[473]) ).
thf(585,plain,
! [SV199: $i,SV89: $i] :
( ( ! [SY725: $i] :
( ! [SY726: $i] :
~ ( separation @ SV89 @ SV199 @ SY725 @ SY726 )
| ( equal_sets @ ( union_of_sets @ SV89 @ SV199 ) @ SY725 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[474]) ).
thf(586,plain,
! [SV200: $i,SV90: $i] :
( ( ! [SY727: $i,SY728: $i] :
~ ( separation @ SV90 @ SV200 @ SY727 @ SY728 )
| ( disjoint_s @ SV90 @ SV200 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[475]) ).
thf(587,plain,
! [SV201: $i,SV91: $i] :
( ( ! [SY729: $i,SY730: $i] :
( ( separation @ SV91 @ SV201 @ SY729 @ SY730 )
| ~ ( topological_space @ SY729 @ SY730 )
| ( equal_sets @ SV91 @ empty_set )
| ( equal_sets @ SV201 @ empty_set )
| ~ ( element_of_collection @ SV91 @ SY730 )
| ~ ( element_of_collection @ SV201 @ SY730 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SY729 )
| ~ ( disjoint_s @ SV91 @ SV201 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[476]) ).
thf(588,plain,
! [SV202: $i,SV92: $i] :
( ( ~ ( connected_space @ SV92 @ SV202 )
| ( topological_space @ SV92 @ SV202 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[477]) ).
thf(589,plain,
! [SV203: $i,SV93: $i] :
( ( ~ ( connected_space @ SV93 @ SV203 )
| ! [SY731: $i,SY732: $i] :
~ ( separation @ SY731 @ SY732 @ SV93 @ SV203 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[478]) ).
thf(590,plain,
! [SV204: $i,SV94: $i] :
( ( ( connected_space @ SV94 @ SV204 )
| ~ ( topological_space @ SV94 @ SV204 )
| ( separation @ ( f21 @ SV94 @ SV204 ) @ ( f22 @ SV94 @ SV204 ) @ SV94 @ SV204 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[479]) ).
thf(591,plain,
! [SV205: $i,SV95: $i] :
( ( ! [SY733: $i] :
( ~ ( connected_set @ SV95 @ SV205 @ SY733 )
| ( topological_space @ SV205 @ SY733 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[480]) ).
thf(592,plain,
! [SV206: $i,SV96: $i] :
( ( ! [SY734: $i] :
~ ( connected_set @ SV96 @ SV206 @ SY734 )
| ( subset_sets @ SV96 @ SV206 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[481]) ).
thf(593,plain,
! [SV207: $i,SV97: $i] :
( ( ! [SY735: $i] :
( ~ ( connected_set @ SV97 @ SV207 @ SY735 )
| ( connected_space @ SV97 @ ( subspace_topology @ SV207 @ SY735 @ SV97 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[482]) ).
thf(594,plain,
! [SV208: $i,SV98: $i] :
( ( ! [SY736: $i] :
( ( connected_set @ SV98 @ SV208 @ SY736 )
| ~ ( topological_space @ SV208 @ SY736 )
| ~ ( subset_sets @ SV98 @ SV208 )
| ~ ( connected_space @ SV98 @ ( subspace_topology @ SV208 @ SY736 @ SV98 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[483]) ).
thf(595,plain,
! [SV209: $i,SV99: $i] :
( ( ! [SY737: $i] :
( ~ ( open_covering @ SV99 @ SV209 @ SY737 )
| ( topological_space @ SV209 @ SY737 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[484]) ).
thf(596,plain,
! [SV210: $i,SV100: $i] :
( ( ! [SY738: $i] :
( ~ ( open_covering @ SV100 @ SV210 @ SY738 )
| ( subset_collections @ SV100 @ SY738 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[485]) ).
thf(597,plain,
! [SV211: $i,SV101: $i] :
( ( ! [SY739: $i] :
~ ( open_covering @ SV101 @ SV211 @ SY739 )
| ( equal_sets @ ( union_of_members @ SV101 ) @ SV211 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[486]) ).
thf(598,plain,
! [SV212: $i,SV102: $i] :
( ( ! [SY740: $i] :
( ( open_covering @ SV102 @ SV212 @ SY740 )
| ~ ( topological_space @ SV212 @ SY740 )
| ~ ( subset_collections @ SV102 @ SY740 )
| ~ ( equal_sets @ ( union_of_members @ SV102 ) @ SV212 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[487]) ).
thf(599,plain,
! [SV213: $i,SV103: $i] :
( ( ~ ( compact_space @ SV103 @ SV213 )
| ( topological_space @ SV103 @ SV213 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[488]) ).
thf(600,plain,
! [SV214: $i,SV104: $i] :
( ( ~ ( compact_space @ SV104 @ SV214 )
| ! [SY741: $i] :
( ~ ( open_covering @ SY741 @ SV104 @ SV214 )
| ( finite @ ( f23 @ SV104 @ SV214 @ SY741 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[489]) ).
thf(601,plain,
! [SV215: $i,SV105: $i] :
( ( ~ ( compact_space @ SV105 @ SV215 )
| ! [SY742: $i] :
( ~ ( open_covering @ SY742 @ SV105 @ SV215 )
| ( subset_collections @ ( f23 @ SV105 @ SV215 @ SY742 ) @ SY742 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[490]) ).
thf(602,plain,
! [SV216: $i,SV106: $i] :
( ( ~ ( compact_space @ SV106 @ SV216 )
| ! [SY743: $i] :
( ~ ( open_covering @ SY743 @ SV106 @ SV216 )
| ( open_covering @ ( f23 @ SV106 @ SV216 @ SY743 ) @ SV106 @ SV216 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[491]) ).
thf(603,plain,
! [SV217: $i,SV107: $i] :
( ( ( compact_space @ SV107 @ SV217 )
| ~ ( topological_space @ SV107 @ SV217 )
| ( open_covering @ ( f24 @ SV107 @ SV217 ) @ SV107 @ SV217 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[492]) ).
thf(604,plain,
! [SV218: $i,SV108: $i] :
( ( ( compact_space @ SV108 @ SV218 )
| ~ ( topological_space @ SV108 @ SV218 )
| ! [SY744: $i] :
( ~ ( finite @ SY744 )
| ~ ( subset_collections @ SY744 @ ( f24 @ SV108 @ SV218 ) )
| ~ ( open_covering @ SY744 @ SV108 @ SV218 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[493]) ).
thf(605,plain,
! [SV219: $i,SV109: $i] :
( ( ! [SY745: $i] :
( ~ ( compact_set @ SV109 @ SV219 @ SY745 )
| ( topological_space @ SV219 @ SY745 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[494]) ).
thf(606,plain,
! [SV220: $i,SV110: $i] :
( ( ! [SY746: $i] :
~ ( compact_set @ SV110 @ SV220 @ SY746 )
| ( subset_sets @ SV110 @ SV220 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[495]) ).
thf(607,plain,
! [SV221: $i,SV111: $i] :
( ( ! [SY747: $i] :
( ~ ( compact_set @ SV111 @ SV221 @ SY747 )
| ( compact_space @ SV111 @ ( subspace_topology @ SV221 @ SY747 @ SV111 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[496]) ).
thf(608,plain,
! [SV222: $i,SV112: $i] :
( ( ! [SY748: $i] :
( ( compact_set @ SV112 @ SV222 @ SY748 )
| ~ ( topological_space @ SV222 @ SY748 )
| ~ ( subset_sets @ SV112 @ SV222 )
| ~ ( compact_space @ SV112 @ ( subspace_topology @ SV222 @ SY748 @ SV112 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[497]) ).
thf(609,plain,
! [SV113: $i,SV3: $i] :
( ( element_of_collection @ ( intersection_of_sets @ SV3 @ SV113 ) @ ( top_of_basis @ f ) )
= $false ),
inference(extcnf_not_pos,[status(thm)],[499]) ).
thf(610,plain,
! [SV114: $i,SV4: $i] :
( ( ( ~ ( element_of_set @ SV4 @ ( union_of_members @ SV114 ) ) )
= $true )
| ( ( element_of_set @ SV4 @ ( f1 @ SV114 @ SV4 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[500]) ).
thf(611,plain,
! [SV115: $i,SV5: $i] :
( ( ( ~ ( element_of_set @ SV5 @ ( union_of_members @ SV115 ) ) )
= $true )
| ( ( element_of_collection @ ( f1 @ SV115 @ SV5 ) @ SV115 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[501]) ).
thf(612,plain,
! [SV116: $i,SV6: $i] :
( ( ( element_of_set @ SV6 @ ( union_of_members @ SV116 ) )
= $true )
| ( ( ! [SY610: $i] :
( ~ ( element_of_set @ SV6 @ SY610 )
| ~ ( element_of_collection @ SY610 @ SV116 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[502]) ).
thf(613,plain,
! [SV117: $i,SV7: $i] :
( ( ( ~ ( element_of_set @ SV7 @ ( intersection_of_members @ SV117 ) ) )
= $true )
| ( ( ! [SY611: $i] :
( ~ ( element_of_collection @ SY611 @ SV117 )
| ( element_of_set @ SV7 @ SY611 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[503]) ).
thf(614,plain,
! [SV8: $i,SV118: $i] :
( ( ( element_of_collection @ ( f2 @ SV118 @ SV8 ) @ SV118 )
= $true )
| ( ( element_of_set @ SV8 @ ( intersection_of_members @ SV118 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[504]) ).
thf(615,plain,
! [SV119: $i,SV9: $i] :
( ( ( element_of_set @ SV9 @ ( intersection_of_members @ SV119 ) )
= $true )
| ( ( ~ ( element_of_set @ SV9 @ ( f2 @ SV119 @ SV9 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[505]) ).
thf(616,plain,
! [SV120: $i,SV10: $i] :
( ( ( ~ ( topological_space @ SV10 @ SV120 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV120 ) @ SV10 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[506]) ).
thf(617,plain,
! [SV121: $i,SV11: $i] :
( ( ( ~ ( topological_space @ SV11 @ SV121 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV121 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[507]) ).
thf(618,plain,
! [SV122: $i,SV12: $i] :
( ( ( ~ ( topological_space @ SV12 @ SV122 ) )
= $true )
| ( ( element_of_collection @ SV12 @ SV122 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[508]) ).
thf(619,plain,
! [SV123: $i,SV13: $i] :
( ( ( ~ ( topological_space @ SV13 @ SV123 ) )
= $true )
| ( ( ! [SY612: $i] :
( ~ ( element_of_collection @ SY612 @ SV123 )
| ! [SY613: $i] :
( ~ ( element_of_collection @ SY613 @ SV123 )
| ( element_of_collection @ ( intersection_of_sets @ SY612 @ SY613 ) @ SV123 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[509]) ).
thf(620,plain,
! [SV124: $i,SV14: $i] :
( ( ( ~ ( topological_space @ SV14 @ SV124 ) )
= $true )
| ( ( ! [SY614: $i] :
( ~ ( subset_collections @ SY614 @ SV124 )
| ( element_of_collection @ ( union_of_members @ SY614 ) @ SV124 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[510]) ).
thf(621,plain,
! [SV125: $i,SV15: $i] :
( ( ( topological_space @ SV15 @ SV125 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV125 ) @ SV15 )
| ~ ( element_of_collection @ empty_set @ SV125 )
| ~ ( element_of_collection @ SV15 @ SV125 )
| ( element_of_collection @ ( f3 @ SV15 @ SV125 ) @ SV125 )
| ( subset_collections @ ( f5 @ SV15 @ SV125 ) @ SV125 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[511]) ).
thf(622,plain,
! [SV126: $i,SV16: $i] :
( ( ( topological_space @ SV16 @ SV126 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 )
| ~ ( element_of_collection @ empty_set @ SV126 )
| ~ ( element_of_collection @ SV16 @ SV126 )
| ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[512]) ).
thf(623,plain,
! [SV127: $i,SV17: $i] :
( ( ( topological_space @ SV17 @ SV127 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV127 ) @ SV17 )
| ~ ( element_of_collection @ empty_set @ SV127 )
| ~ ( element_of_collection @ SV17 @ SV127 )
| ( element_of_collection @ ( f4 @ SV17 @ SV127 ) @ SV127 )
| ( subset_collections @ ( f5 @ SV17 @ SV127 ) @ SV127 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[513]) ).
thf(624,plain,
! [SV128: $i,SV18: $i] :
( ( ( topological_space @ SV18 @ SV128 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 )
| ~ ( element_of_collection @ empty_set @ SV128 )
| ~ ( element_of_collection @ SV18 @ SV128 )
| ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[514]) ).
thf(625,plain,
! [SV129: $i,SV19: $i] :
( ( ( topological_space @ SV19 @ SV129 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 )
| ~ ( element_of_collection @ empty_set @ SV129 )
| ~ ( element_of_collection @ SV19 @ SV129 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 )
| ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[515]) ).
thf(626,plain,
! [SV130: $i,SV20: $i] :
( ( ( topological_space @ SV20 @ SV130 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
| ~ ( element_of_collection @ empty_set @ SV130 )
| ~ ( element_of_collection @ SV20 @ SV130 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[516]) ).
thf(627,plain,
! [SV223: $i,SV131: $i,SV21: $i] :
( ( ~ ( open @ SV21 @ SV131 @ SV223 )
| ( topological_space @ SV131 @ SV223 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[517]) ).
thf(628,plain,
! [SV224: $i,SV132: $i,SV22: $i] :
( ( ~ ( open @ SV22 @ SV132 @ SV224 )
| ( element_of_collection @ SV22 @ SV224 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[518]) ).
thf(629,plain,
! [SV225: $i,SV133: $i,SV23: $i] :
( ( ( open @ SV23 @ SV133 @ SV225 )
| ~ ( topological_space @ SV133 @ SV225 )
| ~ ( element_of_collection @ SV23 @ SV225 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[519]) ).
thf(630,plain,
! [SV226: $i,SV134: $i,SV24: $i] :
( ( ~ ( closed @ SV24 @ SV134 @ SV226 )
| ( topological_space @ SV134 @ SV226 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[520]) ).
thf(631,plain,
! [SV227: $i,SV135: $i,SV25: $i] :
( ( ~ ( closed @ SV25 @ SV135 @ SV227 )
| ( open @ ( relative_complement_sets @ SV25 @ SV135 ) @ SV135 @ SV227 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[521]) ).
thf(632,plain,
! [SV228: $i,SV136: $i,SV26: $i] :
( ( ( closed @ SV26 @ SV136 @ SV228 )
| ~ ( topological_space @ SV136 @ SV228 )
| ~ ( open @ ( relative_complement_sets @ SV26 @ SV136 ) @ SV136 @ SV228 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[522]) ).
thf(633,plain,
! [SV229: $i,SV137: $i,SV27: $i] :
( ( ~ ( finer @ SV27 @ SV137 @ SV229 )
| ( topological_space @ SV229 @ SV27 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[523]) ).
thf(634,plain,
! [SV230: $i,SV138: $i,SV28: $i] :
( ( ~ ( finer @ SV28 @ SV138 @ SV230 )
| ( topological_space @ SV230 @ SV138 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[524]) ).
thf(635,plain,
! [SV139: $i,SV29: $i] :
( ( ( ! [SY623: $i] :
~ ( finer @ SV29 @ SV139 @ SY623 ) )
= $true )
| ( ( subset_collections @ SV139 @ SV29 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[525]) ).
thf(636,plain,
! [SV231: $i,SV140: $i,SV30: $i] :
( ( ( finer @ SV30 @ SV140 @ SV231 )
| ~ ( topological_space @ SV231 @ SV30 )
| ~ ( topological_space @ SV231 @ SV140 )
| ~ ( subset_collections @ SV140 @ SV30 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[526]) ).
thf(637,plain,
! [SV141: $i,SV31: $i] :
( ( ( ~ ( basis @ SV31 @ SV141 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV141 ) @ SV31 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[527]) ).
thf(638,plain,
! [SV232: $i,SV142: $i,SV32: $i] :
( ( ~ ( basis @ SV32 @ SV142 )
| ~ ( element_of_set @ SV232 @ SV32 )
| ! [SY749: $i] :
( ~ ( element_of_collection @ SY749 @ SV142 )
| ! [SY750: $i] :
( ~ ( element_of_collection @ SY750 @ SV142 )
| ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SY749 @ SY750 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SY749 @ SY750 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[528]) ).
thf(639,plain,
! [SV233: $i,SV143: $i,SV33: $i] :
( ( ~ ( basis @ SV33 @ SV143 )
| ~ ( element_of_set @ SV233 @ SV33 )
| ! [SY751: $i] :
( ~ ( element_of_collection @ SY751 @ SV143 )
| ! [SY752: $i] :
( ~ ( element_of_collection @ SY752 @ SV143 )
| ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SY751 @ SY752 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SY751 @ SY752 ) @ SV143 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[529]) ).
thf(640,plain,
! [SV234: $i,SV144: $i,SV34: $i] :
( ( ~ ( basis @ SV34 @ SV144 )
| ~ ( element_of_set @ SV234 @ SV34 )
| ! [SY753: $i] :
( ~ ( element_of_collection @ SY753 @ SV144 )
| ! [SY754: $i] :
( ~ ( element_of_collection @ SY754 @ SV144 )
| ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SY753 @ SY754 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SY753 @ SY754 ) @ ( intersection_of_sets @ SY753 @ SY754 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[530]) ).
thf(641,plain,
! [SV145: $i,SV35: $i] :
( ( ( basis @ SV35 @ SV145 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV145 ) @ SV35 )
| ( element_of_set @ ( f7 @ SV35 @ SV145 ) @ SV35 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[531]) ).
thf(642,plain,
! [SV146: $i,SV36: $i] :
( ( ( basis @ SV36 @ SV146 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV146 ) @ SV36 )
| ( element_of_collection @ ( f8 @ SV36 @ SV146 ) @ SV146 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[532]) ).
thf(643,plain,
! [SV147: $i,SV37: $i] :
( ( ( basis @ SV37 @ SV147 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV147 ) @ SV37 )
| ( element_of_collection @ ( f9 @ SV37 @ SV147 ) @ SV147 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[533]) ).
thf(644,plain,
! [SV148: $i,SV38: $i] :
( ( ( basis @ SV38 @ SV148 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV148 ) @ SV38 )
| ( element_of_set @ ( f7 @ SV38 @ SV148 ) @ ( intersection_of_sets @ ( f8 @ SV38 @ SV148 ) @ ( f9 @ SV38 @ SV148 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[534]) ).
thf(645,plain,
! [SV149: $i,SV39: $i] :
( ( ( basis @ SV39 @ SV149 )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 )
| ! [SY634: $i] :
( ~ ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SY634 )
| ~ ( element_of_collection @ SY634 @ SV149 )
| ~ ( subset_sets @ SY634 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[535]) ).
thf(646,plain,
! [SV150: $i,SV40: $i] :
( ( ( ~ ( element_of_collection @ SV40 @ ( top_of_basis @ SV150 ) ) )
= $true )
| ( ( ! [SY635: $i] :
( ~ ( element_of_set @ SY635 @ SV40 )
| ( element_of_set @ SY635 @ ( f10 @ SV150 @ SV40 @ SY635 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[536]) ).
thf(647,plain,
! [SV151: $i,SV41: $i] :
( ( ( ~ ( element_of_collection @ SV41 @ ( top_of_basis @ SV151 ) ) )
= $true )
| ( ( ! [SY636: $i] :
( ~ ( element_of_set @ SY636 @ SV41 )
| ( element_of_collection @ ( f10 @ SV151 @ SV41 @ SY636 ) @ SV151 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[537]) ).
thf(648,plain,
! [SV152: $i,SV42: $i] :
( ( ( ~ ( element_of_collection @ SV42 @ ( top_of_basis @ SV152 ) ) )
= $true )
| ( ( ! [SY637: $i] :
( ~ ( element_of_set @ SY637 @ SV42 )
| ( subset_sets @ ( f10 @ SV152 @ SV42 @ SY637 ) @ SV42 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[538]) ).
thf(649,plain,
! [SV153: $i,SV43: $i] :
( ( ( element_of_collection @ SV43 @ ( top_of_basis @ SV153 ) )
= $true )
| ( ( element_of_set @ ( f11 @ SV153 @ SV43 ) @ SV43 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[539]) ).
thf(650,plain,
! [SV154: $i,SV44: $i] :
( ( ( element_of_collection @ SV44 @ ( top_of_basis @ SV154 ) )
= $true )
| ( ( ! [SY638: $i] :
( ~ ( element_of_set @ ( f11 @ SV154 @ SV44 ) @ SY638 )
| ~ ( element_of_collection @ SY638 @ SV154 )
| ~ ( subset_sets @ SY638 @ SV44 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[540]) ).
thf(651,plain,
! [SV235: $i,SV155: $i,SV45: $i] :
( ( ! [SY755: $i] :
~ ( element_of_collection @ SV45 @ ( subspace_topology @ SV155 @ SV235 @ SY755 ) )
| ( topological_space @ SV155 @ SV235 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[541]) ).
thf(652,plain,
! [SV236: $i,SV156: $i,SV46: $i] :
( ( ! [SY756: $i] :
( ~ ( element_of_collection @ SV46 @ ( subspace_topology @ SV156 @ SV236 @ SY756 ) )
| ( subset_sets @ SY756 @ SV156 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[542]) ).
thf(653,plain,
! [SV237: $i,SV157: $i,SV47: $i] :
( ( ! [SY757: $i] :
( ~ ( element_of_collection @ SV47 @ ( subspace_topology @ SV157 @ SV237 @ SY757 ) )
| ( element_of_collection @ ( f12 @ SV157 @ SV237 @ SY757 @ SV47 ) @ SV237 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[543]) ).
thf(654,plain,
! [SV238: $i,SV158: $i,SV48: $i] :
( ( ! [SY758: $i] :
( ~ ( element_of_collection @ SV48 @ ( subspace_topology @ SV158 @ SV238 @ SY758 ) )
| ( equal_sets @ SV48 @ ( intersection_of_sets @ SY758 @ ( f12 @ SV158 @ SV238 @ SY758 @ SV48 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[544]) ).
thf(655,plain,
! [SV239: $i,SV159: $i,SV49: $i] :
( ( ! [SY759: $i] :
( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SY759 ) )
| ~ ( topological_space @ SV159 @ SV239 )
| ~ ( subset_sets @ SY759 @ SV159 )
| ! [SY760: $i] :
( ~ ( element_of_collection @ SY760 @ SV239 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SY759 @ SY760 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[545]) ).
thf(656,plain,
! [SV240: $i,SV160: $i,SV50: $i] :
( ( ! [SY761: $i] :
( ~ ( element_of_set @ SV50 @ ( interior @ SV160 @ SV240 @ SY761 ) )
| ( topological_space @ SV240 @ SY761 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[546]) ).
thf(657,plain,
! [SV241: $i,SV161: $i,SV51: $i] :
( ( ! [SY762: $i] :
~ ( element_of_set @ SV51 @ ( interior @ SV161 @ SV241 @ SY762 ) )
| ( subset_sets @ SV161 @ SV241 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[547]) ).
thf(658,plain,
! [SV242: $i,SV162: $i,SV52: $i] :
( ( ! [SY763: $i] :
( ~ ( element_of_set @ SV52 @ ( interior @ SV162 @ SV242 @ SY763 ) )
| ( element_of_set @ SV52 @ ( f13 @ SV162 @ SV242 @ SY763 @ SV52 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[548]) ).
thf(659,plain,
! [SV243: $i,SV163: $i,SV53: $i] :
( ( ! [SY764: $i] :
( ~ ( element_of_set @ SV53 @ ( interior @ SV163 @ SV243 @ SY764 ) )
| ( subset_sets @ ( f13 @ SV163 @ SV243 @ SY764 @ SV53 ) @ SV163 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[549]) ).
thf(660,plain,
! [SV244: $i,SV164: $i,SV54: $i] :
( ( ! [SY765: $i] :
( ~ ( element_of_set @ SV54 @ ( interior @ SV164 @ SV244 @ SY765 ) )
| ( open @ ( f13 @ SV164 @ SV244 @ SY765 @ SV54 ) @ SV244 @ SY765 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[550]) ).
thf(661,plain,
! [SV245: $i,SV165: $i,SV55: $i] :
( ( ! [SY766: $i] :
( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SY766 ) )
| ~ ( topological_space @ SV245 @ SY766 )
| ~ ( subset_sets @ SV165 @ SV245 )
| ! [SY767: $i] :
( ~ ( element_of_set @ SV55 @ SY767 )
| ~ ( subset_sets @ SY767 @ SV165 )
| ~ ( open @ SY767 @ SV245 @ SY766 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[551]) ).
thf(662,plain,
! [SV246: $i,SV166: $i,SV56: $i] :
( ( ! [SY768: $i] :
( ~ ( element_of_set @ SV56 @ ( closure @ SV166 @ SV246 @ SY768 ) )
| ( topological_space @ SV246 @ SY768 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[552]) ).
thf(663,plain,
! [SV247: $i,SV167: $i,SV57: $i] :
( ( ! [SY769: $i] :
~ ( element_of_set @ SV57 @ ( closure @ SV167 @ SV247 @ SY769 ) )
| ( subset_sets @ SV167 @ SV247 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[553]) ).
thf(664,plain,
! [SV248: $i,SV168: $i,SV58: $i] :
( ( ! [SY770: $i] :
( ~ ( element_of_set @ SV58 @ ( closure @ SV168 @ SV248 @ SY770 ) )
| ! [SY771: $i] :
( ~ ( subset_sets @ SV168 @ SY771 )
| ~ ( closed @ SY771 @ SV248 @ SY770 )
| ( element_of_set @ SV58 @ SY771 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[554]) ).
thf(665,plain,
! [SV249: $i,SV169: $i,SV59: $i] :
( ( ! [SY772: $i] :
( ( element_of_set @ SV59 @ ( closure @ SV169 @ SV249 @ SY772 ) )
| ~ ( topological_space @ SV249 @ SY772 )
| ~ ( subset_sets @ SV169 @ SV249 )
| ( subset_sets @ SV169 @ ( f14 @ SV169 @ SV249 @ SY772 @ SV59 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[555]) ).
thf(666,plain,
! [SV250: $i,SV170: $i,SV60: $i] :
( ( ! [SY773: $i] :
( ( element_of_set @ SV60 @ ( closure @ SV170 @ SV250 @ SY773 ) )
| ~ ( topological_space @ SV250 @ SY773 )
| ~ ( subset_sets @ SV170 @ SV250 )
| ( closed @ ( f14 @ SV170 @ SV250 @ SY773 @ SV60 ) @ SV250 @ SY773 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[556]) ).
thf(667,plain,
! [SV251: $i,SV171: $i,SV61: $i] :
( ( ! [SY774: $i] :
( ( element_of_set @ SV61 @ ( closure @ SV171 @ SV251 @ SY774 ) )
| ~ ( topological_space @ SV251 @ SY774 )
| ~ ( subset_sets @ SV171 @ SV251 )
| ~ ( element_of_set @ SV61 @ ( f14 @ SV171 @ SV251 @ SY774 @ SV61 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[557]) ).
thf(668,plain,
! [SV252: $i,SV172: $i,SV62: $i] :
( ( ! [SY775: $i] :
( ~ ( neighborhood @ SV62 @ SV172 @ SV252 @ SY775 )
| ( topological_space @ SV252 @ SY775 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[558]) ).
thf(669,plain,
! [SV253: $i,SV173: $i,SV63: $i] :
( ( ! [SY776: $i] :
( ~ ( neighborhood @ SV63 @ SV173 @ SV253 @ SY776 )
| ( open @ SV63 @ SV253 @ SY776 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[559]) ).
thf(670,plain,
! [SV174: $i,SV64: $i] :
( ( ( ! [SY680: $i,SY681: $i] :
~ ( neighborhood @ SV64 @ SV174 @ SY680 @ SY681 ) )
= $true )
| ( ( element_of_set @ SV174 @ SV64 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[560]) ).
thf(671,plain,
! [SV254: $i,SV175: $i,SV65: $i] :
( ( ! [SY777: $i] :
( ( neighborhood @ SV65 @ SV175 @ SV254 @ SY777 )
| ~ ( topological_space @ SV254 @ SY777 )
| ~ ( open @ SV65 @ SV254 @ SY777 )
| ~ ( element_of_set @ SV175 @ SV65 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[561]) ).
thf(672,plain,
! [SV255: $i,SV176: $i,SV66: $i] :
( ( ! [SY778: $i] :
( ~ ( limit_point @ SV66 @ SV176 @ SV255 @ SY778 )
| ( topological_space @ SV255 @ SY778 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[562]) ).
thf(673,plain,
! [SV256: $i,SV177: $i,SV67: $i] :
( ( ! [SY779: $i] :
~ ( limit_point @ SV67 @ SV177 @ SV256 @ SY779 )
| ( subset_sets @ SV177 @ SV256 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[563]) ).
thf(674,plain,
! [SV257: $i,SV178: $i,SV68: $i] :
( ( ! [SY780: $i] :
( ~ ( limit_point @ SV68 @ SV178 @ SV257 @ SY780 )
| ! [SY781: $i] :
( ~ ( neighborhood @ SY781 @ SV68 @ SV257 @ SY780 )
| ( element_of_set @ ( f15 @ SV68 @ SV178 @ SV257 @ SY780 @ SY781 ) @ ( intersection_of_sets @ SY781 @ SV178 ) ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[564]) ).
thf(675,plain,
! [SV258: $i,SV179: $i,SV69: $i] :
( ( ! [SY782: $i] :
( ~ ( limit_point @ SV69 @ SV179 @ SV258 @ SY782 )
| ! [SY783: $i] :
( ~ ( neighborhood @ SY783 @ SV69 @ SV258 @ SY782 )
| ~ ( eq_p @ ( f15 @ SV69 @ SV179 @ SV258 @ SY782 @ SY783 ) @ SV69 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[565]) ).
thf(676,plain,
! [SV259: $i,SV180: $i,SV70: $i] :
( ( ! [SY784: $i] :
( ( limit_point @ SV70 @ SV180 @ SV259 @ SY784 )
| ~ ( topological_space @ SV259 @ SY784 )
| ~ ( subset_sets @ SV180 @ SV259 )
| ( neighborhood @ ( f16 @ SV70 @ SV180 @ SV259 @ SY784 ) @ SV70 @ SV259 @ SY784 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[566]) ).
thf(677,plain,
! [SV260: $i,SV181: $i,SV71: $i] :
( ( ! [SY785: $i] :
( ( limit_point @ SV71 @ SV181 @ SV260 @ SY785 )
| ~ ( topological_space @ SV260 @ SY785 )
| ~ ( subset_sets @ SV181 @ SV260 )
| ! [SY786: $i] :
( ~ ( element_of_set @ SY786 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SY785 ) @ SV181 ) )
| ( eq_p @ SY786 @ SV71 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[567]) ).
thf(678,plain,
! [SV261: $i,SV182: $i,SV72: $i] :
( ( ! [SY787: $i] :
( ~ ( element_of_set @ SV72 @ ( boundary @ SV182 @ SV261 @ SY787 ) )
| ( topological_space @ SV261 @ SY787 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[568]) ).
thf(679,plain,
! [SV262: $i,SV183: $i,SV73: $i] :
( ( ! [SY788: $i] :
( ~ ( element_of_set @ SV73 @ ( boundary @ SV183 @ SV262 @ SY788 ) )
| ( element_of_set @ SV73 @ ( closure @ SV183 @ SV262 @ SY788 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[569]) ).
thf(680,plain,
! [SV263: $i,SV184: $i,SV74: $i] :
( ( ! [SY789: $i] :
( ~ ( element_of_set @ SV74 @ ( boundary @ SV184 @ SV263 @ SY789 ) )
| ( element_of_set @ SV74 @ ( closure @ ( relative_complement_sets @ SV184 @ SV263 ) @ SV263 @ SY789 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[570]) ).
thf(681,plain,
! [SV264: $i,SV185: $i,SV75: $i] :
( ( ! [SY790: $i] :
( ( element_of_set @ SV75 @ ( boundary @ SV185 @ SV264 @ SY790 ) )
| ~ ( topological_space @ SV264 @ SY790 )
| ~ ( element_of_set @ SV75 @ ( closure @ SV185 @ SV264 @ SY790 ) )
| ~ ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SV185 @ SV264 ) @ SV264 @ SY790 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[571]) ).
thf(682,plain,
! [SV186: $i,SV76: $i] :
( ( ( ~ ( hausdorff @ SV76 @ SV186 ) )
= $true )
| ( ( topological_space @ SV76 @ SV186 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[572]) ).
thf(683,plain,
! [SV187: $i,SV77: $i] :
( ( ( ~ ( hausdorff @ SV77 @ SV187 ) )
= $true )
| ( ( ! [SY707: $i] :
( ~ ( element_of_set @ SY707 @ SV77 )
| ! [SY708: $i] :
( ~ ( element_of_set @ SY708 @ SV77 )
| ( eq_p @ SY707 @ SY708 )
| ( neighborhood @ ( f17 @ SV77 @ SV187 @ SY707 @ SY708 ) @ SY707 @ SV77 @ SV187 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[573]) ).
thf(684,plain,
! [SV188: $i,SV78: $i] :
( ( ( ~ ( hausdorff @ SV78 @ SV188 ) )
= $true )
| ( ( ! [SY709: $i] :
( ~ ( element_of_set @ SY709 @ SV78 )
| ! [SY710: $i] :
( ~ ( element_of_set @ SY710 @ SV78 )
| ( eq_p @ SY709 @ SY710 )
| ( neighborhood @ ( f18 @ SV78 @ SV188 @ SY709 @ SY710 ) @ SY710 @ SV78 @ SV188 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[574]) ).
thf(685,plain,
! [SV189: $i,SV79: $i] :
( ( ( ~ ( hausdorff @ SV79 @ SV189 ) )
= $true )
| ( ( ! [SY711: $i] :
( ~ ( element_of_set @ SY711 @ SV79 )
| ! [SY712: $i] :
( ~ ( element_of_set @ SY712 @ SV79 )
| ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SY711 @ SY712 ) @ ( f18 @ SV79 @ SV189 @ SY711 @ SY712 ) )
| ( eq_p @ SY711 @ SY712 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[575]) ).
thf(686,plain,
! [SV190: $i,SV80: $i] :
( ( ( hausdorff @ SV80 @ SV190 )
= $true )
| ( ( ~ ( topological_space @ SV80 @ SV190 )
| ( element_of_set @ ( f19 @ SV80 @ SV190 ) @ SV80 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[576]) ).
thf(687,plain,
! [SV191: $i,SV81: $i] :
( ( ( hausdorff @ SV81 @ SV191 )
= $true )
| ( ( ~ ( topological_space @ SV81 @ SV191 )
| ( element_of_set @ ( f20 @ SV81 @ SV191 ) @ SV81 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[577]) ).
thf(688,plain,
! [SV192: $i,SV82: $i] :
( ( ( hausdorff @ SV82 @ SV192 )
= $true )
| ( ( ~ ( topological_space @ SV82 @ SV192 )
| ~ ( eq_p @ ( f19 @ SV82 @ SV192 ) @ ( f20 @ SV82 @ SV192 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[578]) ).
thf(689,plain,
! [SV193: $i,SV83: $i] :
( ( ( hausdorff @ SV83 @ SV193 )
= $true )
| ( ( ~ ( topological_space @ SV83 @ SV193 )
| ! [SY713: $i] :
( ~ ( neighborhood @ SY713 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ! [SY714: $i] :
( ~ ( neighborhood @ SY714 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ~ ( disjoint_s @ SY713 @ SY714 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[579]) ).
thf(690,plain,
! [SV265: $i,SV194: $i,SV84: $i] :
( ( ! [SY791: $i] :
( ~ ( separation @ SV84 @ SV194 @ SV265 @ SY791 )
| ( topological_space @ SV265 @ SY791 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[580]) ).
thf(691,plain,
! [SV195: $i,SV85: $i] :
( ( ( ! [SY717: $i,SY718: $i] :
~ ( separation @ SV85 @ SV195 @ SY717 @ SY718 ) )
= $true )
| ( ( ~ ( equal_sets @ SV85 @ empty_set ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[581]) ).
thf(692,plain,
! [SV196: $i,SV86: $i] :
( ( ( ! [SY719: $i,SY720: $i] :
~ ( separation @ SV86 @ SV196 @ SY719 @ SY720 ) )
= $true )
| ( ( ~ ( equal_sets @ SV196 @ empty_set ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[582]) ).
thf(693,plain,
! [SV266: $i,SV197: $i,SV87: $i] :
( ( ! [SY792: $i] :
( ~ ( separation @ SV87 @ SV197 @ SV266 @ SY792 )
| ( element_of_collection @ SV87 @ SY792 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[583]) ).
thf(694,plain,
! [SV267: $i,SV198: $i,SV88: $i] :
( ( ! [SY793: $i] :
( ~ ( separation @ SV88 @ SV198 @ SV267 @ SY793 )
| ( element_of_collection @ SV198 @ SY793 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[584]) ).
thf(695,plain,
! [SV268: $i,SV199: $i,SV89: $i] :
( ( ! [SY794: $i] :
~ ( separation @ SV89 @ SV199 @ SV268 @ SY794 )
| ( equal_sets @ ( union_of_sets @ SV89 @ SV199 ) @ SV268 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[585]) ).
thf(696,plain,
! [SV200: $i,SV90: $i] :
( ( ( ! [SY727: $i,SY728: $i] :
~ ( separation @ SV90 @ SV200 @ SY727 @ SY728 ) )
= $true )
| ( ( disjoint_s @ SV90 @ SV200 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[586]) ).
thf(697,plain,
! [SV269: $i,SV201: $i,SV91: $i] :
( ( ! [SY795: $i] :
( ( separation @ SV91 @ SV201 @ SV269 @ SY795 )
| ~ ( topological_space @ SV269 @ SY795 )
| ( equal_sets @ SV91 @ empty_set )
| ( equal_sets @ SV201 @ empty_set )
| ~ ( element_of_collection @ SV91 @ SY795 )
| ~ ( element_of_collection @ SV201 @ SY795 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[587]) ).
thf(698,plain,
! [SV202: $i,SV92: $i] :
( ( ( ~ ( connected_space @ SV92 @ SV202 ) )
= $true )
| ( ( topological_space @ SV92 @ SV202 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[588]) ).
thf(699,plain,
! [SV203: $i,SV93: $i] :
( ( ( ~ ( connected_space @ SV93 @ SV203 ) )
= $true )
| ( ( ! [SY731: $i,SY732: $i] :
~ ( separation @ SY731 @ SY732 @ SV93 @ SV203 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[589]) ).
thf(700,plain,
! [SV204: $i,SV94: $i] :
( ( ( connected_space @ SV94 @ SV204 )
= $true )
| ( ( ~ ( topological_space @ SV94 @ SV204 )
| ( separation @ ( f21 @ SV94 @ SV204 ) @ ( f22 @ SV94 @ SV204 ) @ SV94 @ SV204 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[590]) ).
thf(701,plain,
! [SV270: $i,SV205: $i,SV95: $i] :
( ( ~ ( connected_set @ SV95 @ SV205 @ SV270 )
| ( topological_space @ SV205 @ SV270 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[591]) ).
thf(702,plain,
! [SV206: $i,SV96: $i] :
( ( ( ! [SY734: $i] :
~ ( connected_set @ SV96 @ SV206 @ SY734 ) )
= $true )
| ( ( subset_sets @ SV96 @ SV206 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[592]) ).
thf(703,plain,
! [SV271: $i,SV207: $i,SV97: $i] :
( ( ~ ( connected_set @ SV97 @ SV207 @ SV271 )
| ( connected_space @ SV97 @ ( subspace_topology @ SV207 @ SV271 @ SV97 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[593]) ).
thf(704,plain,
! [SV272: $i,SV208: $i,SV98: $i] :
( ( ( connected_set @ SV98 @ SV208 @ SV272 )
| ~ ( topological_space @ SV208 @ SV272 )
| ~ ( subset_sets @ SV98 @ SV208 )
| ~ ( connected_space @ SV98 @ ( subspace_topology @ SV208 @ SV272 @ SV98 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[594]) ).
thf(705,plain,
! [SV273: $i,SV209: $i,SV99: $i] :
( ( ~ ( open_covering @ SV99 @ SV209 @ SV273 )
| ( topological_space @ SV209 @ SV273 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[595]) ).
thf(706,plain,
! [SV274: $i,SV210: $i,SV100: $i] :
( ( ~ ( open_covering @ SV100 @ SV210 @ SV274 )
| ( subset_collections @ SV100 @ SV274 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[596]) ).
thf(707,plain,
! [SV211: $i,SV101: $i] :
( ( ( ! [SY739: $i] :
~ ( open_covering @ SV101 @ SV211 @ SY739 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV101 ) @ SV211 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[597]) ).
thf(708,plain,
! [SV275: $i,SV212: $i,SV102: $i] :
( ( ( open_covering @ SV102 @ SV212 @ SV275 )
| ~ ( topological_space @ SV212 @ SV275 )
| ~ ( subset_collections @ SV102 @ SV275 )
| ~ ( equal_sets @ ( union_of_members @ SV102 ) @ SV212 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[598]) ).
thf(709,plain,
! [SV213: $i,SV103: $i] :
( ( ( ~ ( compact_space @ SV103 @ SV213 ) )
= $true )
| ( ( topological_space @ SV103 @ SV213 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[599]) ).
thf(710,plain,
! [SV214: $i,SV104: $i] :
( ( ( ~ ( compact_space @ SV104 @ SV214 ) )
= $true )
| ( ( ! [SY741: $i] :
( ~ ( open_covering @ SY741 @ SV104 @ SV214 )
| ( finite @ ( f23 @ SV104 @ SV214 @ SY741 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[600]) ).
thf(711,plain,
! [SV215: $i,SV105: $i] :
( ( ( ~ ( compact_space @ SV105 @ SV215 ) )
= $true )
| ( ( ! [SY742: $i] :
( ~ ( open_covering @ SY742 @ SV105 @ SV215 )
| ( subset_collections @ ( f23 @ SV105 @ SV215 @ SY742 ) @ SY742 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[601]) ).
thf(712,plain,
! [SV216: $i,SV106: $i] :
( ( ( ~ ( compact_space @ SV106 @ SV216 ) )
= $true )
| ( ( ! [SY743: $i] :
( ~ ( open_covering @ SY743 @ SV106 @ SV216 )
| ( open_covering @ ( f23 @ SV106 @ SV216 @ SY743 ) @ SV106 @ SV216 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[602]) ).
thf(713,plain,
! [SV217: $i,SV107: $i] :
( ( ( compact_space @ SV107 @ SV217 )
= $true )
| ( ( ~ ( topological_space @ SV107 @ SV217 )
| ( open_covering @ ( f24 @ SV107 @ SV217 ) @ SV107 @ SV217 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[603]) ).
thf(714,plain,
! [SV218: $i,SV108: $i] :
( ( ( compact_space @ SV108 @ SV218 )
= $true )
| ( ( ~ ( topological_space @ SV108 @ SV218 )
| ! [SY744: $i] :
( ~ ( finite @ SY744 )
| ~ ( subset_collections @ SY744 @ ( f24 @ SV108 @ SV218 ) )
| ~ ( open_covering @ SY744 @ SV108 @ SV218 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[604]) ).
thf(715,plain,
! [SV276: $i,SV219: $i,SV109: $i] :
( ( ~ ( compact_set @ SV109 @ SV219 @ SV276 )
| ( topological_space @ SV219 @ SV276 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[605]) ).
thf(716,plain,
! [SV220: $i,SV110: $i] :
( ( ( ! [SY746: $i] :
~ ( compact_set @ SV110 @ SV220 @ SY746 ) )
= $true )
| ( ( subset_sets @ SV110 @ SV220 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[606]) ).
thf(717,plain,
! [SV277: $i,SV221: $i,SV111: $i] :
( ( ~ ( compact_set @ SV111 @ SV221 @ SV277 )
| ( compact_space @ SV111 @ ( subspace_topology @ SV221 @ SV277 @ SV111 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[607]) ).
thf(718,plain,
! [SV278: $i,SV222: $i,SV112: $i] :
( ( ( compact_set @ SV112 @ SV222 @ SV278 )
| ~ ( topological_space @ SV222 @ SV278 )
| ~ ( subset_sets @ SV112 @ SV222 )
| ~ ( compact_space @ SV112 @ ( subspace_topology @ SV222 @ SV278 @ SV112 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[608]) ).
thf(719,plain,
! [SV114: $i,SV4: $i] :
( ( ( element_of_set @ SV4 @ ( union_of_members @ SV114 ) )
= $false )
| ( ( element_of_set @ SV4 @ ( f1 @ SV114 @ SV4 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[610]) ).
thf(720,plain,
! [SV115: $i,SV5: $i] :
( ( ( element_of_set @ SV5 @ ( union_of_members @ SV115 ) )
= $false )
| ( ( element_of_collection @ ( f1 @ SV115 @ SV5 ) @ SV115 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[611]) ).
thf(721,plain,
! [SV116: $i,SV279: $i,SV6: $i] :
( ( ( ~ ( element_of_set @ SV6 @ SV279 )
| ~ ( element_of_collection @ SV279 @ SV116 ) )
= $true )
| ( ( element_of_set @ SV6 @ ( union_of_members @ SV116 ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[612]) ).
thf(722,plain,
! [SV117: $i,SV7: $i] :
( ( ( element_of_set @ SV7 @ ( intersection_of_members @ SV117 ) )
= $false )
| ( ( ! [SY611: $i] :
( ~ ( element_of_collection @ SY611 @ SV117 )
| ( element_of_set @ SV7 @ SY611 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[613]) ).
thf(723,plain,
! [SV119: $i,SV9: $i] :
( ( ( element_of_set @ SV9 @ ( f2 @ SV119 @ SV9 ) )
= $false )
| ( ( element_of_set @ SV9 @ ( intersection_of_members @ SV119 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[615]) ).
thf(724,plain,
! [SV120: $i,SV10: $i] :
( ( ( topological_space @ SV10 @ SV120 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV120 ) @ SV10 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[616]) ).
thf(725,plain,
! [SV121: $i,SV11: $i] :
( ( ( topological_space @ SV11 @ SV121 )
= $false )
| ( ( element_of_collection @ empty_set @ SV121 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[617]) ).
thf(726,plain,
! [SV122: $i,SV12: $i] :
( ( ( topological_space @ SV12 @ SV122 )
= $false )
| ( ( element_of_collection @ SV12 @ SV122 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[618]) ).
thf(727,plain,
! [SV123: $i,SV13: $i] :
( ( ( topological_space @ SV13 @ SV123 )
= $false )
| ( ( ! [SY612: $i] :
( ~ ( element_of_collection @ SY612 @ SV123 )
| ! [SY613: $i] :
( ~ ( element_of_collection @ SY613 @ SV123 )
| ( element_of_collection @ ( intersection_of_sets @ SY612 @ SY613 ) @ SV123 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[619]) ).
thf(728,plain,
! [SV124: $i,SV14: $i] :
( ( ( topological_space @ SV14 @ SV124 )
= $false )
| ( ( ! [SY614: $i] :
( ~ ( subset_collections @ SY614 @ SV124 )
| ( element_of_collection @ ( union_of_members @ SY614 ) @ SV124 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[620]) ).
thf(729,plain,
! [SV15: $i,SV125: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV125 ) @ SV15 ) )
= $true )
| ( ( ~ ( element_of_collection @ empty_set @ SV125 )
| ~ ( element_of_collection @ SV15 @ SV125 )
| ( element_of_collection @ ( f3 @ SV15 @ SV125 ) @ SV125 )
| ( subset_collections @ ( f5 @ SV15 @ SV125 ) @ SV125 ) )
= $true )
| ( ( topological_space @ SV15 @ SV125 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[621]) ).
thf(730,plain,
! [SV16: $i,SV126: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 ) )
= $true )
| ( ( ~ ( element_of_collection @ empty_set @ SV126 )
| ~ ( element_of_collection @ SV16 @ SV126 )
| ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 ) )
= $true )
| ( ( topological_space @ SV16 @ SV126 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[622]) ).
thf(731,plain,
! [SV17: $i,SV127: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV127 ) @ SV17 ) )
= $true )
| ( ( ~ ( element_of_collection @ empty_set @ SV127 )
| ~ ( element_of_collection @ SV17 @ SV127 )
| ( element_of_collection @ ( f4 @ SV17 @ SV127 ) @ SV127 )
| ( subset_collections @ ( f5 @ SV17 @ SV127 ) @ SV127 ) )
= $true )
| ( ( topological_space @ SV17 @ SV127 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[623]) ).
thf(732,plain,
! [SV18: $i,SV128: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 ) )
= $true )
| ( ( ~ ( element_of_collection @ empty_set @ SV128 )
| ~ ( element_of_collection @ SV18 @ SV128 )
| ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 ) )
= $true )
| ( ( topological_space @ SV18 @ SV128 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[624]) ).
thf(733,plain,
! [SV19: $i,SV129: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 ) )
= $true )
| ( ( ~ ( element_of_collection @ empty_set @ SV129 )
| ~ ( element_of_collection @ SV19 @ SV129 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 )
| ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 ) )
= $true )
| ( ( topological_space @ SV19 @ SV129 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[625]) ).
thf(734,plain,
! [SV20: $i,SV130: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 ) )
= $true )
| ( ( ~ ( element_of_collection @ empty_set @ SV130 )
| ~ ( element_of_collection @ SV20 @ SV130 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true )
| ( ( topological_space @ SV20 @ SV130 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[626]) ).
thf(735,plain,
! [SV223: $i,SV131: $i,SV21: $i] :
( ( ( ~ ( open @ SV21 @ SV131 @ SV223 ) )
= $true )
| ( ( topological_space @ SV131 @ SV223 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[627]) ).
thf(736,plain,
! [SV224: $i,SV132: $i,SV22: $i] :
( ( ( ~ ( open @ SV22 @ SV132 @ SV224 ) )
= $true )
| ( ( element_of_collection @ SV22 @ SV224 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[628]) ).
thf(737,plain,
! [SV225: $i,SV133: $i,SV23: $i] :
( ( ( open @ SV23 @ SV133 @ SV225 )
= $true )
| ( ( ~ ( topological_space @ SV133 @ SV225 )
| ~ ( element_of_collection @ SV23 @ SV225 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[629]) ).
thf(738,plain,
! [SV226: $i,SV134: $i,SV24: $i] :
( ( ( ~ ( closed @ SV24 @ SV134 @ SV226 ) )
= $true )
| ( ( topological_space @ SV134 @ SV226 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[630]) ).
thf(739,plain,
! [SV227: $i,SV135: $i,SV25: $i] :
( ( ( ~ ( closed @ SV25 @ SV135 @ SV227 ) )
= $true )
| ( ( open @ ( relative_complement_sets @ SV25 @ SV135 ) @ SV135 @ SV227 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[631]) ).
thf(740,plain,
! [SV228: $i,SV136: $i,SV26: $i] :
( ( ( closed @ SV26 @ SV136 @ SV228 )
= $true )
| ( ( ~ ( topological_space @ SV136 @ SV228 )
| ~ ( open @ ( relative_complement_sets @ SV26 @ SV136 ) @ SV136 @ SV228 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[632]) ).
thf(741,plain,
! [SV229: $i,SV137: $i,SV27: $i] :
( ( ( ~ ( finer @ SV27 @ SV137 @ SV229 ) )
= $true )
| ( ( topological_space @ SV229 @ SV27 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[633]) ).
thf(742,plain,
! [SV230: $i,SV138: $i,SV28: $i] :
( ( ( ~ ( finer @ SV28 @ SV138 @ SV230 ) )
= $true )
| ( ( topological_space @ SV230 @ SV138 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[634]) ).
thf(743,plain,
! [SV280: $i,SV139: $i,SV29: $i] :
( ( ( ~ ( finer @ SV29 @ SV139 @ SV280 ) )
= $true )
| ( ( subset_collections @ SV139 @ SV29 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[635]) ).
thf(744,plain,
! [SV231: $i,SV140: $i,SV30: $i] :
( ( ( finer @ SV30 @ SV140 @ SV231 )
= $true )
| ( ( ~ ( topological_space @ SV231 @ SV30 )
| ~ ( topological_space @ SV231 @ SV140 )
| ~ ( subset_collections @ SV140 @ SV30 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[636]) ).
thf(745,plain,
! [SV141: $i,SV31: $i] :
( ( ( basis @ SV31 @ SV141 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV141 ) @ SV31 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[637]) ).
thf(746,plain,
! [SV232: $i,SV142: $i,SV32: $i] :
( ( ( ~ ( basis @ SV32 @ SV142 ) )
= $true )
| ( ( ~ ( element_of_set @ SV232 @ SV32 )
| ! [SY749: $i] :
( ~ ( element_of_collection @ SY749 @ SV142 )
| ! [SY750: $i] :
( ~ ( element_of_collection @ SY750 @ SV142 )
| ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SY749 @ SY750 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SY749 @ SY750 ) ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[638]) ).
thf(747,plain,
! [SV233: $i,SV143: $i,SV33: $i] :
( ( ( ~ ( basis @ SV33 @ SV143 ) )
= $true )
| ( ( ~ ( element_of_set @ SV233 @ SV33 )
| ! [SY751: $i] :
( ~ ( element_of_collection @ SY751 @ SV143 )
| ! [SY752: $i] :
( ~ ( element_of_collection @ SY752 @ SV143 )
| ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SY751 @ SY752 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SY751 @ SY752 ) @ SV143 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[639]) ).
thf(748,plain,
! [SV234: $i,SV144: $i,SV34: $i] :
( ( ( ~ ( basis @ SV34 @ SV144 ) )
= $true )
| ( ( ~ ( element_of_set @ SV234 @ SV34 )
| ! [SY753: $i] :
( ~ ( element_of_collection @ SY753 @ SV144 )
| ! [SY754: $i] :
( ~ ( element_of_collection @ SY754 @ SV144 )
| ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SY753 @ SY754 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SY753 @ SY754 ) @ ( intersection_of_sets @ SY753 @ SY754 ) ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[640]) ).
thf(749,plain,
! [SV35: $i,SV145: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV145 ) @ SV35 ) )
= $true )
| ( ( element_of_set @ ( f7 @ SV35 @ SV145 ) @ SV35 )
= $true )
| ( ( basis @ SV35 @ SV145 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[641]) ).
thf(750,plain,
! [SV36: $i,SV146: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV146 ) @ SV36 ) )
= $true )
| ( ( element_of_collection @ ( f8 @ SV36 @ SV146 ) @ SV146 )
= $true )
| ( ( basis @ SV36 @ SV146 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[642]) ).
thf(751,plain,
! [SV37: $i,SV147: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV147 ) @ SV37 ) )
= $true )
| ( ( element_of_collection @ ( f9 @ SV37 @ SV147 ) @ SV147 )
= $true )
| ( ( basis @ SV37 @ SV147 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[643]) ).
thf(752,plain,
! [SV38: $i,SV148: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV148 ) @ SV38 ) )
= $true )
| ( ( element_of_set @ ( f7 @ SV38 @ SV148 ) @ ( intersection_of_sets @ ( f8 @ SV38 @ SV148 ) @ ( f9 @ SV38 @ SV148 ) ) )
= $true )
| ( ( basis @ SV38 @ SV148 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[644]) ).
thf(753,plain,
! [SV39: $i,SV149: $i] :
( ( ( ~ ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 ) )
= $true )
| ( ( ! [SY634: $i] :
( ~ ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SY634 )
| ~ ( element_of_collection @ SY634 @ SV149 )
| ~ ( subset_sets @ SY634 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) ) ) )
= $true )
| ( ( basis @ SV39 @ SV149 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[645]) ).
thf(754,plain,
! [SV150: $i,SV40: $i] :
( ( ( element_of_collection @ SV40 @ ( top_of_basis @ SV150 ) )
= $false )
| ( ( ! [SY635: $i] :
( ~ ( element_of_set @ SY635 @ SV40 )
| ( element_of_set @ SY635 @ ( f10 @ SV150 @ SV40 @ SY635 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[646]) ).
thf(755,plain,
! [SV151: $i,SV41: $i] :
( ( ( element_of_collection @ SV41 @ ( top_of_basis @ SV151 ) )
= $false )
| ( ( ! [SY636: $i] :
( ~ ( element_of_set @ SY636 @ SV41 )
| ( element_of_collection @ ( f10 @ SV151 @ SV41 @ SY636 ) @ SV151 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[647]) ).
thf(756,plain,
! [SV152: $i,SV42: $i] :
( ( ( element_of_collection @ SV42 @ ( top_of_basis @ SV152 ) )
= $false )
| ( ( ! [SY637: $i] :
( ~ ( element_of_set @ SY637 @ SV42 )
| ( subset_sets @ ( f10 @ SV152 @ SV42 @ SY637 ) @ SV42 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[648]) ).
thf(757,plain,
! [SV281: $i,SV44: $i,SV154: $i] :
( ( ( ~ ( element_of_set @ ( f11 @ SV154 @ SV44 ) @ SV281 )
| ~ ( element_of_collection @ SV281 @ SV154 )
| ~ ( subset_sets @ SV281 @ SV44 ) )
= $true )
| ( ( element_of_collection @ SV44 @ ( top_of_basis @ SV154 ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[650]) ).
thf(758,plain,
! [SV235: $i,SV155: $i,SV45: $i] :
( ( ( ! [SY755: $i] :
~ ( element_of_collection @ SV45 @ ( subspace_topology @ SV155 @ SV235 @ SY755 ) ) )
= $true )
| ( ( topological_space @ SV155 @ SV235 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[651]) ).
thf(759,plain,
! [SV282: $i,SV236: $i,SV156: $i,SV46: $i] :
( ( ~ ( element_of_collection @ SV46 @ ( subspace_topology @ SV156 @ SV236 @ SV282 ) )
| ( subset_sets @ SV282 @ SV156 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[652]) ).
thf(760,plain,
! [SV283: $i,SV237: $i,SV157: $i,SV47: $i] :
( ( ~ ( element_of_collection @ SV47 @ ( subspace_topology @ SV157 @ SV237 @ SV283 ) )
| ( element_of_collection @ ( f12 @ SV157 @ SV237 @ SV283 @ SV47 ) @ SV237 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[653]) ).
thf(761,plain,
! [SV284: $i,SV238: $i,SV158: $i,SV48: $i] :
( ( ~ ( element_of_collection @ SV48 @ ( subspace_topology @ SV158 @ SV238 @ SV284 ) )
| ( equal_sets @ SV48 @ ( intersection_of_sets @ SV284 @ ( f12 @ SV158 @ SV238 @ SV284 @ SV48 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[654]) ).
thf(762,plain,
! [SV285: $i,SV239: $i,SV159: $i,SV49: $i] :
( ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
| ~ ( topological_space @ SV159 @ SV239 )
| ~ ( subset_sets @ SV285 @ SV159 )
| ! [SY796: $i] :
( ~ ( element_of_collection @ SY796 @ SV239 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SY796 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[655]) ).
thf(763,plain,
! [SV286: $i,SV240: $i,SV160: $i,SV50: $i] :
( ( ~ ( element_of_set @ SV50 @ ( interior @ SV160 @ SV240 @ SV286 ) )
| ( topological_space @ SV240 @ SV286 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[656]) ).
thf(764,plain,
! [SV241: $i,SV161: $i,SV51: $i] :
( ( ( ! [SY762: $i] :
~ ( element_of_set @ SV51 @ ( interior @ SV161 @ SV241 @ SY762 ) ) )
= $true )
| ( ( subset_sets @ SV161 @ SV241 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[657]) ).
thf(765,plain,
! [SV287: $i,SV242: $i,SV162: $i,SV52: $i] :
( ( ~ ( element_of_set @ SV52 @ ( interior @ SV162 @ SV242 @ SV287 ) )
| ( element_of_set @ SV52 @ ( f13 @ SV162 @ SV242 @ SV287 @ SV52 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[658]) ).
thf(766,plain,
! [SV288: $i,SV243: $i,SV163: $i,SV53: $i] :
( ( ~ ( element_of_set @ SV53 @ ( interior @ SV163 @ SV243 @ SV288 ) )
| ( subset_sets @ ( f13 @ SV163 @ SV243 @ SV288 @ SV53 ) @ SV163 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[659]) ).
thf(767,plain,
! [SV289: $i,SV244: $i,SV164: $i,SV54: $i] :
( ( ~ ( element_of_set @ SV54 @ ( interior @ SV164 @ SV244 @ SV289 ) )
| ( open @ ( f13 @ SV164 @ SV244 @ SV289 @ SV54 ) @ SV244 @ SV289 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[660]) ).
thf(768,plain,
! [SV290: $i,SV245: $i,SV165: $i,SV55: $i] :
( ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
| ~ ( topological_space @ SV245 @ SV290 )
| ~ ( subset_sets @ SV165 @ SV245 )
| ! [SY797: $i] :
( ~ ( element_of_set @ SV55 @ SY797 )
| ~ ( subset_sets @ SY797 @ SV165 )
| ~ ( open @ SY797 @ SV245 @ SV290 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[661]) ).
thf(769,plain,
! [SV291: $i,SV246: $i,SV166: $i,SV56: $i] :
( ( ~ ( element_of_set @ SV56 @ ( closure @ SV166 @ SV246 @ SV291 ) )
| ( topological_space @ SV246 @ SV291 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[662]) ).
thf(770,plain,
! [SV247: $i,SV167: $i,SV57: $i] :
( ( ( ! [SY769: $i] :
~ ( element_of_set @ SV57 @ ( closure @ SV167 @ SV247 @ SY769 ) ) )
= $true )
| ( ( subset_sets @ SV167 @ SV247 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[663]) ).
thf(771,plain,
! [SV292: $i,SV248: $i,SV168: $i,SV58: $i] :
( ( ~ ( element_of_set @ SV58 @ ( closure @ SV168 @ SV248 @ SV292 ) )
| ! [SY798: $i] :
( ~ ( subset_sets @ SV168 @ SY798 )
| ~ ( closed @ SY798 @ SV248 @ SV292 )
| ( element_of_set @ SV58 @ SY798 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[664]) ).
thf(772,plain,
! [SV293: $i,SV249: $i,SV169: $i,SV59: $i] :
( ( ( element_of_set @ SV59 @ ( closure @ SV169 @ SV249 @ SV293 ) )
| ~ ( topological_space @ SV249 @ SV293 )
| ~ ( subset_sets @ SV169 @ SV249 )
| ( subset_sets @ SV169 @ ( f14 @ SV169 @ SV249 @ SV293 @ SV59 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[665]) ).
thf(773,plain,
! [SV294: $i,SV250: $i,SV170: $i,SV60: $i] :
( ( ( element_of_set @ SV60 @ ( closure @ SV170 @ SV250 @ SV294 ) )
| ~ ( topological_space @ SV250 @ SV294 )
| ~ ( subset_sets @ SV170 @ SV250 )
| ( closed @ ( f14 @ SV170 @ SV250 @ SV294 @ SV60 ) @ SV250 @ SV294 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[666]) ).
thf(774,plain,
! [SV295: $i,SV251: $i,SV171: $i,SV61: $i] :
( ( ( element_of_set @ SV61 @ ( closure @ SV171 @ SV251 @ SV295 ) )
| ~ ( topological_space @ SV251 @ SV295 )
| ~ ( subset_sets @ SV171 @ SV251 )
| ~ ( element_of_set @ SV61 @ ( f14 @ SV171 @ SV251 @ SV295 @ SV61 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[667]) ).
thf(775,plain,
! [SV296: $i,SV252: $i,SV172: $i,SV62: $i] :
( ( ~ ( neighborhood @ SV62 @ SV172 @ SV252 @ SV296 )
| ( topological_space @ SV252 @ SV296 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[668]) ).
thf(776,plain,
! [SV297: $i,SV253: $i,SV173: $i,SV63: $i] :
( ( ~ ( neighborhood @ SV63 @ SV173 @ SV253 @ SV297 )
| ( open @ SV63 @ SV253 @ SV297 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[669]) ).
thf(777,plain,
! [SV298: $i,SV174: $i,SV64: $i] :
( ( ( ! [SY799: $i] :
~ ( neighborhood @ SV64 @ SV174 @ SV298 @ SY799 ) )
= $true )
| ( ( element_of_set @ SV174 @ SV64 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[670]) ).
thf(778,plain,
! [SV299: $i,SV254: $i,SV175: $i,SV65: $i] :
( ( ( neighborhood @ SV65 @ SV175 @ SV254 @ SV299 )
| ~ ( topological_space @ SV254 @ SV299 )
| ~ ( open @ SV65 @ SV254 @ SV299 )
| ~ ( element_of_set @ SV175 @ SV65 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[671]) ).
thf(779,plain,
! [SV300: $i,SV255: $i,SV176: $i,SV66: $i] :
( ( ~ ( limit_point @ SV66 @ SV176 @ SV255 @ SV300 )
| ( topological_space @ SV255 @ SV300 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[672]) ).
thf(780,plain,
! [SV256: $i,SV177: $i,SV67: $i] :
( ( ( ! [SY779: $i] :
~ ( limit_point @ SV67 @ SV177 @ SV256 @ SY779 ) )
= $true )
| ( ( subset_sets @ SV177 @ SV256 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[673]) ).
thf(781,plain,
! [SV301: $i,SV257: $i,SV178: $i,SV68: $i] :
( ( ~ ( limit_point @ SV68 @ SV178 @ SV257 @ SV301 )
| ! [SY800: $i] :
( ~ ( neighborhood @ SY800 @ SV68 @ SV257 @ SV301 )
| ( element_of_set @ ( f15 @ SV68 @ SV178 @ SV257 @ SV301 @ SY800 ) @ ( intersection_of_sets @ SY800 @ SV178 ) ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[674]) ).
thf(782,plain,
! [SV302: $i,SV258: $i,SV179: $i,SV69: $i] :
( ( ~ ( limit_point @ SV69 @ SV179 @ SV258 @ SV302 )
| ! [SY801: $i] :
( ~ ( neighborhood @ SY801 @ SV69 @ SV258 @ SV302 )
| ~ ( eq_p @ ( f15 @ SV69 @ SV179 @ SV258 @ SV302 @ SY801 ) @ SV69 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[675]) ).
thf(783,plain,
! [SV303: $i,SV259: $i,SV180: $i,SV70: $i] :
( ( ( limit_point @ SV70 @ SV180 @ SV259 @ SV303 )
| ~ ( topological_space @ SV259 @ SV303 )
| ~ ( subset_sets @ SV180 @ SV259 )
| ( neighborhood @ ( f16 @ SV70 @ SV180 @ SV259 @ SV303 ) @ SV70 @ SV259 @ SV303 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[676]) ).
thf(784,plain,
! [SV304: $i,SV260: $i,SV181: $i,SV71: $i] :
( ( ( limit_point @ SV71 @ SV181 @ SV260 @ SV304 )
| ~ ( topological_space @ SV260 @ SV304 )
| ~ ( subset_sets @ SV181 @ SV260 )
| ! [SY802: $i] :
( ~ ( element_of_set @ SY802 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SV304 ) @ SV181 ) )
| ( eq_p @ SY802 @ SV71 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[677]) ).
thf(785,plain,
! [SV305: $i,SV261: $i,SV182: $i,SV72: $i] :
( ( ~ ( element_of_set @ SV72 @ ( boundary @ SV182 @ SV261 @ SV305 ) )
| ( topological_space @ SV261 @ SV305 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[678]) ).
thf(786,plain,
! [SV306: $i,SV262: $i,SV183: $i,SV73: $i] :
( ( ~ ( element_of_set @ SV73 @ ( boundary @ SV183 @ SV262 @ SV306 ) )
| ( element_of_set @ SV73 @ ( closure @ SV183 @ SV262 @ SV306 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[679]) ).
thf(787,plain,
! [SV307: $i,SV263: $i,SV184: $i,SV74: $i] :
( ( ~ ( element_of_set @ SV74 @ ( boundary @ SV184 @ SV263 @ SV307 ) )
| ( element_of_set @ SV74 @ ( closure @ ( relative_complement_sets @ SV184 @ SV263 ) @ SV263 @ SV307 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[680]) ).
thf(788,plain,
! [SV308: $i,SV264: $i,SV185: $i,SV75: $i] :
( ( ( element_of_set @ SV75 @ ( boundary @ SV185 @ SV264 @ SV308 ) )
| ~ ( topological_space @ SV264 @ SV308 )
| ~ ( element_of_set @ SV75 @ ( closure @ SV185 @ SV264 @ SV308 ) )
| ~ ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SV185 @ SV264 ) @ SV264 @ SV308 ) ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[681]) ).
thf(789,plain,
! [SV186: $i,SV76: $i] :
( ( ( hausdorff @ SV76 @ SV186 )
= $false )
| ( ( topological_space @ SV76 @ SV186 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[682]) ).
thf(790,plain,
! [SV187: $i,SV77: $i] :
( ( ( hausdorff @ SV77 @ SV187 )
= $false )
| ( ( ! [SY707: $i] :
( ~ ( element_of_set @ SY707 @ SV77 )
| ! [SY708: $i] :
( ~ ( element_of_set @ SY708 @ SV77 )
| ( eq_p @ SY707 @ SY708 )
| ( neighborhood @ ( f17 @ SV77 @ SV187 @ SY707 @ SY708 ) @ SY707 @ SV77 @ SV187 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[683]) ).
thf(791,plain,
! [SV188: $i,SV78: $i] :
( ( ( hausdorff @ SV78 @ SV188 )
= $false )
| ( ( ! [SY709: $i] :
( ~ ( element_of_set @ SY709 @ SV78 )
| ! [SY710: $i] :
( ~ ( element_of_set @ SY710 @ SV78 )
| ( eq_p @ SY709 @ SY710 )
| ( neighborhood @ ( f18 @ SV78 @ SV188 @ SY709 @ SY710 ) @ SY710 @ SV78 @ SV188 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[684]) ).
thf(792,plain,
! [SV189: $i,SV79: $i] :
( ( ( hausdorff @ SV79 @ SV189 )
= $false )
| ( ( ! [SY711: $i] :
( ~ ( element_of_set @ SY711 @ SV79 )
| ! [SY712: $i] :
( ~ ( element_of_set @ SY712 @ SV79 )
| ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SY711 @ SY712 ) @ ( f18 @ SV79 @ SV189 @ SY711 @ SY712 ) )
| ( eq_p @ SY711 @ SY712 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[685]) ).
thf(793,plain,
! [SV190: $i,SV80: $i] :
( ( ( ~ ( topological_space @ SV80 @ SV190 ) )
= $true )
| ( ( element_of_set @ ( f19 @ SV80 @ SV190 ) @ SV80 )
= $true )
| ( ( hausdorff @ SV80 @ SV190 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[686]) ).
thf(794,plain,
! [SV191: $i,SV81: $i] :
( ( ( ~ ( topological_space @ SV81 @ SV191 ) )
= $true )
| ( ( element_of_set @ ( f20 @ SV81 @ SV191 ) @ SV81 )
= $true )
| ( ( hausdorff @ SV81 @ SV191 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[687]) ).
thf(795,plain,
! [SV192: $i,SV82: $i] :
( ( ( ~ ( topological_space @ SV82 @ SV192 ) )
= $true )
| ( ( ~ ( eq_p @ ( f19 @ SV82 @ SV192 ) @ ( f20 @ SV82 @ SV192 ) ) )
= $true )
| ( ( hausdorff @ SV82 @ SV192 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[688]) ).
thf(796,plain,
! [SV193: $i,SV83: $i] :
( ( ( ~ ( topological_space @ SV83 @ SV193 ) )
= $true )
| ( ( ! [SY713: $i] :
( ~ ( neighborhood @ SY713 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ! [SY714: $i] :
( ~ ( neighborhood @ SY714 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ~ ( disjoint_s @ SY713 @ SY714 ) ) ) )
= $true )
| ( ( hausdorff @ SV83 @ SV193 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[689]) ).
thf(797,plain,
! [SV309: $i,SV265: $i,SV194: $i,SV84: $i] :
( ( ~ ( separation @ SV84 @ SV194 @ SV265 @ SV309 )
| ( topological_space @ SV265 @ SV309 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[690]) ).
thf(798,plain,
! [SV310: $i,SV195: $i,SV85: $i] :
( ( ( ! [SY803: $i] :
~ ( separation @ SV85 @ SV195 @ SV310 @ SY803 ) )
= $true )
| ( ( ~ ( equal_sets @ SV85 @ empty_set ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[691]) ).
thf(799,plain,
! [SV311: $i,SV196: $i,SV86: $i] :
( ( ( ! [SY804: $i] :
~ ( separation @ SV86 @ SV196 @ SV311 @ SY804 ) )
= $true )
| ( ( ~ ( equal_sets @ SV196 @ empty_set ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[692]) ).
thf(800,plain,
! [SV312: $i,SV266: $i,SV197: $i,SV87: $i] :
( ( ~ ( separation @ SV87 @ SV197 @ SV266 @ SV312 )
| ( element_of_collection @ SV87 @ SV312 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[693]) ).
thf(801,plain,
! [SV313: $i,SV267: $i,SV198: $i,SV88: $i] :
( ( ~ ( separation @ SV88 @ SV198 @ SV267 @ SV313 )
| ( element_of_collection @ SV198 @ SV313 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[694]) ).
thf(802,plain,
! [SV268: $i,SV199: $i,SV89: $i] :
( ( ( ! [SY794: $i] :
~ ( separation @ SV89 @ SV199 @ SV268 @ SY794 ) )
= $true )
| ( ( equal_sets @ ( union_of_sets @ SV89 @ SV199 ) @ SV268 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[695]) ).
thf(803,plain,
! [SV314: $i,SV200: $i,SV90: $i] :
( ( ( ! [SY805: $i] :
~ ( separation @ SV90 @ SV200 @ SV314 @ SY805 ) )
= $true )
| ( ( disjoint_s @ SV90 @ SV200 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[696]) ).
thf(804,plain,
! [SV315: $i,SV269: $i,SV201: $i,SV91: $i] :
( ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
| ~ ( topological_space @ SV269 @ SV315 )
| ( equal_sets @ SV91 @ empty_set )
| ( equal_sets @ SV201 @ empty_set )
| ~ ( element_of_collection @ SV91 @ SV315 )
| ~ ( element_of_collection @ SV201 @ SV315 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true ),
inference(extcnf_forall_pos,[status(thm)],[697]) ).
thf(805,plain,
! [SV202: $i,SV92: $i] :
( ( ( connected_space @ SV92 @ SV202 )
= $false )
| ( ( topological_space @ SV92 @ SV202 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[698]) ).
thf(806,plain,
! [SV203: $i,SV93: $i] :
( ( ( connected_space @ SV93 @ SV203 )
= $false )
| ( ( ! [SY731: $i,SY732: $i] :
~ ( separation @ SY731 @ SY732 @ SV93 @ SV203 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[699]) ).
thf(807,plain,
! [SV204: $i,SV94: $i] :
( ( ( ~ ( topological_space @ SV94 @ SV204 ) )
= $true )
| ( ( separation @ ( f21 @ SV94 @ SV204 ) @ ( f22 @ SV94 @ SV204 ) @ SV94 @ SV204 )
= $true )
| ( ( connected_space @ SV94 @ SV204 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[700]) ).
thf(808,plain,
! [SV270: $i,SV205: $i,SV95: $i] :
( ( ( ~ ( connected_set @ SV95 @ SV205 @ SV270 ) )
= $true )
| ( ( topological_space @ SV205 @ SV270 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[701]) ).
thf(809,plain,
! [SV316: $i,SV206: $i,SV96: $i] :
( ( ( ~ ( connected_set @ SV96 @ SV206 @ SV316 ) )
= $true )
| ( ( subset_sets @ SV96 @ SV206 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[702]) ).
thf(810,plain,
! [SV271: $i,SV207: $i,SV97: $i] :
( ( ( ~ ( connected_set @ SV97 @ SV207 @ SV271 ) )
= $true )
| ( ( connected_space @ SV97 @ ( subspace_topology @ SV207 @ SV271 @ SV97 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[703]) ).
thf(811,plain,
! [SV272: $i,SV208: $i,SV98: $i] :
( ( ( connected_set @ SV98 @ SV208 @ SV272 )
= $true )
| ( ( ~ ( topological_space @ SV208 @ SV272 )
| ~ ( subset_sets @ SV98 @ SV208 )
| ~ ( connected_space @ SV98 @ ( subspace_topology @ SV208 @ SV272 @ SV98 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[704]) ).
thf(812,plain,
! [SV273: $i,SV209: $i,SV99: $i] :
( ( ( ~ ( open_covering @ SV99 @ SV209 @ SV273 ) )
= $true )
| ( ( topological_space @ SV209 @ SV273 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[705]) ).
thf(813,plain,
! [SV274: $i,SV210: $i,SV100: $i] :
( ( ( ~ ( open_covering @ SV100 @ SV210 @ SV274 ) )
= $true )
| ( ( subset_collections @ SV100 @ SV274 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[706]) ).
thf(814,plain,
! [SV317: $i,SV211: $i,SV101: $i] :
( ( ( ~ ( open_covering @ SV101 @ SV211 @ SV317 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV101 ) @ SV211 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[707]) ).
thf(815,plain,
! [SV275: $i,SV212: $i,SV102: $i] :
( ( ( open_covering @ SV102 @ SV212 @ SV275 )
= $true )
| ( ( ~ ( topological_space @ SV212 @ SV275 )
| ~ ( subset_collections @ SV102 @ SV275 )
| ~ ( equal_sets @ ( union_of_members @ SV102 ) @ SV212 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[708]) ).
thf(816,plain,
! [SV213: $i,SV103: $i] :
( ( ( compact_space @ SV103 @ SV213 )
= $false )
| ( ( topological_space @ SV103 @ SV213 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[709]) ).
thf(817,plain,
! [SV214: $i,SV104: $i] :
( ( ( compact_space @ SV104 @ SV214 )
= $false )
| ( ( ! [SY741: $i] :
( ~ ( open_covering @ SY741 @ SV104 @ SV214 )
| ( finite @ ( f23 @ SV104 @ SV214 @ SY741 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[710]) ).
thf(818,plain,
! [SV215: $i,SV105: $i] :
( ( ( compact_space @ SV105 @ SV215 )
= $false )
| ( ( ! [SY742: $i] :
( ~ ( open_covering @ SY742 @ SV105 @ SV215 )
| ( subset_collections @ ( f23 @ SV105 @ SV215 @ SY742 ) @ SY742 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[711]) ).
thf(819,plain,
! [SV216: $i,SV106: $i] :
( ( ( compact_space @ SV106 @ SV216 )
= $false )
| ( ( ! [SY743: $i] :
( ~ ( open_covering @ SY743 @ SV106 @ SV216 )
| ( open_covering @ ( f23 @ SV106 @ SV216 @ SY743 ) @ SV106 @ SV216 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[712]) ).
thf(820,plain,
! [SV217: $i,SV107: $i] :
( ( ( ~ ( topological_space @ SV107 @ SV217 ) )
= $true )
| ( ( open_covering @ ( f24 @ SV107 @ SV217 ) @ SV107 @ SV217 )
= $true )
| ( ( compact_space @ SV107 @ SV217 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[713]) ).
thf(821,plain,
! [SV218: $i,SV108: $i] :
( ( ( ~ ( topological_space @ SV108 @ SV218 ) )
= $true )
| ( ( ! [SY744: $i] :
( ~ ( finite @ SY744 )
| ~ ( subset_collections @ SY744 @ ( f24 @ SV108 @ SV218 ) )
| ~ ( open_covering @ SY744 @ SV108 @ SV218 ) ) )
= $true )
| ( ( compact_space @ SV108 @ SV218 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[714]) ).
thf(822,plain,
! [SV276: $i,SV219: $i,SV109: $i] :
( ( ( ~ ( compact_set @ SV109 @ SV219 @ SV276 ) )
= $true )
| ( ( topological_space @ SV219 @ SV276 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[715]) ).
thf(823,plain,
! [SV318: $i,SV220: $i,SV110: $i] :
( ( ( ~ ( compact_set @ SV110 @ SV220 @ SV318 ) )
= $true )
| ( ( subset_sets @ SV110 @ SV220 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[716]) ).
thf(824,plain,
! [SV277: $i,SV221: $i,SV111: $i] :
( ( ( ~ ( compact_set @ SV111 @ SV221 @ SV277 ) )
= $true )
| ( ( compact_space @ SV111 @ ( subspace_topology @ SV221 @ SV277 @ SV111 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[717]) ).
thf(825,plain,
! [SV278: $i,SV222: $i,SV112: $i] :
( ( ( compact_set @ SV112 @ SV222 @ SV278 )
= $true )
| ( ( ~ ( topological_space @ SV222 @ SV278 )
| ~ ( subset_sets @ SV112 @ SV222 )
| ~ ( compact_space @ SV112 @ ( subspace_topology @ SV222 @ SV278 @ SV112 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[718]) ).
thf(826,plain,
! [SV116: $i,SV279: $i,SV6: $i] :
( ( ( ~ ( element_of_set @ SV6 @ SV279 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV279 @ SV116 ) )
= $true )
| ( ( element_of_set @ SV6 @ ( union_of_members @ SV116 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[721]) ).
thf(827,plain,
! [SV7: $i,SV117: $i,SV319: $i] :
( ( ( ~ ( element_of_collection @ SV319 @ SV117 )
| ( element_of_set @ SV7 @ SV319 ) )
= $true )
| ( ( element_of_set @ SV7 @ ( intersection_of_members @ SV117 ) )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[722]) ).
thf(828,plain,
! [SV13: $i,SV123: $i,SV320: $i] :
( ( ( ~ ( element_of_collection @ SV320 @ SV123 )
| ! [SY806: $i] :
( ~ ( element_of_collection @ SY806 @ SV123 )
| ( element_of_collection @ ( intersection_of_sets @ SV320 @ SY806 ) @ SV123 ) ) )
= $true )
| ( ( topological_space @ SV13 @ SV123 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[727]) ).
thf(829,plain,
! [SV14: $i,SV124: $i,SV321: $i] :
( ( ( ~ ( subset_collections @ SV321 @ SV124 )
| ( element_of_collection @ ( union_of_members @ SV321 ) @ SV124 ) )
= $true )
| ( ( topological_space @ SV14 @ SV124 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[728]) ).
thf(830,plain,
! [SV15: $i,SV125: $i] :
( ( ( equal_sets @ ( union_of_members @ SV125 ) @ SV15 )
= $false )
| ( ( ~ ( element_of_collection @ empty_set @ SV125 )
| ~ ( element_of_collection @ SV15 @ SV125 )
| ( element_of_collection @ ( f3 @ SV15 @ SV125 ) @ SV125 )
| ( subset_collections @ ( f5 @ SV15 @ SV125 ) @ SV125 ) )
= $true )
| ( ( topological_space @ SV15 @ SV125 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[729]) ).
thf(831,plain,
! [SV16: $i,SV126: $i] :
( ( ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 )
= $false )
| ( ( ~ ( element_of_collection @ empty_set @ SV126 )
| ~ ( element_of_collection @ SV16 @ SV126 )
| ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 ) )
= $true )
| ( ( topological_space @ SV16 @ SV126 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[730]) ).
thf(832,plain,
! [SV17: $i,SV127: $i] :
( ( ( equal_sets @ ( union_of_members @ SV127 ) @ SV17 )
= $false )
| ( ( ~ ( element_of_collection @ empty_set @ SV127 )
| ~ ( element_of_collection @ SV17 @ SV127 )
| ( element_of_collection @ ( f4 @ SV17 @ SV127 ) @ SV127 )
| ( subset_collections @ ( f5 @ SV17 @ SV127 ) @ SV127 ) )
= $true )
| ( ( topological_space @ SV17 @ SV127 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[731]) ).
thf(833,plain,
! [SV18: $i,SV128: $i] :
( ( ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 )
= $false )
| ( ( ~ ( element_of_collection @ empty_set @ SV128 )
| ~ ( element_of_collection @ SV18 @ SV128 )
| ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 ) )
= $true )
| ( ( topological_space @ SV18 @ SV128 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[732]) ).
thf(834,plain,
! [SV19: $i,SV129: $i] :
( ( ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 )
= $false )
| ( ( ~ ( element_of_collection @ empty_set @ SV129 )
| ~ ( element_of_collection @ SV19 @ SV129 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 )
| ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 ) )
= $true )
| ( ( topological_space @ SV19 @ SV129 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[733]) ).
thf(835,plain,
! [SV20: $i,SV130: $i] :
( ( ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
= $false )
| ( ( ~ ( element_of_collection @ empty_set @ SV130 )
| ~ ( element_of_collection @ SV20 @ SV130 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true )
| ( ( topological_space @ SV20 @ SV130 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[734]) ).
thf(836,plain,
! [SV223: $i,SV131: $i,SV21: $i] :
( ( ( open @ SV21 @ SV131 @ SV223 )
= $false )
| ( ( topological_space @ SV131 @ SV223 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[735]) ).
thf(837,plain,
! [SV224: $i,SV132: $i,SV22: $i] :
( ( ( open @ SV22 @ SV132 @ SV224 )
= $false )
| ( ( element_of_collection @ SV22 @ SV224 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[736]) ).
thf(838,plain,
! [SV23: $i,SV225: $i,SV133: $i] :
( ( ( ~ ( topological_space @ SV133 @ SV225 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV23 @ SV225 ) )
= $true )
| ( ( open @ SV23 @ SV133 @ SV225 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[737]) ).
thf(839,plain,
! [SV226: $i,SV134: $i,SV24: $i] :
( ( ( closed @ SV24 @ SV134 @ SV226 )
= $false )
| ( ( topological_space @ SV134 @ SV226 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[738]) ).
thf(840,plain,
! [SV227: $i,SV135: $i,SV25: $i] :
( ( ( closed @ SV25 @ SV135 @ SV227 )
= $false )
| ( ( open @ ( relative_complement_sets @ SV25 @ SV135 ) @ SV135 @ SV227 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[739]) ).
thf(841,plain,
! [SV26: $i,SV228: $i,SV136: $i] :
( ( ( ~ ( topological_space @ SV136 @ SV228 ) )
= $true )
| ( ( ~ ( open @ ( relative_complement_sets @ SV26 @ SV136 ) @ SV136 @ SV228 ) )
= $true )
| ( ( closed @ SV26 @ SV136 @ SV228 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[740]) ).
thf(842,plain,
! [SV229: $i,SV137: $i,SV27: $i] :
( ( ( finer @ SV27 @ SV137 @ SV229 )
= $false )
| ( ( topological_space @ SV229 @ SV27 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[741]) ).
thf(843,plain,
! [SV230: $i,SV138: $i,SV28: $i] :
( ( ( finer @ SV28 @ SV138 @ SV230 )
= $false )
| ( ( topological_space @ SV230 @ SV138 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[742]) ).
thf(844,plain,
! [SV280: $i,SV139: $i,SV29: $i] :
( ( ( finer @ SV29 @ SV139 @ SV280 )
= $false )
| ( ( subset_collections @ SV139 @ SV29 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[743]) ).
thf(845,plain,
! [SV140: $i,SV30: $i,SV231: $i] :
( ( ( ~ ( topological_space @ SV231 @ SV30 ) )
= $true )
| ( ( ~ ( topological_space @ SV231 @ SV140 )
| ~ ( subset_collections @ SV140 @ SV30 ) )
= $true )
| ( ( finer @ SV30 @ SV140 @ SV231 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[744]) ).
thf(846,plain,
! [SV232: $i,SV142: $i,SV32: $i] :
( ( ( basis @ SV32 @ SV142 )
= $false )
| ( ( ~ ( element_of_set @ SV232 @ SV32 )
| ! [SY749: $i] :
( ~ ( element_of_collection @ SY749 @ SV142 )
| ! [SY750: $i] :
( ~ ( element_of_collection @ SY750 @ SV142 )
| ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SY749 @ SY750 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SY749 @ SY750 ) ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[746]) ).
thf(847,plain,
! [SV233: $i,SV143: $i,SV33: $i] :
( ( ( basis @ SV33 @ SV143 )
= $false )
| ( ( ~ ( element_of_set @ SV233 @ SV33 )
| ! [SY751: $i] :
( ~ ( element_of_collection @ SY751 @ SV143 )
| ! [SY752: $i] :
( ~ ( element_of_collection @ SY752 @ SV143 )
| ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SY751 @ SY752 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SY751 @ SY752 ) @ SV143 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[747]) ).
thf(848,plain,
! [SV234: $i,SV144: $i,SV34: $i] :
( ( ( basis @ SV34 @ SV144 )
= $false )
| ( ( ~ ( element_of_set @ SV234 @ SV34 )
| ! [SY753: $i] :
( ~ ( element_of_collection @ SY753 @ SV144 )
| ! [SY754: $i] :
( ~ ( element_of_collection @ SY754 @ SV144 )
| ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SY753 @ SY754 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SY753 @ SY754 ) @ ( intersection_of_sets @ SY753 @ SY754 ) ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[748]) ).
thf(849,plain,
! [SV35: $i,SV145: $i] :
( ( ( equal_sets @ ( union_of_members @ SV145 ) @ SV35 )
= $false )
| ( ( element_of_set @ ( f7 @ SV35 @ SV145 ) @ SV35 )
= $true )
| ( ( basis @ SV35 @ SV145 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[749]) ).
thf(850,plain,
! [SV36: $i,SV146: $i] :
( ( ( equal_sets @ ( union_of_members @ SV146 ) @ SV36 )
= $false )
| ( ( element_of_collection @ ( f8 @ SV36 @ SV146 ) @ SV146 )
= $true )
| ( ( basis @ SV36 @ SV146 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[750]) ).
thf(851,plain,
! [SV37: $i,SV147: $i] :
( ( ( equal_sets @ ( union_of_members @ SV147 ) @ SV37 )
= $false )
| ( ( element_of_collection @ ( f9 @ SV37 @ SV147 ) @ SV147 )
= $true )
| ( ( basis @ SV37 @ SV147 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[751]) ).
thf(852,plain,
! [SV38: $i,SV148: $i] :
( ( ( equal_sets @ ( union_of_members @ SV148 ) @ SV38 )
= $false )
| ( ( element_of_set @ ( f7 @ SV38 @ SV148 ) @ ( intersection_of_sets @ ( f8 @ SV38 @ SV148 ) @ ( f9 @ SV38 @ SV148 ) ) )
= $true )
| ( ( basis @ SV38 @ SV148 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[752]) ).
thf(853,plain,
! [SV39: $i,SV149: $i] :
( ( ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 )
= $false )
| ( ( ! [SY634: $i] :
( ~ ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SY634 )
| ~ ( element_of_collection @ SY634 @ SV149 )
| ~ ( subset_sets @ SY634 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) ) ) )
= $true )
| ( ( basis @ SV39 @ SV149 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[753]) ).
thf(854,plain,
! [SV150: $i,SV40: $i,SV322: $i] :
( ( ( ~ ( element_of_set @ SV322 @ SV40 )
| ( element_of_set @ SV322 @ ( f10 @ SV150 @ SV40 @ SV322 ) ) )
= $true )
| ( ( element_of_collection @ SV40 @ ( top_of_basis @ SV150 ) )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[754]) ).
thf(855,plain,
! [SV151: $i,SV41: $i,SV323: $i] :
( ( ( ~ ( element_of_set @ SV323 @ SV41 )
| ( element_of_collection @ ( f10 @ SV151 @ SV41 @ SV323 ) @ SV151 ) )
= $true )
| ( ( element_of_collection @ SV41 @ ( top_of_basis @ SV151 ) )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[755]) ).
thf(856,plain,
! [SV152: $i,SV42: $i,SV324: $i] :
( ( ( ~ ( element_of_set @ SV324 @ SV42 )
| ( subset_sets @ ( f10 @ SV152 @ SV42 @ SV324 ) @ SV42 ) )
= $true )
| ( ( element_of_collection @ SV42 @ ( top_of_basis @ SV152 ) )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[756]) ).
thf(857,plain,
! [SV281: $i,SV44: $i,SV154: $i] :
( ( ( ~ ( element_of_set @ ( f11 @ SV154 @ SV44 ) @ SV281 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV281 @ SV154 )
| ~ ( subset_sets @ SV281 @ SV44 ) )
= $true )
| ( ( element_of_collection @ SV44 @ ( top_of_basis @ SV154 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[757]) ).
thf(858,plain,
! [SV325: $i,SV235: $i,SV155: $i,SV45: $i] :
( ( ( ~ ( element_of_collection @ SV45 @ ( subspace_topology @ SV155 @ SV235 @ SV325 ) ) )
= $true )
| ( ( topological_space @ SV155 @ SV235 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[758]) ).
thf(859,plain,
! [SV282: $i,SV236: $i,SV156: $i,SV46: $i] :
( ( ( ~ ( element_of_collection @ SV46 @ ( subspace_topology @ SV156 @ SV236 @ SV282 ) ) )
= $true )
| ( ( subset_sets @ SV282 @ SV156 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[759]) ).
thf(860,plain,
! [SV283: $i,SV237: $i,SV157: $i,SV47: $i] :
( ( ( ~ ( element_of_collection @ SV47 @ ( subspace_topology @ SV157 @ SV237 @ SV283 ) ) )
= $true )
| ( ( element_of_collection @ ( f12 @ SV157 @ SV237 @ SV283 @ SV47 ) @ SV237 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[760]) ).
thf(861,plain,
! [SV284: $i,SV238: $i,SV158: $i,SV48: $i] :
( ( ( ~ ( element_of_collection @ SV48 @ ( subspace_topology @ SV158 @ SV238 @ SV284 ) ) )
= $true )
| ( ( equal_sets @ SV48 @ ( intersection_of_sets @ SV284 @ ( f12 @ SV158 @ SV238 @ SV284 @ SV48 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[761]) ).
thf(862,plain,
! [SV285: $i,SV239: $i,SV159: $i,SV49: $i] :
( ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
= $true )
| ( ( ~ ( topological_space @ SV159 @ SV239 )
| ~ ( subset_sets @ SV285 @ SV159 )
| ! [SY796: $i] :
( ~ ( element_of_collection @ SY796 @ SV239 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SY796 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[762]) ).
thf(863,plain,
! [SV286: $i,SV240: $i,SV160: $i,SV50: $i] :
( ( ( ~ ( element_of_set @ SV50 @ ( interior @ SV160 @ SV240 @ SV286 ) ) )
= $true )
| ( ( topological_space @ SV240 @ SV286 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[763]) ).
thf(864,plain,
! [SV326: $i,SV241: $i,SV161: $i,SV51: $i] :
( ( ( ~ ( element_of_set @ SV51 @ ( interior @ SV161 @ SV241 @ SV326 ) ) )
= $true )
| ( ( subset_sets @ SV161 @ SV241 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[764]) ).
thf(865,plain,
! [SV287: $i,SV242: $i,SV162: $i,SV52: $i] :
( ( ( ~ ( element_of_set @ SV52 @ ( interior @ SV162 @ SV242 @ SV287 ) ) )
= $true )
| ( ( element_of_set @ SV52 @ ( f13 @ SV162 @ SV242 @ SV287 @ SV52 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[765]) ).
thf(866,plain,
! [SV288: $i,SV243: $i,SV163: $i,SV53: $i] :
( ( ( ~ ( element_of_set @ SV53 @ ( interior @ SV163 @ SV243 @ SV288 ) ) )
= $true )
| ( ( subset_sets @ ( f13 @ SV163 @ SV243 @ SV288 @ SV53 ) @ SV163 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[766]) ).
thf(867,plain,
! [SV289: $i,SV244: $i,SV164: $i,SV54: $i] :
( ( ( ~ ( element_of_set @ SV54 @ ( interior @ SV164 @ SV244 @ SV289 ) ) )
= $true )
| ( ( open @ ( f13 @ SV164 @ SV244 @ SV289 @ SV54 ) @ SV244 @ SV289 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[767]) ).
thf(868,plain,
! [SV290: $i,SV245: $i,SV165: $i,SV55: $i] :
( ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true )
| ( ( ~ ( topological_space @ SV245 @ SV290 )
| ~ ( subset_sets @ SV165 @ SV245 )
| ! [SY797: $i] :
( ~ ( element_of_set @ SV55 @ SY797 )
| ~ ( subset_sets @ SY797 @ SV165 )
| ~ ( open @ SY797 @ SV245 @ SV290 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[768]) ).
thf(869,plain,
! [SV291: $i,SV246: $i,SV166: $i,SV56: $i] :
( ( ( ~ ( element_of_set @ SV56 @ ( closure @ SV166 @ SV246 @ SV291 ) ) )
= $true )
| ( ( topological_space @ SV246 @ SV291 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[769]) ).
thf(870,plain,
! [SV327: $i,SV247: $i,SV167: $i,SV57: $i] :
( ( ( ~ ( element_of_set @ SV57 @ ( closure @ SV167 @ SV247 @ SV327 ) ) )
= $true )
| ( ( subset_sets @ SV167 @ SV247 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[770]) ).
thf(871,plain,
! [SV292: $i,SV248: $i,SV168: $i,SV58: $i] :
( ( ( ~ ( element_of_set @ SV58 @ ( closure @ SV168 @ SV248 @ SV292 ) ) )
= $true )
| ( ( ! [SY798: $i] :
( ~ ( subset_sets @ SV168 @ SY798 )
| ~ ( closed @ SY798 @ SV248 @ SV292 )
| ( element_of_set @ SV58 @ SY798 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[771]) ).
thf(872,plain,
! [SV293: $i,SV249: $i,SV169: $i,SV59: $i] :
( ( ( element_of_set @ SV59 @ ( closure @ SV169 @ SV249 @ SV293 ) )
= $true )
| ( ( ~ ( topological_space @ SV249 @ SV293 )
| ~ ( subset_sets @ SV169 @ SV249 )
| ( subset_sets @ SV169 @ ( f14 @ SV169 @ SV249 @ SV293 @ SV59 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[772]) ).
thf(873,plain,
! [SV294: $i,SV250: $i,SV170: $i,SV60: $i] :
( ( ( element_of_set @ SV60 @ ( closure @ SV170 @ SV250 @ SV294 ) )
= $true )
| ( ( ~ ( topological_space @ SV250 @ SV294 )
| ~ ( subset_sets @ SV170 @ SV250 )
| ( closed @ ( f14 @ SV170 @ SV250 @ SV294 @ SV60 ) @ SV250 @ SV294 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[773]) ).
thf(874,plain,
! [SV295: $i,SV251: $i,SV171: $i,SV61: $i] :
( ( ( element_of_set @ SV61 @ ( closure @ SV171 @ SV251 @ SV295 ) )
= $true )
| ( ( ~ ( topological_space @ SV251 @ SV295 )
| ~ ( subset_sets @ SV171 @ SV251 )
| ~ ( element_of_set @ SV61 @ ( f14 @ SV171 @ SV251 @ SV295 @ SV61 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[774]) ).
thf(875,plain,
! [SV296: $i,SV252: $i,SV172: $i,SV62: $i] :
( ( ( ~ ( neighborhood @ SV62 @ SV172 @ SV252 @ SV296 ) )
= $true )
| ( ( topological_space @ SV252 @ SV296 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[775]) ).
thf(876,plain,
! [SV297: $i,SV253: $i,SV173: $i,SV63: $i] :
( ( ( ~ ( neighborhood @ SV63 @ SV173 @ SV253 @ SV297 ) )
= $true )
| ( ( open @ SV63 @ SV253 @ SV297 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[776]) ).
thf(877,plain,
! [SV328: $i,SV298: $i,SV174: $i,SV64: $i] :
( ( ( ~ ( neighborhood @ SV64 @ SV174 @ SV298 @ SV328 ) )
= $true )
| ( ( element_of_set @ SV174 @ SV64 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[777]) ).
thf(878,plain,
! [SV299: $i,SV254: $i,SV175: $i,SV65: $i] :
( ( ( neighborhood @ SV65 @ SV175 @ SV254 @ SV299 )
= $true )
| ( ( ~ ( topological_space @ SV254 @ SV299 )
| ~ ( open @ SV65 @ SV254 @ SV299 )
| ~ ( element_of_set @ SV175 @ SV65 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[778]) ).
thf(879,plain,
! [SV300: $i,SV255: $i,SV176: $i,SV66: $i] :
( ( ( ~ ( limit_point @ SV66 @ SV176 @ SV255 @ SV300 ) )
= $true )
| ( ( topological_space @ SV255 @ SV300 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[779]) ).
thf(880,plain,
! [SV329: $i,SV256: $i,SV177: $i,SV67: $i] :
( ( ( ~ ( limit_point @ SV67 @ SV177 @ SV256 @ SV329 ) )
= $true )
| ( ( subset_sets @ SV177 @ SV256 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[780]) ).
thf(881,plain,
! [SV301: $i,SV257: $i,SV178: $i,SV68: $i] :
( ( ( ~ ( limit_point @ SV68 @ SV178 @ SV257 @ SV301 ) )
= $true )
| ( ( ! [SY800: $i] :
( ~ ( neighborhood @ SY800 @ SV68 @ SV257 @ SV301 )
| ( element_of_set @ ( f15 @ SV68 @ SV178 @ SV257 @ SV301 @ SY800 ) @ ( intersection_of_sets @ SY800 @ SV178 ) ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[781]) ).
thf(882,plain,
! [SV302: $i,SV258: $i,SV179: $i,SV69: $i] :
( ( ( ~ ( limit_point @ SV69 @ SV179 @ SV258 @ SV302 ) )
= $true )
| ( ( ! [SY801: $i] :
( ~ ( neighborhood @ SY801 @ SV69 @ SV258 @ SV302 )
| ~ ( eq_p @ ( f15 @ SV69 @ SV179 @ SV258 @ SV302 @ SY801 ) @ SV69 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[782]) ).
thf(883,plain,
! [SV303: $i,SV259: $i,SV180: $i,SV70: $i] :
( ( ( limit_point @ SV70 @ SV180 @ SV259 @ SV303 )
= $true )
| ( ( ~ ( topological_space @ SV259 @ SV303 )
| ~ ( subset_sets @ SV180 @ SV259 )
| ( neighborhood @ ( f16 @ SV70 @ SV180 @ SV259 @ SV303 ) @ SV70 @ SV259 @ SV303 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[783]) ).
thf(884,plain,
! [SV304: $i,SV260: $i,SV181: $i,SV71: $i] :
( ( ( limit_point @ SV71 @ SV181 @ SV260 @ SV304 )
= $true )
| ( ( ~ ( topological_space @ SV260 @ SV304 )
| ~ ( subset_sets @ SV181 @ SV260 )
| ! [SY802: $i] :
( ~ ( element_of_set @ SY802 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SV304 ) @ SV181 ) )
| ( eq_p @ SY802 @ SV71 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[784]) ).
thf(885,plain,
! [SV305: $i,SV261: $i,SV182: $i,SV72: $i] :
( ( ( ~ ( element_of_set @ SV72 @ ( boundary @ SV182 @ SV261 @ SV305 ) ) )
= $true )
| ( ( topological_space @ SV261 @ SV305 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[785]) ).
thf(886,plain,
! [SV306: $i,SV262: $i,SV183: $i,SV73: $i] :
( ( ( ~ ( element_of_set @ SV73 @ ( boundary @ SV183 @ SV262 @ SV306 ) ) )
= $true )
| ( ( element_of_set @ SV73 @ ( closure @ SV183 @ SV262 @ SV306 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[786]) ).
thf(887,plain,
! [SV307: $i,SV263: $i,SV184: $i,SV74: $i] :
( ( ( ~ ( element_of_set @ SV74 @ ( boundary @ SV184 @ SV263 @ SV307 ) ) )
= $true )
| ( ( element_of_set @ SV74 @ ( closure @ ( relative_complement_sets @ SV184 @ SV263 ) @ SV263 @ SV307 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[787]) ).
thf(888,plain,
! [SV308: $i,SV264: $i,SV185: $i,SV75: $i] :
( ( ( element_of_set @ SV75 @ ( boundary @ SV185 @ SV264 @ SV308 ) )
= $true )
| ( ( ~ ( topological_space @ SV264 @ SV308 )
| ~ ( element_of_set @ SV75 @ ( closure @ SV185 @ SV264 @ SV308 ) )
| ~ ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SV185 @ SV264 ) @ SV264 @ SV308 ) ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[788]) ).
thf(889,plain,
! [SV187: $i,SV77: $i,SV330: $i] :
( ( ( ~ ( element_of_set @ SV330 @ SV77 )
| ! [SY807: $i] :
( ~ ( element_of_set @ SY807 @ SV77 )
| ( eq_p @ SV330 @ SY807 )
| ( neighborhood @ ( f17 @ SV77 @ SV187 @ SV330 @ SY807 ) @ SV330 @ SV77 @ SV187 ) ) )
= $true )
| ( ( hausdorff @ SV77 @ SV187 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[790]) ).
thf(890,plain,
! [SV188: $i,SV78: $i,SV331: $i] :
( ( ( ~ ( element_of_set @ SV331 @ SV78 )
| ! [SY808: $i] :
( ~ ( element_of_set @ SY808 @ SV78 )
| ( eq_p @ SV331 @ SY808 )
| ( neighborhood @ ( f18 @ SV78 @ SV188 @ SV331 @ SY808 ) @ SY808 @ SV78 @ SV188 ) ) )
= $true )
| ( ( hausdorff @ SV78 @ SV188 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[791]) ).
thf(891,plain,
! [SV189: $i,SV79: $i,SV332: $i] :
( ( ( ~ ( element_of_set @ SV332 @ SV79 )
| ! [SY809: $i] :
( ~ ( element_of_set @ SY809 @ SV79 )
| ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SV332 @ SY809 ) @ ( f18 @ SV79 @ SV189 @ SV332 @ SY809 ) )
| ( eq_p @ SV332 @ SY809 ) ) )
= $true )
| ( ( hausdorff @ SV79 @ SV189 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[792]) ).
thf(892,plain,
! [SV190: $i,SV80: $i] :
( ( ( topological_space @ SV80 @ SV190 )
= $false )
| ( ( element_of_set @ ( f19 @ SV80 @ SV190 ) @ SV80 )
= $true )
| ( ( hausdorff @ SV80 @ SV190 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[793]) ).
thf(893,plain,
! [SV191: $i,SV81: $i] :
( ( ( topological_space @ SV81 @ SV191 )
= $false )
| ( ( element_of_set @ ( f20 @ SV81 @ SV191 ) @ SV81 )
= $true )
| ( ( hausdorff @ SV81 @ SV191 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[794]) ).
thf(894,plain,
! [SV192: $i,SV82: $i] :
( ( ( topological_space @ SV82 @ SV192 )
= $false )
| ( ( ~ ( eq_p @ ( f19 @ SV82 @ SV192 ) @ ( f20 @ SV82 @ SV192 ) ) )
= $true )
| ( ( hausdorff @ SV82 @ SV192 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[795]) ).
thf(895,plain,
! [SV193: $i,SV83: $i] :
( ( ( topological_space @ SV83 @ SV193 )
= $false )
| ( ( ! [SY713: $i] :
( ~ ( neighborhood @ SY713 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ! [SY714: $i] :
( ~ ( neighborhood @ SY714 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ~ ( disjoint_s @ SY713 @ SY714 ) ) ) )
= $true )
| ( ( hausdorff @ SV83 @ SV193 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[796]) ).
thf(896,plain,
! [SV309: $i,SV265: $i,SV194: $i,SV84: $i] :
( ( ( ~ ( separation @ SV84 @ SV194 @ SV265 @ SV309 ) )
= $true )
| ( ( topological_space @ SV265 @ SV309 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[797]) ).
thf(897,plain,
! [SV333: $i,SV310: $i,SV195: $i,SV85: $i] :
( ( ( ~ ( separation @ SV85 @ SV195 @ SV310 @ SV333 ) )
= $true )
| ( ( ~ ( equal_sets @ SV85 @ empty_set ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[798]) ).
thf(898,plain,
! [SV334: $i,SV311: $i,SV196: $i,SV86: $i] :
( ( ( ~ ( separation @ SV86 @ SV196 @ SV311 @ SV334 ) )
= $true )
| ( ( ~ ( equal_sets @ SV196 @ empty_set ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[799]) ).
thf(899,plain,
! [SV312: $i,SV266: $i,SV197: $i,SV87: $i] :
( ( ( ~ ( separation @ SV87 @ SV197 @ SV266 @ SV312 ) )
= $true )
| ( ( element_of_collection @ SV87 @ SV312 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[800]) ).
thf(900,plain,
! [SV313: $i,SV267: $i,SV198: $i,SV88: $i] :
( ( ( ~ ( separation @ SV88 @ SV198 @ SV267 @ SV313 ) )
= $true )
| ( ( element_of_collection @ SV198 @ SV313 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[801]) ).
thf(901,plain,
! [SV335: $i,SV268: $i,SV199: $i,SV89: $i] :
( ( ( ~ ( separation @ SV89 @ SV199 @ SV268 @ SV335 ) )
= $true )
| ( ( equal_sets @ ( union_of_sets @ SV89 @ SV199 ) @ SV268 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[802]) ).
thf(902,plain,
! [SV336: $i,SV314: $i,SV200: $i,SV90: $i] :
( ( ( ~ ( separation @ SV90 @ SV200 @ SV314 @ SV336 ) )
= $true )
| ( ( disjoint_s @ SV90 @ SV200 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[803]) ).
thf(903,plain,
! [SV315: $i,SV269: $i,SV201: $i,SV91: $i] :
( ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true )
| ( ( ~ ( topological_space @ SV269 @ SV315 )
| ( equal_sets @ SV91 @ empty_set )
| ( equal_sets @ SV201 @ empty_set )
| ~ ( element_of_collection @ SV91 @ SV315 )
| ~ ( element_of_collection @ SV201 @ SV315 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[804]) ).
thf(904,plain,
! [SV203: $i,SV93: $i,SV337: $i] :
( ( ( ! [SY810: $i] :
~ ( separation @ SV337 @ SY810 @ SV93 @ SV203 ) )
= $true )
| ( ( connected_space @ SV93 @ SV203 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[806]) ).
thf(905,plain,
! [SV204: $i,SV94: $i] :
( ( ( topological_space @ SV94 @ SV204 )
= $false )
| ( ( separation @ ( f21 @ SV94 @ SV204 ) @ ( f22 @ SV94 @ SV204 ) @ SV94 @ SV204 )
= $true )
| ( ( connected_space @ SV94 @ SV204 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[807]) ).
thf(906,plain,
! [SV270: $i,SV205: $i,SV95: $i] :
( ( ( connected_set @ SV95 @ SV205 @ SV270 )
= $false )
| ( ( topological_space @ SV205 @ SV270 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[808]) ).
thf(907,plain,
! [SV316: $i,SV206: $i,SV96: $i] :
( ( ( connected_set @ SV96 @ SV206 @ SV316 )
= $false )
| ( ( subset_sets @ SV96 @ SV206 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[809]) ).
thf(908,plain,
! [SV271: $i,SV207: $i,SV97: $i] :
( ( ( connected_set @ SV97 @ SV207 @ SV271 )
= $false )
| ( ( connected_space @ SV97 @ ( subspace_topology @ SV207 @ SV271 @ SV97 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[810]) ).
thf(909,plain,
! [SV98: $i,SV272: $i,SV208: $i] :
( ( ( ~ ( topological_space @ SV208 @ SV272 ) )
= $true )
| ( ( ~ ( subset_sets @ SV98 @ SV208 )
| ~ ( connected_space @ SV98 @ ( subspace_topology @ SV208 @ SV272 @ SV98 ) ) )
= $true )
| ( ( connected_set @ SV98 @ SV208 @ SV272 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[811]) ).
thf(910,plain,
! [SV273: $i,SV209: $i,SV99: $i] :
( ( ( open_covering @ SV99 @ SV209 @ SV273 )
= $false )
| ( ( topological_space @ SV209 @ SV273 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[812]) ).
thf(911,plain,
! [SV274: $i,SV210: $i,SV100: $i] :
( ( ( open_covering @ SV100 @ SV210 @ SV274 )
= $false )
| ( ( subset_collections @ SV100 @ SV274 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[813]) ).
thf(912,plain,
! [SV317: $i,SV211: $i,SV101: $i] :
( ( ( open_covering @ SV101 @ SV211 @ SV317 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV101 ) @ SV211 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[814]) ).
thf(913,plain,
! [SV102: $i,SV275: $i,SV212: $i] :
( ( ( ~ ( topological_space @ SV212 @ SV275 ) )
= $true )
| ( ( ~ ( subset_collections @ SV102 @ SV275 )
| ~ ( equal_sets @ ( union_of_members @ SV102 ) @ SV212 ) )
= $true )
| ( ( open_covering @ SV102 @ SV212 @ SV275 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[815]) ).
thf(914,plain,
! [SV214: $i,SV104: $i,SV338: $i] :
( ( ( ~ ( open_covering @ SV338 @ SV104 @ SV214 )
| ( finite @ ( f23 @ SV104 @ SV214 @ SV338 ) ) )
= $true )
| ( ( compact_space @ SV104 @ SV214 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[817]) ).
thf(915,plain,
! [SV215: $i,SV105: $i,SV339: $i] :
( ( ( ~ ( open_covering @ SV339 @ SV105 @ SV215 )
| ( subset_collections @ ( f23 @ SV105 @ SV215 @ SV339 ) @ SV339 ) )
= $true )
| ( ( compact_space @ SV105 @ SV215 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[818]) ).
thf(916,plain,
! [SV216: $i,SV106: $i,SV340: $i] :
( ( ( ~ ( open_covering @ SV340 @ SV106 @ SV216 )
| ( open_covering @ ( f23 @ SV106 @ SV216 @ SV340 ) @ SV106 @ SV216 ) )
= $true )
| ( ( compact_space @ SV106 @ SV216 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[819]) ).
thf(917,plain,
! [SV217: $i,SV107: $i] :
( ( ( topological_space @ SV107 @ SV217 )
= $false )
| ( ( open_covering @ ( f24 @ SV107 @ SV217 ) @ SV107 @ SV217 )
= $true )
| ( ( compact_space @ SV107 @ SV217 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[820]) ).
thf(918,plain,
! [SV218: $i,SV108: $i] :
( ( ( topological_space @ SV108 @ SV218 )
= $false )
| ( ( ! [SY744: $i] :
( ~ ( finite @ SY744 )
| ~ ( subset_collections @ SY744 @ ( f24 @ SV108 @ SV218 ) )
| ~ ( open_covering @ SY744 @ SV108 @ SV218 ) ) )
= $true )
| ( ( compact_space @ SV108 @ SV218 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[821]) ).
thf(919,plain,
! [SV276: $i,SV219: $i,SV109: $i] :
( ( ( compact_set @ SV109 @ SV219 @ SV276 )
= $false )
| ( ( topological_space @ SV219 @ SV276 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[822]) ).
thf(920,plain,
! [SV318: $i,SV220: $i,SV110: $i] :
( ( ( compact_set @ SV110 @ SV220 @ SV318 )
= $false )
| ( ( subset_sets @ SV110 @ SV220 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[823]) ).
thf(921,plain,
! [SV277: $i,SV221: $i,SV111: $i] :
( ( ( compact_set @ SV111 @ SV221 @ SV277 )
= $false )
| ( ( compact_space @ SV111 @ ( subspace_topology @ SV221 @ SV277 @ SV111 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[824]) ).
thf(922,plain,
! [SV112: $i,SV278: $i,SV222: $i] :
( ( ( ~ ( topological_space @ SV222 @ SV278 ) )
= $true )
| ( ( ~ ( subset_sets @ SV112 @ SV222 )
| ~ ( compact_space @ SV112 @ ( subspace_topology @ SV222 @ SV278 @ SV112 ) ) )
= $true )
| ( ( compact_set @ SV112 @ SV222 @ SV278 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[825]) ).
thf(923,plain,
! [SV116: $i,SV279: $i,SV6: $i] :
( ( ( element_of_set @ SV6 @ SV279 )
= $false )
| ( ( ~ ( element_of_collection @ SV279 @ SV116 ) )
= $true )
| ( ( element_of_set @ SV6 @ ( union_of_members @ SV116 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[826]) ).
thf(924,plain,
! [SV7: $i,SV117: $i,SV319: $i] :
( ( ( ~ ( element_of_collection @ SV319 @ SV117 ) )
= $true )
| ( ( element_of_set @ SV7 @ SV319 )
= $true )
| ( ( element_of_set @ SV7 @ ( intersection_of_members @ SV117 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[827]) ).
thf(925,plain,
! [SV13: $i,SV123: $i,SV320: $i] :
( ( ( ~ ( element_of_collection @ SV320 @ SV123 ) )
= $true )
| ( ( ! [SY806: $i] :
( ~ ( element_of_collection @ SY806 @ SV123 )
| ( element_of_collection @ ( intersection_of_sets @ SV320 @ SY806 ) @ SV123 ) ) )
= $true )
| ( ( topological_space @ SV13 @ SV123 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[828]) ).
thf(926,plain,
! [SV14: $i,SV124: $i,SV321: $i] :
( ( ( ~ ( subset_collections @ SV321 @ SV124 ) )
= $true )
| ( ( element_of_collection @ ( union_of_members @ SV321 ) @ SV124 )
= $true )
| ( ( topological_space @ SV14 @ SV124 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[829]) ).
thf(927,plain,
! [SV15: $i,SV125: $i] :
( ( ( ~ ( element_of_collection @ empty_set @ SV125 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV15 @ SV125 )
| ( element_of_collection @ ( f3 @ SV15 @ SV125 ) @ SV125 )
| ( subset_collections @ ( f5 @ SV15 @ SV125 ) @ SV125 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV125 ) @ SV15 )
= $false )
| ( ( topological_space @ SV15 @ SV125 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[830]) ).
thf(928,plain,
! [SV16: $i,SV126: $i] :
( ( ( ~ ( element_of_collection @ empty_set @ SV126 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV16 @ SV126 )
| ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 )
= $false )
| ( ( topological_space @ SV16 @ SV126 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[831]) ).
thf(929,plain,
! [SV17: $i,SV127: $i] :
( ( ( ~ ( element_of_collection @ empty_set @ SV127 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV17 @ SV127 )
| ( element_of_collection @ ( f4 @ SV17 @ SV127 ) @ SV127 )
| ( subset_collections @ ( f5 @ SV17 @ SV127 ) @ SV127 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV127 ) @ SV17 )
= $false )
| ( ( topological_space @ SV17 @ SV127 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[832]) ).
thf(930,plain,
! [SV18: $i,SV128: $i] :
( ( ( ~ ( element_of_collection @ empty_set @ SV128 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV18 @ SV128 )
| ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 )
= $false )
| ( ( topological_space @ SV18 @ SV128 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[833]) ).
thf(931,plain,
! [SV19: $i,SV129: $i] :
( ( ( ~ ( element_of_collection @ empty_set @ SV129 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV19 @ SV129 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 )
| ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 )
= $false )
| ( ( topological_space @ SV19 @ SV129 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[834]) ).
thf(932,plain,
! [SV20: $i,SV130: $i] :
( ( ( ~ ( element_of_collection @ empty_set @ SV130 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV20 @ SV130 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
= $false )
| ( ( topological_space @ SV20 @ SV130 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[835]) ).
thf(933,plain,
! [SV23: $i,SV225: $i,SV133: $i] :
( ( ( topological_space @ SV133 @ SV225 )
= $false )
| ( ( ~ ( element_of_collection @ SV23 @ SV225 ) )
= $true )
| ( ( open @ SV23 @ SV133 @ SV225 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[838]) ).
thf(934,plain,
! [SV26: $i,SV228: $i,SV136: $i] :
( ( ( topological_space @ SV136 @ SV228 )
= $false )
| ( ( ~ ( open @ ( relative_complement_sets @ SV26 @ SV136 ) @ SV136 @ SV228 ) )
= $true )
| ( ( closed @ SV26 @ SV136 @ SV228 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[841]) ).
thf(935,plain,
! [SV140: $i,SV30: $i,SV231: $i] :
( ( ( topological_space @ SV231 @ SV30 )
= $false )
| ( ( ~ ( topological_space @ SV231 @ SV140 )
| ~ ( subset_collections @ SV140 @ SV30 ) )
= $true )
| ( ( finer @ SV30 @ SV140 @ SV231 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[845]) ).
thf(936,plain,
! [SV142: $i,SV32: $i,SV232: $i] :
( ( ( ~ ( element_of_set @ SV232 @ SV32 ) )
= $true )
| ( ( ! [SY749: $i] :
( ~ ( element_of_collection @ SY749 @ SV142 )
| ! [SY750: $i] :
( ~ ( element_of_collection @ SY750 @ SV142 )
| ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SY749 @ SY750 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SY749 @ SY750 ) ) ) ) )
= $true )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[846]) ).
thf(937,plain,
! [SV143: $i,SV33: $i,SV233: $i] :
( ( ( ~ ( element_of_set @ SV233 @ SV33 ) )
= $true )
| ( ( ! [SY751: $i] :
( ~ ( element_of_collection @ SY751 @ SV143 )
| ! [SY752: $i] :
( ~ ( element_of_collection @ SY752 @ SV143 )
| ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SY751 @ SY752 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SY751 @ SY752 ) @ SV143 ) ) ) )
= $true )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[847]) ).
thf(938,plain,
! [SV144: $i,SV34: $i,SV234: $i] :
( ( ( ~ ( element_of_set @ SV234 @ SV34 ) )
= $true )
| ( ( ! [SY753: $i] :
( ~ ( element_of_collection @ SY753 @ SV144 )
| ! [SY754: $i] :
( ~ ( element_of_collection @ SY754 @ SV144 )
| ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SY753 @ SY754 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SY753 @ SY754 ) @ ( intersection_of_sets @ SY753 @ SY754 ) ) ) ) )
= $true )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[848]) ).
thf(939,plain,
! [SV341: $i,SV149: $i,SV39: $i] :
( ( ( ~ ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SV341 )
| ~ ( element_of_collection @ SV341 @ SV149 )
| ~ ( subset_sets @ SV341 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 )
= $false )
| ( ( basis @ SV39 @ SV149 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[853]) ).
thf(940,plain,
! [SV150: $i,SV40: $i,SV322: $i] :
( ( ( ~ ( element_of_set @ SV322 @ SV40 ) )
= $true )
| ( ( element_of_set @ SV322 @ ( f10 @ SV150 @ SV40 @ SV322 ) )
= $true )
| ( ( element_of_collection @ SV40 @ ( top_of_basis @ SV150 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[854]) ).
thf(941,plain,
! [SV151: $i,SV41: $i,SV323: $i] :
( ( ( ~ ( element_of_set @ SV323 @ SV41 ) )
= $true )
| ( ( element_of_collection @ ( f10 @ SV151 @ SV41 @ SV323 ) @ SV151 )
= $true )
| ( ( element_of_collection @ SV41 @ ( top_of_basis @ SV151 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[855]) ).
thf(942,plain,
! [SV152: $i,SV42: $i,SV324: $i] :
( ( ( ~ ( element_of_set @ SV324 @ SV42 ) )
= $true )
| ( ( subset_sets @ ( f10 @ SV152 @ SV42 @ SV324 ) @ SV42 )
= $true )
| ( ( element_of_collection @ SV42 @ ( top_of_basis @ SV152 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[856]) ).
thf(943,plain,
! [SV281: $i,SV44: $i,SV154: $i] :
( ( ( element_of_set @ ( f11 @ SV154 @ SV44 ) @ SV281 )
= $false )
| ( ( ~ ( element_of_collection @ SV281 @ SV154 )
| ~ ( subset_sets @ SV281 @ SV44 ) )
= $true )
| ( ( element_of_collection @ SV44 @ ( top_of_basis @ SV154 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[857]) ).
thf(944,plain,
! [SV325: $i,SV235: $i,SV155: $i,SV45: $i] :
( ( ( element_of_collection @ SV45 @ ( subspace_topology @ SV155 @ SV235 @ SV325 ) )
= $false )
| ( ( topological_space @ SV155 @ SV235 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[858]) ).
thf(945,plain,
! [SV282: $i,SV236: $i,SV156: $i,SV46: $i] :
( ( ( element_of_collection @ SV46 @ ( subspace_topology @ SV156 @ SV236 @ SV282 ) )
= $false )
| ( ( subset_sets @ SV282 @ SV156 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[859]) ).
thf(946,plain,
! [SV283: $i,SV237: $i,SV157: $i,SV47: $i] :
( ( ( element_of_collection @ SV47 @ ( subspace_topology @ SV157 @ SV237 @ SV283 ) )
= $false )
| ( ( element_of_collection @ ( f12 @ SV157 @ SV237 @ SV283 @ SV47 ) @ SV237 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[860]) ).
thf(947,plain,
! [SV284: $i,SV238: $i,SV158: $i,SV48: $i] :
( ( ( element_of_collection @ SV48 @ ( subspace_topology @ SV158 @ SV238 @ SV284 ) )
= $false )
| ( ( equal_sets @ SV48 @ ( intersection_of_sets @ SV284 @ ( f12 @ SV158 @ SV238 @ SV284 @ SV48 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[861]) ).
thf(948,plain,
! [SV49: $i,SV285: $i,SV239: $i,SV159: $i] :
( ( ( ~ ( topological_space @ SV159 @ SV239 ) )
= $true )
| ( ( ~ ( subset_sets @ SV285 @ SV159 )
| ! [SY796: $i] :
( ~ ( element_of_collection @ SY796 @ SV239 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SY796 ) ) ) )
= $true )
| ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[862]) ).
thf(949,plain,
! [SV286: $i,SV240: $i,SV160: $i,SV50: $i] :
( ( ( element_of_set @ SV50 @ ( interior @ SV160 @ SV240 @ SV286 ) )
= $false )
| ( ( topological_space @ SV240 @ SV286 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[863]) ).
thf(950,plain,
! [SV326: $i,SV241: $i,SV161: $i,SV51: $i] :
( ( ( element_of_set @ SV51 @ ( interior @ SV161 @ SV241 @ SV326 ) )
= $false )
| ( ( subset_sets @ SV161 @ SV241 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[864]) ).
thf(951,plain,
! [SV287: $i,SV242: $i,SV162: $i,SV52: $i] :
( ( ( element_of_set @ SV52 @ ( interior @ SV162 @ SV242 @ SV287 ) )
= $false )
| ( ( element_of_set @ SV52 @ ( f13 @ SV162 @ SV242 @ SV287 @ SV52 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[865]) ).
thf(952,plain,
! [SV288: $i,SV243: $i,SV163: $i,SV53: $i] :
( ( ( element_of_set @ SV53 @ ( interior @ SV163 @ SV243 @ SV288 ) )
= $false )
| ( ( subset_sets @ ( f13 @ SV163 @ SV243 @ SV288 @ SV53 ) @ SV163 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[866]) ).
thf(953,plain,
! [SV289: $i,SV244: $i,SV164: $i,SV54: $i] :
( ( ( element_of_set @ SV54 @ ( interior @ SV164 @ SV244 @ SV289 ) )
= $false )
| ( ( open @ ( f13 @ SV164 @ SV244 @ SV289 @ SV54 ) @ SV244 @ SV289 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[867]) ).
thf(954,plain,
! [SV55: $i,SV165: $i,SV290: $i,SV245: $i] :
( ( ( ~ ( topological_space @ SV245 @ SV290 ) )
= $true )
| ( ( ~ ( subset_sets @ SV165 @ SV245 )
| ! [SY797: $i] :
( ~ ( element_of_set @ SV55 @ SY797 )
| ~ ( subset_sets @ SY797 @ SV165 )
| ~ ( open @ SY797 @ SV245 @ SV290 ) ) )
= $true )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[868]) ).
thf(955,plain,
! [SV291: $i,SV246: $i,SV166: $i,SV56: $i] :
( ( ( element_of_set @ SV56 @ ( closure @ SV166 @ SV246 @ SV291 ) )
= $false )
| ( ( topological_space @ SV246 @ SV291 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[869]) ).
thf(956,plain,
! [SV327: $i,SV247: $i,SV167: $i,SV57: $i] :
( ( ( element_of_set @ SV57 @ ( closure @ SV167 @ SV247 @ SV327 ) )
= $false )
| ( ( subset_sets @ SV167 @ SV247 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[870]) ).
thf(957,plain,
! [SV292: $i,SV248: $i,SV168: $i,SV58: $i] :
( ( ( element_of_set @ SV58 @ ( closure @ SV168 @ SV248 @ SV292 ) )
= $false )
| ( ( ! [SY798: $i] :
( ~ ( subset_sets @ SV168 @ SY798 )
| ~ ( closed @ SY798 @ SV248 @ SV292 )
| ( element_of_set @ SV58 @ SY798 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[871]) ).
thf(958,plain,
! [SV59: $i,SV169: $i,SV293: $i,SV249: $i] :
( ( ( ~ ( topological_space @ SV249 @ SV293 ) )
= $true )
| ( ( ~ ( subset_sets @ SV169 @ SV249 )
| ( subset_sets @ SV169 @ ( f14 @ SV169 @ SV249 @ SV293 @ SV59 ) ) )
= $true )
| ( ( element_of_set @ SV59 @ ( closure @ SV169 @ SV249 @ SV293 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[872]) ).
thf(959,plain,
! [SV60: $i,SV170: $i,SV294: $i,SV250: $i] :
( ( ( ~ ( topological_space @ SV250 @ SV294 ) )
= $true )
| ( ( ~ ( subset_sets @ SV170 @ SV250 )
| ( closed @ ( f14 @ SV170 @ SV250 @ SV294 @ SV60 ) @ SV250 @ SV294 ) )
= $true )
| ( ( element_of_set @ SV60 @ ( closure @ SV170 @ SV250 @ SV294 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[873]) ).
thf(960,plain,
! [SV61: $i,SV171: $i,SV295: $i,SV251: $i] :
( ( ( ~ ( topological_space @ SV251 @ SV295 ) )
= $true )
| ( ( ~ ( subset_sets @ SV171 @ SV251 )
| ~ ( element_of_set @ SV61 @ ( f14 @ SV171 @ SV251 @ SV295 @ SV61 ) ) )
= $true )
| ( ( element_of_set @ SV61 @ ( closure @ SV171 @ SV251 @ SV295 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[874]) ).
thf(961,plain,
! [SV296: $i,SV252: $i,SV172: $i,SV62: $i] :
( ( ( neighborhood @ SV62 @ SV172 @ SV252 @ SV296 )
= $false )
| ( ( topological_space @ SV252 @ SV296 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[875]) ).
thf(962,plain,
! [SV297: $i,SV253: $i,SV173: $i,SV63: $i] :
( ( ( neighborhood @ SV63 @ SV173 @ SV253 @ SV297 )
= $false )
| ( ( open @ SV63 @ SV253 @ SV297 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[876]) ).
thf(963,plain,
! [SV328: $i,SV298: $i,SV174: $i,SV64: $i] :
( ( ( neighborhood @ SV64 @ SV174 @ SV298 @ SV328 )
= $false )
| ( ( element_of_set @ SV174 @ SV64 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[877]) ).
thf(964,plain,
! [SV175: $i,SV65: $i,SV299: $i,SV254: $i] :
( ( ( ~ ( topological_space @ SV254 @ SV299 ) )
= $true )
| ( ( ~ ( open @ SV65 @ SV254 @ SV299 )
| ~ ( element_of_set @ SV175 @ SV65 ) )
= $true )
| ( ( neighborhood @ SV65 @ SV175 @ SV254 @ SV299 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[878]) ).
thf(965,plain,
! [SV300: $i,SV255: $i,SV176: $i,SV66: $i] :
( ( ( limit_point @ SV66 @ SV176 @ SV255 @ SV300 )
= $false )
| ( ( topological_space @ SV255 @ SV300 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[879]) ).
thf(966,plain,
! [SV329: $i,SV256: $i,SV177: $i,SV67: $i] :
( ( ( limit_point @ SV67 @ SV177 @ SV256 @ SV329 )
= $false )
| ( ( subset_sets @ SV177 @ SV256 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[880]) ).
thf(967,plain,
! [SV301: $i,SV257: $i,SV178: $i,SV68: $i] :
( ( ( limit_point @ SV68 @ SV178 @ SV257 @ SV301 )
= $false )
| ( ( ! [SY800: $i] :
( ~ ( neighborhood @ SY800 @ SV68 @ SV257 @ SV301 )
| ( element_of_set @ ( f15 @ SV68 @ SV178 @ SV257 @ SV301 @ SY800 ) @ ( intersection_of_sets @ SY800 @ SV178 ) ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[881]) ).
thf(968,plain,
! [SV302: $i,SV258: $i,SV179: $i,SV69: $i] :
( ( ( limit_point @ SV69 @ SV179 @ SV258 @ SV302 )
= $false )
| ( ( ! [SY801: $i] :
( ~ ( neighborhood @ SY801 @ SV69 @ SV258 @ SV302 )
| ~ ( eq_p @ ( f15 @ SV69 @ SV179 @ SV258 @ SV302 @ SY801 ) @ SV69 ) ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[882]) ).
thf(969,plain,
! [SV70: $i,SV180: $i,SV303: $i,SV259: $i] :
( ( ( ~ ( topological_space @ SV259 @ SV303 ) )
= $true )
| ( ( ~ ( subset_sets @ SV180 @ SV259 )
| ( neighborhood @ ( f16 @ SV70 @ SV180 @ SV259 @ SV303 ) @ SV70 @ SV259 @ SV303 ) )
= $true )
| ( ( limit_point @ SV70 @ SV180 @ SV259 @ SV303 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[883]) ).
thf(970,plain,
! [SV71: $i,SV181: $i,SV304: $i,SV260: $i] :
( ( ( ~ ( topological_space @ SV260 @ SV304 ) )
= $true )
| ( ( ~ ( subset_sets @ SV181 @ SV260 )
| ! [SY802: $i] :
( ~ ( element_of_set @ SY802 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SV304 ) @ SV181 ) )
| ( eq_p @ SY802 @ SV71 ) ) )
= $true )
| ( ( limit_point @ SV71 @ SV181 @ SV260 @ SV304 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[884]) ).
thf(971,plain,
! [SV305: $i,SV261: $i,SV182: $i,SV72: $i] :
( ( ( element_of_set @ SV72 @ ( boundary @ SV182 @ SV261 @ SV305 ) )
= $false )
| ( ( topological_space @ SV261 @ SV305 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[885]) ).
thf(972,plain,
! [SV306: $i,SV262: $i,SV183: $i,SV73: $i] :
( ( ( element_of_set @ SV73 @ ( boundary @ SV183 @ SV262 @ SV306 ) )
= $false )
| ( ( element_of_set @ SV73 @ ( closure @ SV183 @ SV262 @ SV306 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[886]) ).
thf(973,plain,
! [SV307: $i,SV263: $i,SV184: $i,SV74: $i] :
( ( ( element_of_set @ SV74 @ ( boundary @ SV184 @ SV263 @ SV307 ) )
= $false )
| ( ( element_of_set @ SV74 @ ( closure @ ( relative_complement_sets @ SV184 @ SV263 ) @ SV263 @ SV307 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[887]) ).
thf(974,plain,
! [SV185: $i,SV75: $i,SV308: $i,SV264: $i] :
( ( ( ~ ( topological_space @ SV264 @ SV308 ) )
= $true )
| ( ( ~ ( element_of_set @ SV75 @ ( closure @ SV185 @ SV264 @ SV308 ) )
| ~ ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SV185 @ SV264 ) @ SV264 @ SV308 ) ) )
= $true )
| ( ( element_of_set @ SV75 @ ( boundary @ SV185 @ SV264 @ SV308 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[888]) ).
thf(975,plain,
! [SV187: $i,SV77: $i,SV330: $i] :
( ( ( ~ ( element_of_set @ SV330 @ SV77 ) )
= $true )
| ( ( ! [SY807: $i] :
( ~ ( element_of_set @ SY807 @ SV77 )
| ( eq_p @ SV330 @ SY807 )
| ( neighborhood @ ( f17 @ SV77 @ SV187 @ SV330 @ SY807 ) @ SV330 @ SV77 @ SV187 ) ) )
= $true )
| ( ( hausdorff @ SV77 @ SV187 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[889]) ).
thf(976,plain,
! [SV188: $i,SV78: $i,SV331: $i] :
( ( ( ~ ( element_of_set @ SV331 @ SV78 ) )
= $true )
| ( ( ! [SY808: $i] :
( ~ ( element_of_set @ SY808 @ SV78 )
| ( eq_p @ SV331 @ SY808 )
| ( neighborhood @ ( f18 @ SV78 @ SV188 @ SV331 @ SY808 ) @ SY808 @ SV78 @ SV188 ) ) )
= $true )
| ( ( hausdorff @ SV78 @ SV188 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[890]) ).
thf(977,plain,
! [SV189: $i,SV79: $i,SV332: $i] :
( ( ( ~ ( element_of_set @ SV332 @ SV79 ) )
= $true )
| ( ( ! [SY809: $i] :
( ~ ( element_of_set @ SY809 @ SV79 )
| ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SV332 @ SY809 ) @ ( f18 @ SV79 @ SV189 @ SV332 @ SY809 ) )
| ( eq_p @ SV332 @ SY809 ) ) )
= $true )
| ( ( hausdorff @ SV79 @ SV189 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[891]) ).
thf(978,plain,
! [SV192: $i,SV82: $i] :
( ( ( eq_p @ ( f19 @ SV82 @ SV192 ) @ ( f20 @ SV82 @ SV192 ) )
= $false )
| ( ( topological_space @ SV82 @ SV192 )
= $false )
| ( ( hausdorff @ SV82 @ SV192 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[894]) ).
thf(979,plain,
! [SV193: $i,SV83: $i,SV342: $i] :
( ( ( ~ ( neighborhood @ SV342 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ! [SY811: $i] :
( ~ ( neighborhood @ SY811 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ~ ( disjoint_s @ SV342 @ SY811 ) ) )
= $true )
| ( ( topological_space @ SV83 @ SV193 )
= $false )
| ( ( hausdorff @ SV83 @ SV193 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[895]) ).
thf(980,plain,
! [SV309: $i,SV265: $i,SV194: $i,SV84: $i] :
( ( ( separation @ SV84 @ SV194 @ SV265 @ SV309 )
= $false )
| ( ( topological_space @ SV265 @ SV309 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[896]) ).
thf(981,plain,
! [SV333: $i,SV310: $i,SV195: $i,SV85: $i] :
( ( ( separation @ SV85 @ SV195 @ SV310 @ SV333 )
= $false )
| ( ( ~ ( equal_sets @ SV85 @ empty_set ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[897]) ).
thf(982,plain,
! [SV334: $i,SV311: $i,SV196: $i,SV86: $i] :
( ( ( separation @ SV86 @ SV196 @ SV311 @ SV334 )
= $false )
| ( ( ~ ( equal_sets @ SV196 @ empty_set ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[898]) ).
thf(983,plain,
! [SV312: $i,SV266: $i,SV197: $i,SV87: $i] :
( ( ( separation @ SV87 @ SV197 @ SV266 @ SV312 )
= $false )
| ( ( element_of_collection @ SV87 @ SV312 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[899]) ).
thf(984,plain,
! [SV313: $i,SV267: $i,SV198: $i,SV88: $i] :
( ( ( separation @ SV88 @ SV198 @ SV267 @ SV313 )
= $false )
| ( ( element_of_collection @ SV198 @ SV313 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[900]) ).
thf(985,plain,
! [SV335: $i,SV268: $i,SV199: $i,SV89: $i] :
( ( ( separation @ SV89 @ SV199 @ SV268 @ SV335 )
= $false )
| ( ( equal_sets @ ( union_of_sets @ SV89 @ SV199 ) @ SV268 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[901]) ).
thf(986,plain,
! [SV336: $i,SV314: $i,SV200: $i,SV90: $i] :
( ( ( separation @ SV90 @ SV200 @ SV314 @ SV336 )
= $false )
| ( ( disjoint_s @ SV90 @ SV200 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[902]) ).
thf(987,plain,
! [SV201: $i,SV91: $i,SV315: $i,SV269: $i] :
( ( ( ~ ( topological_space @ SV269 @ SV315 ) )
= $true )
| ( ( ( equal_sets @ SV91 @ empty_set )
| ( equal_sets @ SV201 @ empty_set )
| ~ ( element_of_collection @ SV91 @ SV315 )
| ~ ( element_of_collection @ SV201 @ SV315 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[903]) ).
thf(988,plain,
! [SV203: $i,SV93: $i,SV343: $i,SV337: $i] :
( ( ( ~ ( separation @ SV337 @ SV343 @ SV93 @ SV203 ) )
= $true )
| ( ( connected_space @ SV93 @ SV203 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[904]) ).
thf(989,plain,
! [SV98: $i,SV272: $i,SV208: $i] :
( ( ( topological_space @ SV208 @ SV272 )
= $false )
| ( ( ~ ( subset_sets @ SV98 @ SV208 )
| ~ ( connected_space @ SV98 @ ( subspace_topology @ SV208 @ SV272 @ SV98 ) ) )
= $true )
| ( ( connected_set @ SV98 @ SV208 @ SV272 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[909]) ).
thf(990,plain,
! [SV102: $i,SV275: $i,SV212: $i] :
( ( ( topological_space @ SV212 @ SV275 )
= $false )
| ( ( ~ ( subset_collections @ SV102 @ SV275 )
| ~ ( equal_sets @ ( union_of_members @ SV102 ) @ SV212 ) )
= $true )
| ( ( open_covering @ SV102 @ SV212 @ SV275 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[913]) ).
thf(991,plain,
! [SV214: $i,SV104: $i,SV338: $i] :
( ( ( ~ ( open_covering @ SV338 @ SV104 @ SV214 ) )
= $true )
| ( ( finite @ ( f23 @ SV104 @ SV214 @ SV338 ) )
= $true )
| ( ( compact_space @ SV104 @ SV214 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[914]) ).
thf(992,plain,
! [SV215: $i,SV105: $i,SV339: $i] :
( ( ( ~ ( open_covering @ SV339 @ SV105 @ SV215 ) )
= $true )
| ( ( subset_collections @ ( f23 @ SV105 @ SV215 @ SV339 ) @ SV339 )
= $true )
| ( ( compact_space @ SV105 @ SV215 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[915]) ).
thf(993,plain,
! [SV216: $i,SV106: $i,SV340: $i] :
( ( ( ~ ( open_covering @ SV340 @ SV106 @ SV216 ) )
= $true )
| ( ( open_covering @ ( f23 @ SV106 @ SV216 @ SV340 ) @ SV106 @ SV216 )
= $true )
| ( ( compact_space @ SV106 @ SV216 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[916]) ).
thf(994,plain,
! [SV218: $i,SV108: $i,SV344: $i] :
( ( ( ~ ( finite @ SV344 )
| ~ ( subset_collections @ SV344 @ ( f24 @ SV108 @ SV218 ) )
| ~ ( open_covering @ SV344 @ SV108 @ SV218 ) )
= $true )
| ( ( topological_space @ SV108 @ SV218 )
= $false )
| ( ( compact_space @ SV108 @ SV218 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[918]) ).
thf(995,plain,
! [SV112: $i,SV278: $i,SV222: $i] :
( ( ( topological_space @ SV222 @ SV278 )
= $false )
| ( ( ~ ( subset_sets @ SV112 @ SV222 )
| ~ ( compact_space @ SV112 @ ( subspace_topology @ SV222 @ SV278 @ SV112 ) ) )
= $true )
| ( ( compact_set @ SV112 @ SV222 @ SV278 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[922]) ).
thf(996,plain,
! [SV6: $i,SV116: $i,SV279: $i] :
( ( ( element_of_collection @ SV279 @ SV116 )
= $false )
| ( ( element_of_set @ SV6 @ SV279 )
= $false )
| ( ( element_of_set @ SV6 @ ( union_of_members @ SV116 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[923]) ).
thf(997,plain,
! [SV7: $i,SV117: $i,SV319: $i] :
( ( ( element_of_collection @ SV319 @ SV117 )
= $false )
| ( ( element_of_set @ SV7 @ SV319 )
= $true )
| ( ( element_of_set @ SV7 @ ( intersection_of_members @ SV117 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[924]) ).
thf(998,plain,
! [SV13: $i,SV123: $i,SV320: $i] :
( ( ( element_of_collection @ SV320 @ SV123 )
= $false )
| ( ( ! [SY806: $i] :
( ~ ( element_of_collection @ SY806 @ SV123 )
| ( element_of_collection @ ( intersection_of_sets @ SV320 @ SY806 ) @ SV123 ) ) )
= $true )
| ( ( topological_space @ SV13 @ SV123 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[925]) ).
thf(999,plain,
! [SV14: $i,SV124: $i,SV321: $i] :
( ( ( subset_collections @ SV321 @ SV124 )
= $false )
| ( ( element_of_collection @ ( union_of_members @ SV321 ) @ SV124 )
= $true )
| ( ( topological_space @ SV14 @ SV124 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[926]) ).
thf(1000,plain,
! [SV15: $i,SV125: $i] :
( ( ( element_of_collection @ empty_set @ SV125 )
= $false )
| ( ( ~ ( element_of_collection @ SV15 @ SV125 )
| ( element_of_collection @ ( f3 @ SV15 @ SV125 ) @ SV125 )
| ( subset_collections @ ( f5 @ SV15 @ SV125 ) @ SV125 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV125 ) @ SV15 )
= $false )
| ( ( topological_space @ SV15 @ SV125 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[927]) ).
thf(1001,plain,
! [SV16: $i,SV126: $i] :
( ( ( element_of_collection @ empty_set @ SV126 )
= $false )
| ( ( ~ ( element_of_collection @ SV16 @ SV126 )
| ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 )
= $false )
| ( ( topological_space @ SV16 @ SV126 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[928]) ).
thf(1002,plain,
! [SV17: $i,SV127: $i] :
( ( ( element_of_collection @ empty_set @ SV127 )
= $false )
| ( ( ~ ( element_of_collection @ SV17 @ SV127 )
| ( element_of_collection @ ( f4 @ SV17 @ SV127 ) @ SV127 )
| ( subset_collections @ ( f5 @ SV17 @ SV127 ) @ SV127 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV127 ) @ SV17 )
= $false )
| ( ( topological_space @ SV17 @ SV127 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[929]) ).
thf(1003,plain,
! [SV18: $i,SV128: $i] :
( ( ( element_of_collection @ empty_set @ SV128 )
= $false )
| ( ( ~ ( element_of_collection @ SV18 @ SV128 )
| ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 )
= $false )
| ( ( topological_space @ SV18 @ SV128 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[930]) ).
thf(1004,plain,
! [SV19: $i,SV129: $i] :
( ( ( element_of_collection @ empty_set @ SV129 )
= $false )
| ( ( ~ ( element_of_collection @ SV19 @ SV129 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 )
| ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 )
= $false )
| ( ( topological_space @ SV19 @ SV129 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[931]) ).
thf(1005,plain,
! [SV20: $i,SV130: $i] :
( ( ( element_of_collection @ empty_set @ SV130 )
= $false )
| ( ( ~ ( element_of_collection @ SV20 @ SV130 )
| ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
= $false )
| ( ( topological_space @ SV20 @ SV130 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[932]) ).
thf(1006,plain,
! [SV133: $i,SV225: $i,SV23: $i] :
( ( ( element_of_collection @ SV23 @ SV225 )
= $false )
| ( ( topological_space @ SV133 @ SV225 )
= $false )
| ( ( open @ SV23 @ SV133 @ SV225 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[933]) ).
thf(1007,plain,
! [SV228: $i,SV136: $i,SV26: $i] :
( ( ( open @ ( relative_complement_sets @ SV26 @ SV136 ) @ SV136 @ SV228 )
= $false )
| ( ( topological_space @ SV136 @ SV228 )
= $false )
| ( ( closed @ SV26 @ SV136 @ SV228 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[934]) ).
thf(1008,plain,
! [SV30: $i,SV140: $i,SV231: $i] :
( ( ( ~ ( topological_space @ SV231 @ SV140 ) )
= $true )
| ( ( ~ ( subset_collections @ SV140 @ SV30 ) )
= $true )
| ( ( topological_space @ SV231 @ SV30 )
= $false )
| ( ( finer @ SV30 @ SV140 @ SV231 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[935]) ).
thf(1009,plain,
! [SV142: $i,SV32: $i,SV232: $i] :
( ( ( element_of_set @ SV232 @ SV32 )
= $false )
| ( ( ! [SY749: $i] :
( ~ ( element_of_collection @ SY749 @ SV142 )
| ! [SY750: $i] :
( ~ ( element_of_collection @ SY750 @ SV142 )
| ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SY749 @ SY750 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SY749 @ SY750 ) ) ) ) )
= $true )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[936]) ).
thf(1010,plain,
! [SV143: $i,SV33: $i,SV233: $i] :
( ( ( element_of_set @ SV233 @ SV33 )
= $false )
| ( ( ! [SY751: $i] :
( ~ ( element_of_collection @ SY751 @ SV143 )
| ! [SY752: $i] :
( ~ ( element_of_collection @ SY752 @ SV143 )
| ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SY751 @ SY752 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SY751 @ SY752 ) @ SV143 ) ) ) )
= $true )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[937]) ).
thf(1011,plain,
! [SV144: $i,SV34: $i,SV234: $i] :
( ( ( element_of_set @ SV234 @ SV34 )
= $false )
| ( ( ! [SY753: $i] :
( ~ ( element_of_collection @ SY753 @ SV144 )
| ! [SY754: $i] :
( ~ ( element_of_collection @ SY754 @ SV144 )
| ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SY753 @ SY754 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SY753 @ SY754 ) @ ( intersection_of_sets @ SY753 @ SY754 ) ) ) ) )
= $true )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[938]) ).
thf(1012,plain,
! [SV341: $i,SV149: $i,SV39: $i] :
( ( ( ~ ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SV341 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV341 @ SV149 )
| ~ ( subset_sets @ SV341 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 )
= $false )
| ( ( basis @ SV39 @ SV149 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[939]) ).
thf(1013,plain,
! [SV150: $i,SV40: $i,SV322: $i] :
( ( ( element_of_set @ SV322 @ SV40 )
= $false )
| ( ( element_of_set @ SV322 @ ( f10 @ SV150 @ SV40 @ SV322 ) )
= $true )
| ( ( element_of_collection @ SV40 @ ( top_of_basis @ SV150 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[940]) ).
thf(1014,plain,
! [SV151: $i,SV41: $i,SV323: $i] :
( ( ( element_of_set @ SV323 @ SV41 )
= $false )
| ( ( element_of_collection @ ( f10 @ SV151 @ SV41 @ SV323 ) @ SV151 )
= $true )
| ( ( element_of_collection @ SV41 @ ( top_of_basis @ SV151 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[941]) ).
thf(1015,plain,
! [SV152: $i,SV42: $i,SV324: $i] :
( ( ( element_of_set @ SV324 @ SV42 )
= $false )
| ( ( subset_sets @ ( f10 @ SV152 @ SV42 @ SV324 ) @ SV42 )
= $true )
| ( ( element_of_collection @ SV42 @ ( top_of_basis @ SV152 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[942]) ).
thf(1016,plain,
! [SV44: $i,SV154: $i,SV281: $i] :
( ( ( ~ ( element_of_collection @ SV281 @ SV154 ) )
= $true )
| ( ( ~ ( subset_sets @ SV281 @ SV44 ) )
= $true )
| ( ( element_of_set @ ( f11 @ SV154 @ SV44 ) @ SV281 )
= $false )
| ( ( element_of_collection @ SV44 @ ( top_of_basis @ SV154 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[943]) ).
thf(1017,plain,
! [SV49: $i,SV285: $i,SV239: $i,SV159: $i] :
( ( ( topological_space @ SV159 @ SV239 )
= $false )
| ( ( ~ ( subset_sets @ SV285 @ SV159 )
| ! [SY796: $i] :
( ~ ( element_of_collection @ SY796 @ SV239 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SY796 ) ) ) )
= $true )
| ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[948]) ).
thf(1018,plain,
! [SV55: $i,SV165: $i,SV290: $i,SV245: $i] :
( ( ( topological_space @ SV245 @ SV290 )
= $false )
| ( ( ~ ( subset_sets @ SV165 @ SV245 )
| ! [SY797: $i] :
( ~ ( element_of_set @ SV55 @ SY797 )
| ~ ( subset_sets @ SY797 @ SV165 )
| ~ ( open @ SY797 @ SV245 @ SV290 ) ) )
= $true )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[954]) ).
thf(1019,plain,
! [SV58: $i,SV292: $i,SV248: $i,SV345: $i,SV168: $i] :
( ( ( ~ ( subset_sets @ SV168 @ SV345 )
| ~ ( closed @ SV345 @ SV248 @ SV292 )
| ( element_of_set @ SV58 @ SV345 ) )
= $true )
| ( ( element_of_set @ SV58 @ ( closure @ SV168 @ SV248 @ SV292 ) )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[957]) ).
thf(1020,plain,
! [SV59: $i,SV169: $i,SV293: $i,SV249: $i] :
( ( ( topological_space @ SV249 @ SV293 )
= $false )
| ( ( ~ ( subset_sets @ SV169 @ SV249 )
| ( subset_sets @ SV169 @ ( f14 @ SV169 @ SV249 @ SV293 @ SV59 ) ) )
= $true )
| ( ( element_of_set @ SV59 @ ( closure @ SV169 @ SV249 @ SV293 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[958]) ).
thf(1021,plain,
! [SV60: $i,SV170: $i,SV294: $i,SV250: $i] :
( ( ( topological_space @ SV250 @ SV294 )
= $false )
| ( ( ~ ( subset_sets @ SV170 @ SV250 )
| ( closed @ ( f14 @ SV170 @ SV250 @ SV294 @ SV60 ) @ SV250 @ SV294 ) )
= $true )
| ( ( element_of_set @ SV60 @ ( closure @ SV170 @ SV250 @ SV294 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[959]) ).
thf(1022,plain,
! [SV61: $i,SV171: $i,SV295: $i,SV251: $i] :
( ( ( topological_space @ SV251 @ SV295 )
= $false )
| ( ( ~ ( subset_sets @ SV171 @ SV251 )
| ~ ( element_of_set @ SV61 @ ( f14 @ SV171 @ SV251 @ SV295 @ SV61 ) ) )
= $true )
| ( ( element_of_set @ SV61 @ ( closure @ SV171 @ SV251 @ SV295 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[960]) ).
thf(1023,plain,
! [SV175: $i,SV65: $i,SV299: $i,SV254: $i] :
( ( ( topological_space @ SV254 @ SV299 )
= $false )
| ( ( ~ ( open @ SV65 @ SV254 @ SV299 )
| ~ ( element_of_set @ SV175 @ SV65 ) )
= $true )
| ( ( neighborhood @ SV65 @ SV175 @ SV254 @ SV299 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[964]) ).
thf(1024,plain,
! [SV178: $i,SV301: $i,SV257: $i,SV68: $i,SV346: $i] :
( ( ( ~ ( neighborhood @ SV346 @ SV68 @ SV257 @ SV301 )
| ( element_of_set @ ( f15 @ SV68 @ SV178 @ SV257 @ SV301 @ SV346 ) @ ( intersection_of_sets @ SV346 @ SV178 ) ) )
= $true )
| ( ( limit_point @ SV68 @ SV178 @ SV257 @ SV301 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[967]) ).
thf(1025,plain,
! [SV179: $i,SV302: $i,SV258: $i,SV69: $i,SV347: $i] :
( ( ( ~ ( neighborhood @ SV347 @ SV69 @ SV258 @ SV302 )
| ~ ( eq_p @ ( f15 @ SV69 @ SV179 @ SV258 @ SV302 @ SV347 ) @ SV69 ) )
= $true )
| ( ( limit_point @ SV69 @ SV179 @ SV258 @ SV302 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[968]) ).
thf(1026,plain,
! [SV70: $i,SV180: $i,SV303: $i,SV259: $i] :
( ( ( topological_space @ SV259 @ SV303 )
= $false )
| ( ( ~ ( subset_sets @ SV180 @ SV259 )
| ( neighborhood @ ( f16 @ SV70 @ SV180 @ SV259 @ SV303 ) @ SV70 @ SV259 @ SV303 ) )
= $true )
| ( ( limit_point @ SV70 @ SV180 @ SV259 @ SV303 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[969]) ).
thf(1027,plain,
! [SV71: $i,SV181: $i,SV304: $i,SV260: $i] :
( ( ( topological_space @ SV260 @ SV304 )
= $false )
| ( ( ~ ( subset_sets @ SV181 @ SV260 )
| ! [SY802: $i] :
( ~ ( element_of_set @ SY802 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SV304 ) @ SV181 ) )
| ( eq_p @ SY802 @ SV71 ) ) )
= $true )
| ( ( limit_point @ SV71 @ SV181 @ SV260 @ SV304 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[970]) ).
thf(1028,plain,
! [SV185: $i,SV75: $i,SV308: $i,SV264: $i] :
( ( ( topological_space @ SV264 @ SV308 )
= $false )
| ( ( ~ ( element_of_set @ SV75 @ ( closure @ SV185 @ SV264 @ SV308 ) )
| ~ ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SV185 @ SV264 ) @ SV264 @ SV308 ) ) )
= $true )
| ( ( element_of_set @ SV75 @ ( boundary @ SV185 @ SV264 @ SV308 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[974]) ).
thf(1029,plain,
! [SV187: $i,SV77: $i,SV330: $i] :
( ( ( element_of_set @ SV330 @ SV77 )
= $false )
| ( ( ! [SY807: $i] :
( ~ ( element_of_set @ SY807 @ SV77 )
| ( eq_p @ SV330 @ SY807 )
| ( neighborhood @ ( f17 @ SV77 @ SV187 @ SV330 @ SY807 ) @ SV330 @ SV77 @ SV187 ) ) )
= $true )
| ( ( hausdorff @ SV77 @ SV187 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[975]) ).
thf(1030,plain,
! [SV188: $i,SV78: $i,SV331: $i] :
( ( ( element_of_set @ SV331 @ SV78 )
= $false )
| ( ( ! [SY808: $i] :
( ~ ( element_of_set @ SY808 @ SV78 )
| ( eq_p @ SV331 @ SY808 )
| ( neighborhood @ ( f18 @ SV78 @ SV188 @ SV331 @ SY808 ) @ SY808 @ SV78 @ SV188 ) ) )
= $true )
| ( ( hausdorff @ SV78 @ SV188 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[976]) ).
thf(1031,plain,
! [SV189: $i,SV79: $i,SV332: $i] :
( ( ( element_of_set @ SV332 @ SV79 )
= $false )
| ( ( ! [SY809: $i] :
( ~ ( element_of_set @ SY809 @ SV79 )
| ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SV332 @ SY809 ) @ ( f18 @ SV79 @ SV189 @ SV332 @ SY809 ) )
| ( eq_p @ SV332 @ SY809 ) ) )
= $true )
| ( ( hausdorff @ SV79 @ SV189 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[977]) ).
thf(1032,plain,
! [SV193: $i,SV83: $i,SV342: $i] :
( ( ( ~ ( neighborhood @ SV342 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 ) )
= $true )
| ( ( ! [SY811: $i] :
( ~ ( neighborhood @ SY811 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ~ ( disjoint_s @ SV342 @ SY811 ) ) )
= $true )
| ( ( topological_space @ SV83 @ SV193 )
= $false )
| ( ( hausdorff @ SV83 @ SV193 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[979]) ).
thf(1033,plain,
! [SV333: $i,SV310: $i,SV195: $i,SV85: $i] :
( ( ( equal_sets @ SV85 @ empty_set )
= $false )
| ( ( separation @ SV85 @ SV195 @ SV310 @ SV333 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[981]) ).
thf(1034,plain,
! [SV334: $i,SV311: $i,SV86: $i,SV196: $i] :
( ( ( equal_sets @ SV196 @ empty_set )
= $false )
| ( ( separation @ SV86 @ SV196 @ SV311 @ SV334 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[982]) ).
thf(1035,plain,
! [SV201: $i,SV91: $i,SV315: $i,SV269: $i] :
( ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( ( equal_sets @ SV91 @ empty_set )
| ( equal_sets @ SV201 @ empty_set )
| ~ ( element_of_collection @ SV91 @ SV315 )
| ~ ( element_of_collection @ SV201 @ SV315 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[987]) ).
thf(1036,plain,
! [SV203: $i,SV93: $i,SV343: $i,SV337: $i] :
( ( ( separation @ SV337 @ SV343 @ SV93 @ SV203 )
= $false )
| ( ( connected_space @ SV93 @ SV203 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[988]) ).
thf(1037,plain,
! [SV272: $i,SV208: $i,SV98: $i] :
( ( ( ~ ( subset_sets @ SV98 @ SV208 ) )
= $true )
| ( ( ~ ( connected_space @ SV98 @ ( subspace_topology @ SV208 @ SV272 @ SV98 ) ) )
= $true )
| ( ( topological_space @ SV208 @ SV272 )
= $false )
| ( ( connected_set @ SV98 @ SV208 @ SV272 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[989]) ).
thf(1038,plain,
! [SV212: $i,SV275: $i,SV102: $i] :
( ( ( ~ ( subset_collections @ SV102 @ SV275 ) )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV102 ) @ SV212 ) )
= $true )
| ( ( topological_space @ SV212 @ SV275 )
= $false )
| ( ( open_covering @ SV102 @ SV212 @ SV275 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[990]) ).
thf(1039,plain,
! [SV214: $i,SV104: $i,SV338: $i] :
( ( ( open_covering @ SV338 @ SV104 @ SV214 )
= $false )
| ( ( finite @ ( f23 @ SV104 @ SV214 @ SV338 ) )
= $true )
| ( ( compact_space @ SV104 @ SV214 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[991]) ).
thf(1040,plain,
! [SV215: $i,SV105: $i,SV339: $i] :
( ( ( open_covering @ SV339 @ SV105 @ SV215 )
= $false )
| ( ( subset_collections @ ( f23 @ SV105 @ SV215 @ SV339 ) @ SV339 )
= $true )
| ( ( compact_space @ SV105 @ SV215 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[992]) ).
thf(1041,plain,
! [SV216: $i,SV106: $i,SV340: $i] :
( ( ( open_covering @ SV340 @ SV106 @ SV216 )
= $false )
| ( ( open_covering @ ( f23 @ SV106 @ SV216 @ SV340 ) @ SV106 @ SV216 )
= $true )
| ( ( compact_space @ SV106 @ SV216 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[993]) ).
thf(1042,plain,
! [SV218: $i,SV108: $i,SV344: $i] :
( ( ( ~ ( finite @ SV344 ) )
= $true )
| ( ( ~ ( subset_collections @ SV344 @ ( f24 @ SV108 @ SV218 ) )
| ~ ( open_covering @ SV344 @ SV108 @ SV218 ) )
= $true )
| ( ( topological_space @ SV108 @ SV218 )
= $false )
| ( ( compact_space @ SV108 @ SV218 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[994]) ).
thf(1043,plain,
! [SV278: $i,SV222: $i,SV112: $i] :
( ( ( ~ ( subset_sets @ SV112 @ SV222 ) )
= $true )
| ( ( ~ ( compact_space @ SV112 @ ( subspace_topology @ SV222 @ SV278 @ SV112 ) ) )
= $true )
| ( ( topological_space @ SV222 @ SV278 )
= $false )
| ( ( compact_set @ SV112 @ SV222 @ SV278 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[995]) ).
thf(1044,plain,
! [SV13: $i,SV320: $i,SV123: $i,SV348: $i] :
( ( ( ~ ( element_of_collection @ SV348 @ SV123 )
| ( element_of_collection @ ( intersection_of_sets @ SV320 @ SV348 ) @ SV123 ) )
= $true )
| ( ( element_of_collection @ SV320 @ SV123 )
= $false )
| ( ( topological_space @ SV13 @ SV123 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[998]) ).
thf(1045,plain,
! [SV125: $i,SV15: $i] :
( ( ( ~ ( element_of_collection @ SV15 @ SV125 ) )
= $true )
| ( ( ( element_of_collection @ ( f3 @ SV15 @ SV125 ) @ SV125 )
| ( subset_collections @ ( f5 @ SV15 @ SV125 ) @ SV125 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV125 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV125 ) @ SV15 )
= $false )
| ( ( topological_space @ SV15 @ SV125 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1000]) ).
thf(1046,plain,
! [SV126: $i,SV16: $i] :
( ( ( ~ ( element_of_collection @ SV16 @ SV126 ) )
= $true )
| ( ( ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV126 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 )
= $false )
| ( ( topological_space @ SV16 @ SV126 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1001]) ).
thf(1047,plain,
! [SV127: $i,SV17: $i] :
( ( ( ~ ( element_of_collection @ SV17 @ SV127 ) )
= $true )
| ( ( ( element_of_collection @ ( f4 @ SV17 @ SV127 ) @ SV127 )
| ( subset_collections @ ( f5 @ SV17 @ SV127 ) @ SV127 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV127 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV127 ) @ SV17 )
= $false )
| ( ( topological_space @ SV17 @ SV127 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1002]) ).
thf(1048,plain,
! [SV128: $i,SV18: $i] :
( ( ( ~ ( element_of_collection @ SV18 @ SV128 ) )
= $true )
| ( ( ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV128 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 )
= $false )
| ( ( topological_space @ SV18 @ SV128 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1003]) ).
thf(1049,plain,
! [SV129: $i,SV19: $i] :
( ( ( ~ ( element_of_collection @ SV19 @ SV129 ) )
= $true )
| ( ( ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 )
| ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV129 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 )
= $false )
| ( ( topological_space @ SV19 @ SV129 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1004]) ).
thf(1050,plain,
! [SV130: $i,SV20: $i] :
( ( ( ~ ( element_of_collection @ SV20 @ SV130 ) )
= $true )
| ( ( ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV130 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
= $false )
| ( ( topological_space @ SV20 @ SV130 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1005]) ).
thf(1051,plain,
! [SV30: $i,SV140: $i,SV231: $i] :
( ( ( topological_space @ SV231 @ SV140 )
= $false )
| ( ( ~ ( subset_collections @ SV140 @ SV30 ) )
= $true )
| ( ( topological_space @ SV231 @ SV30 )
= $false )
| ( ( finer @ SV30 @ SV140 @ SV231 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1008]) ).
thf(1052,plain,
! [SV32: $i,SV232: $i,SV142: $i,SV349: $i] :
( ( ( ~ ( element_of_collection @ SV349 @ SV142 )
| ! [SY812: $i] :
( ~ ( element_of_collection @ SY812 @ SV142 )
| ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SV349 @ SY812 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SV349 @ SY812 ) ) ) )
= $true )
| ( ( element_of_set @ SV232 @ SV32 )
= $false )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[1009]) ).
thf(1053,plain,
! [SV33: $i,SV233: $i,SV143: $i,SV350: $i] :
( ( ( ~ ( element_of_collection @ SV350 @ SV143 )
| ! [SY813: $i] :
( ~ ( element_of_collection @ SY813 @ SV143 )
| ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SV350 @ SY813 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SV350 @ SY813 ) @ SV143 ) ) )
= $true )
| ( ( element_of_set @ SV233 @ SV33 )
= $false )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[1010]) ).
thf(1054,plain,
! [SV34: $i,SV234: $i,SV144: $i,SV351: $i] :
( ( ( ~ ( element_of_collection @ SV351 @ SV144 )
| ! [SY814: $i] :
( ~ ( element_of_collection @ SY814 @ SV144 )
| ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SV351 @ SY814 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SV351 @ SY814 ) @ ( intersection_of_sets @ SV351 @ SY814 ) ) ) )
= $true )
| ( ( element_of_set @ SV234 @ SV34 )
= $false )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[1011]) ).
thf(1055,plain,
! [SV341: $i,SV149: $i,SV39: $i] :
( ( ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SV341 )
= $false )
| ( ( ~ ( element_of_collection @ SV341 @ SV149 )
| ~ ( subset_sets @ SV341 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) ) )
= $true )
| ( ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 )
= $false )
| ( ( basis @ SV39 @ SV149 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1012]) ).
thf(1056,plain,
! [SV44: $i,SV154: $i,SV281: $i] :
( ( ( element_of_collection @ SV281 @ SV154 )
= $false )
| ( ( ~ ( subset_sets @ SV281 @ SV44 ) )
= $true )
| ( ( element_of_set @ ( f11 @ SV154 @ SV44 ) @ SV281 )
= $false )
| ( ( element_of_collection @ SV44 @ ( top_of_basis @ SV154 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1016]) ).
thf(1057,plain,
! [SV49: $i,SV239: $i,SV159: $i,SV285: $i] :
( ( ( ~ ( subset_sets @ SV285 @ SV159 ) )
= $true )
| ( ( ! [SY796: $i] :
( ~ ( element_of_collection @ SY796 @ SV239 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SY796 ) ) ) )
= $true )
| ( ( topological_space @ SV159 @ SV239 )
= $false )
| ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1017]) ).
thf(1058,plain,
! [SV290: $i,SV55: $i,SV245: $i,SV165: $i] :
( ( ( ~ ( subset_sets @ SV165 @ SV245 ) )
= $true )
| ( ( ! [SY797: $i] :
( ~ ( element_of_set @ SV55 @ SY797 )
| ~ ( subset_sets @ SY797 @ SV165 )
| ~ ( open @ SY797 @ SV245 @ SV290 ) ) )
= $true )
| ( ( topological_space @ SV245 @ SV290 )
= $false )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1018]) ).
thf(1059,plain,
! [SV58: $i,SV292: $i,SV248: $i,SV345: $i,SV168: $i] :
( ( ( ~ ( subset_sets @ SV168 @ SV345 ) )
= $true )
| ( ( ~ ( closed @ SV345 @ SV248 @ SV292 )
| ( element_of_set @ SV58 @ SV345 ) )
= $true )
| ( ( element_of_set @ SV58 @ ( closure @ SV168 @ SV248 @ SV292 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1019]) ).
thf(1060,plain,
! [SV59: $i,SV293: $i,SV249: $i,SV169: $i] :
( ( ( ~ ( subset_sets @ SV169 @ SV249 ) )
= $true )
| ( ( subset_sets @ SV169 @ ( f14 @ SV169 @ SV249 @ SV293 @ SV59 ) )
= $true )
| ( ( topological_space @ SV249 @ SV293 )
= $false )
| ( ( element_of_set @ SV59 @ ( closure @ SV169 @ SV249 @ SV293 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1020]) ).
thf(1061,plain,
! [SV60: $i,SV294: $i,SV250: $i,SV170: $i] :
( ( ( ~ ( subset_sets @ SV170 @ SV250 ) )
= $true )
| ( ( closed @ ( f14 @ SV170 @ SV250 @ SV294 @ SV60 ) @ SV250 @ SV294 )
= $true )
| ( ( topological_space @ SV250 @ SV294 )
= $false )
| ( ( element_of_set @ SV60 @ ( closure @ SV170 @ SV250 @ SV294 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1021]) ).
thf(1062,plain,
! [SV295: $i,SV61: $i,SV251: $i,SV171: $i] :
( ( ( ~ ( subset_sets @ SV171 @ SV251 ) )
= $true )
| ( ( ~ ( element_of_set @ SV61 @ ( f14 @ SV171 @ SV251 @ SV295 @ SV61 ) ) )
= $true )
| ( ( topological_space @ SV251 @ SV295 )
= $false )
| ( ( element_of_set @ SV61 @ ( closure @ SV171 @ SV251 @ SV295 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1022]) ).
thf(1063,plain,
! [SV175: $i,SV299: $i,SV254: $i,SV65: $i] :
( ( ( ~ ( open @ SV65 @ SV254 @ SV299 ) )
= $true )
| ( ( ~ ( element_of_set @ SV175 @ SV65 ) )
= $true )
| ( ( topological_space @ SV254 @ SV299 )
= $false )
| ( ( neighborhood @ SV65 @ SV175 @ SV254 @ SV299 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1023]) ).
thf(1064,plain,
! [SV178: $i,SV301: $i,SV257: $i,SV68: $i,SV346: $i] :
( ( ( ~ ( neighborhood @ SV346 @ SV68 @ SV257 @ SV301 ) )
= $true )
| ( ( element_of_set @ ( f15 @ SV68 @ SV178 @ SV257 @ SV301 @ SV346 ) @ ( intersection_of_sets @ SV346 @ SV178 ) )
= $true )
| ( ( limit_point @ SV68 @ SV178 @ SV257 @ SV301 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1024]) ).
thf(1065,plain,
! [SV179: $i,SV302: $i,SV258: $i,SV69: $i,SV347: $i] :
( ( ( ~ ( neighborhood @ SV347 @ SV69 @ SV258 @ SV302 ) )
= $true )
| ( ( ~ ( eq_p @ ( f15 @ SV69 @ SV179 @ SV258 @ SV302 @ SV347 ) @ SV69 ) )
= $true )
| ( ( limit_point @ SV69 @ SV179 @ SV258 @ SV302 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1025]) ).
thf(1066,plain,
! [SV303: $i,SV70: $i,SV259: $i,SV180: $i] :
( ( ( ~ ( subset_sets @ SV180 @ SV259 ) )
= $true )
| ( ( neighborhood @ ( f16 @ SV70 @ SV180 @ SV259 @ SV303 ) @ SV70 @ SV259 @ SV303 )
= $true )
| ( ( topological_space @ SV259 @ SV303 )
= $false )
| ( ( limit_point @ SV70 @ SV180 @ SV259 @ SV303 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1026]) ).
thf(1067,plain,
! [SV304: $i,SV71: $i,SV260: $i,SV181: $i] :
( ( ( ~ ( subset_sets @ SV181 @ SV260 ) )
= $true )
| ( ( ! [SY802: $i] :
( ~ ( element_of_set @ SY802 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SV304 ) @ SV181 ) )
| ( eq_p @ SY802 @ SV71 ) ) )
= $true )
| ( ( topological_space @ SV260 @ SV304 )
= $false )
| ( ( limit_point @ SV71 @ SV181 @ SV260 @ SV304 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1027]) ).
thf(1068,plain,
! [SV308: $i,SV264: $i,SV185: $i,SV75: $i] :
( ( ( ~ ( element_of_set @ SV75 @ ( closure @ SV185 @ SV264 @ SV308 ) ) )
= $true )
| ( ( ~ ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SV185 @ SV264 ) @ SV264 @ SV308 ) ) )
= $true )
| ( ( topological_space @ SV264 @ SV308 )
= $false )
| ( ( element_of_set @ SV75 @ ( boundary @ SV185 @ SV264 @ SV308 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1028]) ).
thf(1069,plain,
! [SV187: $i,SV330: $i,SV77: $i,SV352: $i] :
( ( ( ~ ( element_of_set @ SV352 @ SV77 )
| ( eq_p @ SV330 @ SV352 )
| ( neighborhood @ ( f17 @ SV77 @ SV187 @ SV330 @ SV352 ) @ SV330 @ SV77 @ SV187 ) )
= $true )
| ( ( element_of_set @ SV330 @ SV77 )
= $false )
| ( ( hausdorff @ SV77 @ SV187 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[1029]) ).
thf(1070,plain,
! [SV188: $i,SV331: $i,SV78: $i,SV353: $i] :
( ( ( ~ ( element_of_set @ SV353 @ SV78 )
| ( eq_p @ SV331 @ SV353 )
| ( neighborhood @ ( f18 @ SV78 @ SV188 @ SV331 @ SV353 ) @ SV353 @ SV78 @ SV188 ) )
= $true )
| ( ( element_of_set @ SV331 @ SV78 )
= $false )
| ( ( hausdorff @ SV78 @ SV188 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[1030]) ).
thf(1071,plain,
! [SV332: $i,SV189: $i,SV79: $i,SV354: $i] :
( ( ( ~ ( element_of_set @ SV354 @ SV79 )
| ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SV332 @ SV354 ) @ ( f18 @ SV79 @ SV189 @ SV332 @ SV354 ) )
| ( eq_p @ SV332 @ SV354 ) )
= $true )
| ( ( element_of_set @ SV332 @ SV79 )
= $false )
| ( ( hausdorff @ SV79 @ SV189 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[1031]) ).
thf(1072,plain,
! [SV193: $i,SV83: $i,SV342: $i] :
( ( ( neighborhood @ SV342 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
= $false )
| ( ( ! [SY811: $i] :
( ~ ( neighborhood @ SY811 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ~ ( disjoint_s @ SV342 @ SY811 ) ) )
= $true )
| ( ( topological_space @ SV83 @ SV193 )
= $false )
| ( ( hausdorff @ SV83 @ SV193 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1032]) ).
thf(1073,plain,
! [SV269: $i,SV315: $i,SV201: $i,SV91: $i] :
( ( ( equal_sets @ SV91 @ empty_set )
= $true )
| ( ( ( equal_sets @ SV201 @ empty_set )
| ~ ( element_of_collection @ SV91 @ SV315 )
| ~ ( element_of_collection @ SV201 @ SV315 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1035]) ).
thf(1074,plain,
! [SV272: $i,SV208: $i,SV98: $i] :
( ( ( subset_sets @ SV98 @ SV208 )
= $false )
| ( ( ~ ( connected_space @ SV98 @ ( subspace_topology @ SV208 @ SV272 @ SV98 ) ) )
= $true )
| ( ( topological_space @ SV208 @ SV272 )
= $false )
| ( ( connected_set @ SV98 @ SV208 @ SV272 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1037]) ).
thf(1075,plain,
! [SV212: $i,SV275: $i,SV102: $i] :
( ( ( subset_collections @ SV102 @ SV275 )
= $false )
| ( ( ~ ( equal_sets @ ( union_of_members @ SV102 ) @ SV212 ) )
= $true )
| ( ( topological_space @ SV212 @ SV275 )
= $false )
| ( ( open_covering @ SV102 @ SV212 @ SV275 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1038]) ).
thf(1076,plain,
! [SV218: $i,SV108: $i,SV344: $i] :
( ( ( finite @ SV344 )
= $false )
| ( ( ~ ( subset_collections @ SV344 @ ( f24 @ SV108 @ SV218 ) )
| ~ ( open_covering @ SV344 @ SV108 @ SV218 ) )
= $true )
| ( ( topological_space @ SV108 @ SV218 )
= $false )
| ( ( compact_space @ SV108 @ SV218 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1042]) ).
thf(1077,plain,
! [SV278: $i,SV222: $i,SV112: $i] :
( ( ( subset_sets @ SV112 @ SV222 )
= $false )
| ( ( ~ ( compact_space @ SV112 @ ( subspace_topology @ SV222 @ SV278 @ SV112 ) ) )
= $true )
| ( ( topological_space @ SV222 @ SV278 )
= $false )
| ( ( compact_set @ SV112 @ SV222 @ SV278 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1043]) ).
thf(1078,plain,
! [SV13: $i,SV320: $i,SV123: $i,SV348: $i] :
( ( ( ~ ( element_of_collection @ SV348 @ SV123 ) )
= $true )
| ( ( element_of_collection @ ( intersection_of_sets @ SV320 @ SV348 ) @ SV123 )
= $true )
| ( ( element_of_collection @ SV320 @ SV123 )
= $false )
| ( ( topological_space @ SV13 @ SV123 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1044]) ).
thf(1079,plain,
! [SV125: $i,SV15: $i] :
( ( ( element_of_collection @ SV15 @ SV125 )
= $false )
| ( ( ( element_of_collection @ ( f3 @ SV15 @ SV125 ) @ SV125 )
| ( subset_collections @ ( f5 @ SV15 @ SV125 ) @ SV125 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV125 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV125 ) @ SV15 )
= $false )
| ( ( topological_space @ SV15 @ SV125 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1045]) ).
thf(1080,plain,
! [SV126: $i,SV16: $i] :
( ( ( element_of_collection @ SV16 @ SV126 )
= $false )
| ( ( ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV126 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 )
= $false )
| ( ( topological_space @ SV16 @ SV126 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1046]) ).
thf(1081,plain,
! [SV127: $i,SV17: $i] :
( ( ( element_of_collection @ SV17 @ SV127 )
= $false )
| ( ( ( element_of_collection @ ( f4 @ SV17 @ SV127 ) @ SV127 )
| ( subset_collections @ ( f5 @ SV17 @ SV127 ) @ SV127 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV127 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV127 ) @ SV17 )
= $false )
| ( ( topological_space @ SV17 @ SV127 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1047]) ).
thf(1082,plain,
! [SV128: $i,SV18: $i] :
( ( ( element_of_collection @ SV18 @ SV128 )
= $false )
| ( ( ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV128 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 )
= $false )
| ( ( topological_space @ SV18 @ SV128 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1048]) ).
thf(1083,plain,
! [SV129: $i,SV19: $i] :
( ( ( element_of_collection @ SV19 @ SV129 )
= $false )
| ( ( ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 )
| ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV129 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 )
= $false )
| ( ( topological_space @ SV19 @ SV129 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1049]) ).
thf(1084,plain,
! [SV130: $i,SV20: $i] :
( ( ( element_of_collection @ SV20 @ SV130 )
= $false )
| ( ( ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
| ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true )
| ( ( element_of_collection @ empty_set @ SV130 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
= $false )
| ( ( topological_space @ SV20 @ SV130 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1050]) ).
thf(1085,plain,
! [SV231: $i,SV30: $i,SV140: $i] :
( ( ( subset_collections @ SV140 @ SV30 )
= $false )
| ( ( topological_space @ SV231 @ SV140 )
= $false )
| ( ( topological_space @ SV231 @ SV30 )
= $false )
| ( ( finer @ SV30 @ SV140 @ SV231 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1051]) ).
thf(1086,plain,
! [SV32: $i,SV232: $i,SV142: $i,SV349: $i] :
( ( ( ~ ( element_of_collection @ SV349 @ SV142 ) )
= $true )
| ( ( ! [SY812: $i] :
( ~ ( element_of_collection @ SY812 @ SV142 )
| ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SV349 @ SY812 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SV349 @ SY812 ) ) ) )
= $true )
| ( ( element_of_set @ SV232 @ SV32 )
= $false )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1052]) ).
thf(1087,plain,
! [SV33: $i,SV233: $i,SV143: $i,SV350: $i] :
( ( ( ~ ( element_of_collection @ SV350 @ SV143 ) )
= $true )
| ( ( ! [SY813: $i] :
( ~ ( element_of_collection @ SY813 @ SV143 )
| ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SV350 @ SY813 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SV350 @ SY813 ) @ SV143 ) ) )
= $true )
| ( ( element_of_set @ SV233 @ SV33 )
= $false )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1053]) ).
thf(1088,plain,
! [SV34: $i,SV234: $i,SV144: $i,SV351: $i] :
( ( ( ~ ( element_of_collection @ SV351 @ SV144 ) )
= $true )
| ( ( ! [SY814: $i] :
( ~ ( element_of_collection @ SY814 @ SV144 )
| ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SV351 @ SY814 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SV351 @ SY814 ) @ ( intersection_of_sets @ SV351 @ SY814 ) ) ) )
= $true )
| ( ( element_of_set @ SV234 @ SV34 )
= $false )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1054]) ).
thf(1089,plain,
! [SV39: $i,SV149: $i,SV341: $i] :
( ( ( ~ ( element_of_collection @ SV341 @ SV149 ) )
= $true )
| ( ( ~ ( subset_sets @ SV341 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) ) )
= $true )
| ( ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SV341 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 )
= $false )
| ( ( basis @ SV39 @ SV149 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1055]) ).
thf(1090,plain,
! [SV154: $i,SV44: $i,SV281: $i] :
( ( ( subset_sets @ SV281 @ SV44 )
= $false )
| ( ( element_of_collection @ SV281 @ SV154 )
= $false )
| ( ( element_of_set @ ( f11 @ SV154 @ SV44 ) @ SV281 )
= $false )
| ( ( element_of_collection @ SV44 @ ( top_of_basis @ SV154 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1056]) ).
thf(1091,plain,
! [SV49: $i,SV239: $i,SV159: $i,SV285: $i] :
( ( ( subset_sets @ SV285 @ SV159 )
= $false )
| ( ( ! [SY796: $i] :
( ~ ( element_of_collection @ SY796 @ SV239 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SY796 ) ) ) )
= $true )
| ( ( topological_space @ SV159 @ SV239 )
= $false )
| ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1057]) ).
thf(1092,plain,
! [SV290: $i,SV55: $i,SV245: $i,SV165: $i] :
( ( ( subset_sets @ SV165 @ SV245 )
= $false )
| ( ( ! [SY797: $i] :
( ~ ( element_of_set @ SV55 @ SY797 )
| ~ ( subset_sets @ SY797 @ SV165 )
| ~ ( open @ SY797 @ SV245 @ SV290 ) ) )
= $true )
| ( ( topological_space @ SV245 @ SV290 )
= $false )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1058]) ).
thf(1093,plain,
! [SV58: $i,SV292: $i,SV248: $i,SV345: $i,SV168: $i] :
( ( ( subset_sets @ SV168 @ SV345 )
= $false )
| ( ( ~ ( closed @ SV345 @ SV248 @ SV292 )
| ( element_of_set @ SV58 @ SV345 ) )
= $true )
| ( ( element_of_set @ SV58 @ ( closure @ SV168 @ SV248 @ SV292 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1059]) ).
thf(1094,plain,
! [SV59: $i,SV293: $i,SV249: $i,SV169: $i] :
( ( ( subset_sets @ SV169 @ SV249 )
= $false )
| ( ( subset_sets @ SV169 @ ( f14 @ SV169 @ SV249 @ SV293 @ SV59 ) )
= $true )
| ( ( topological_space @ SV249 @ SV293 )
= $false )
| ( ( element_of_set @ SV59 @ ( closure @ SV169 @ SV249 @ SV293 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1060]) ).
thf(1095,plain,
! [SV60: $i,SV294: $i,SV250: $i,SV170: $i] :
( ( ( subset_sets @ SV170 @ SV250 )
= $false )
| ( ( closed @ ( f14 @ SV170 @ SV250 @ SV294 @ SV60 ) @ SV250 @ SV294 )
= $true )
| ( ( topological_space @ SV250 @ SV294 )
= $false )
| ( ( element_of_set @ SV60 @ ( closure @ SV170 @ SV250 @ SV294 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1061]) ).
thf(1096,plain,
! [SV295: $i,SV61: $i,SV251: $i,SV171: $i] :
( ( ( subset_sets @ SV171 @ SV251 )
= $false )
| ( ( ~ ( element_of_set @ SV61 @ ( f14 @ SV171 @ SV251 @ SV295 @ SV61 ) ) )
= $true )
| ( ( topological_space @ SV251 @ SV295 )
= $false )
| ( ( element_of_set @ SV61 @ ( closure @ SV171 @ SV251 @ SV295 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1062]) ).
thf(1097,plain,
! [SV175: $i,SV299: $i,SV254: $i,SV65: $i] :
( ( ( open @ SV65 @ SV254 @ SV299 )
= $false )
| ( ( ~ ( element_of_set @ SV175 @ SV65 ) )
= $true )
| ( ( topological_space @ SV254 @ SV299 )
= $false )
| ( ( neighborhood @ SV65 @ SV175 @ SV254 @ SV299 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1063]) ).
thf(1098,plain,
! [SV178: $i,SV301: $i,SV257: $i,SV68: $i,SV346: $i] :
( ( ( neighborhood @ SV346 @ SV68 @ SV257 @ SV301 )
= $false )
| ( ( element_of_set @ ( f15 @ SV68 @ SV178 @ SV257 @ SV301 @ SV346 ) @ ( intersection_of_sets @ SV346 @ SV178 ) )
= $true )
| ( ( limit_point @ SV68 @ SV178 @ SV257 @ SV301 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1064]) ).
thf(1099,plain,
! [SV179: $i,SV302: $i,SV258: $i,SV69: $i,SV347: $i] :
( ( ( neighborhood @ SV347 @ SV69 @ SV258 @ SV302 )
= $false )
| ( ( ~ ( eq_p @ ( f15 @ SV69 @ SV179 @ SV258 @ SV302 @ SV347 ) @ SV69 ) )
= $true )
| ( ( limit_point @ SV69 @ SV179 @ SV258 @ SV302 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1065]) ).
thf(1100,plain,
! [SV303: $i,SV70: $i,SV259: $i,SV180: $i] :
( ( ( subset_sets @ SV180 @ SV259 )
= $false )
| ( ( neighborhood @ ( f16 @ SV70 @ SV180 @ SV259 @ SV303 ) @ SV70 @ SV259 @ SV303 )
= $true )
| ( ( topological_space @ SV259 @ SV303 )
= $false )
| ( ( limit_point @ SV70 @ SV180 @ SV259 @ SV303 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1066]) ).
thf(1101,plain,
! [SV304: $i,SV71: $i,SV260: $i,SV181: $i] :
( ( ( subset_sets @ SV181 @ SV260 )
= $false )
| ( ( ! [SY802: $i] :
( ~ ( element_of_set @ SY802 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SV304 ) @ SV181 ) )
| ( eq_p @ SY802 @ SV71 ) ) )
= $true )
| ( ( topological_space @ SV260 @ SV304 )
= $false )
| ( ( limit_point @ SV71 @ SV181 @ SV260 @ SV304 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1067]) ).
thf(1102,plain,
! [SV308: $i,SV264: $i,SV185: $i,SV75: $i] :
( ( ( element_of_set @ SV75 @ ( closure @ SV185 @ SV264 @ SV308 ) )
= $false )
| ( ( ~ ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SV185 @ SV264 ) @ SV264 @ SV308 ) ) )
= $true )
| ( ( topological_space @ SV264 @ SV308 )
= $false )
| ( ( element_of_set @ SV75 @ ( boundary @ SV185 @ SV264 @ SV308 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1068]) ).
thf(1103,plain,
! [SV187: $i,SV330: $i,SV77: $i,SV352: $i] :
( ( ( ~ ( element_of_set @ SV352 @ SV77 ) )
= $true )
| ( ( ( eq_p @ SV330 @ SV352 )
| ( neighborhood @ ( f17 @ SV77 @ SV187 @ SV330 @ SV352 ) @ SV330 @ SV77 @ SV187 ) )
= $true )
| ( ( element_of_set @ SV330 @ SV77 )
= $false )
| ( ( hausdorff @ SV77 @ SV187 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1069]) ).
thf(1104,plain,
! [SV188: $i,SV331: $i,SV78: $i,SV353: $i] :
( ( ( ~ ( element_of_set @ SV353 @ SV78 ) )
= $true )
| ( ( ( eq_p @ SV331 @ SV353 )
| ( neighborhood @ ( f18 @ SV78 @ SV188 @ SV331 @ SV353 ) @ SV353 @ SV78 @ SV188 ) )
= $true )
| ( ( element_of_set @ SV331 @ SV78 )
= $false )
| ( ( hausdorff @ SV78 @ SV188 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1070]) ).
thf(1105,plain,
! [SV332: $i,SV189: $i,SV79: $i,SV354: $i] :
( ( ( ~ ( element_of_set @ SV354 @ SV79 ) )
= $true )
| ( ( ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SV332 @ SV354 ) @ ( f18 @ SV79 @ SV189 @ SV332 @ SV354 ) )
| ( eq_p @ SV332 @ SV354 ) )
= $true )
| ( ( element_of_set @ SV332 @ SV79 )
= $false )
| ( ( hausdorff @ SV79 @ SV189 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1071]) ).
thf(1106,plain,
! [SV342: $i,SV193: $i,SV83: $i,SV355: $i] :
( ( ( ~ ( neighborhood @ SV355 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
| ~ ( disjoint_s @ SV342 @ SV355 ) )
= $true )
| ( ( neighborhood @ SV342 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
= $false )
| ( ( topological_space @ SV83 @ SV193 )
= $false )
| ( ( hausdorff @ SV83 @ SV193 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[1072]) ).
thf(1107,plain,
! [SV269: $i,SV315: $i,SV91: $i,SV201: $i] :
( ( ( equal_sets @ SV201 @ empty_set )
= $true )
| ( ( ~ ( element_of_collection @ SV91 @ SV315 )
| ~ ( element_of_collection @ SV201 @ SV315 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( equal_sets @ SV91 @ empty_set )
= $true )
| ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1073]) ).
thf(1108,plain,
! [SV272: $i,SV208: $i,SV98: $i] :
( ( ( connected_space @ SV98 @ ( subspace_topology @ SV208 @ SV272 @ SV98 ) )
= $false )
| ( ( subset_sets @ SV98 @ SV208 )
= $false )
| ( ( topological_space @ SV208 @ SV272 )
= $false )
| ( ( connected_set @ SV98 @ SV208 @ SV272 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1074]) ).
thf(1109,plain,
! [SV275: $i,SV212: $i,SV102: $i] :
( ( ( equal_sets @ ( union_of_members @ SV102 ) @ SV212 )
= $false )
| ( ( subset_collections @ SV102 @ SV275 )
= $false )
| ( ( topological_space @ SV212 @ SV275 )
= $false )
| ( ( open_covering @ SV102 @ SV212 @ SV275 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1075]) ).
thf(1110,plain,
! [SV218: $i,SV108: $i,SV344: $i] :
( ( ( ~ ( subset_collections @ SV344 @ ( f24 @ SV108 @ SV218 ) ) )
= $true )
| ( ( ~ ( open_covering @ SV344 @ SV108 @ SV218 ) )
= $true )
| ( ( finite @ SV344 )
= $false )
| ( ( topological_space @ SV108 @ SV218 )
= $false )
| ( ( compact_space @ SV108 @ SV218 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1076]) ).
thf(1111,plain,
! [SV278: $i,SV222: $i,SV112: $i] :
( ( ( compact_space @ SV112 @ ( subspace_topology @ SV222 @ SV278 @ SV112 ) )
= $false )
| ( ( subset_sets @ SV112 @ SV222 )
= $false )
| ( ( topological_space @ SV222 @ SV278 )
= $false )
| ( ( compact_set @ SV112 @ SV222 @ SV278 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1077]) ).
thf(1112,plain,
! [SV13: $i,SV320: $i,SV123: $i,SV348: $i] :
( ( ( element_of_collection @ SV348 @ SV123 )
= $false )
| ( ( element_of_collection @ ( intersection_of_sets @ SV320 @ SV348 ) @ SV123 )
= $true )
| ( ( element_of_collection @ SV320 @ SV123 )
= $false )
| ( ( topological_space @ SV13 @ SV123 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1078]) ).
thf(1113,plain,
! [SV125: $i,SV15: $i] :
( ( ( element_of_collection @ ( f3 @ SV15 @ SV125 ) @ SV125 )
= $true )
| ( ( subset_collections @ ( f5 @ SV15 @ SV125 ) @ SV125 )
= $true )
| ( ( element_of_collection @ SV15 @ SV125 )
= $false )
| ( ( element_of_collection @ empty_set @ SV125 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV125 ) @ SV15 )
= $false )
| ( ( topological_space @ SV15 @ SV125 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1079]) ).
thf(1114,plain,
! [SV126: $i,SV16: $i] :
( ( ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
= $true )
| ( ( ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 ) )
= $true )
| ( ( element_of_collection @ SV16 @ SV126 )
= $false )
| ( ( element_of_collection @ empty_set @ SV126 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 )
= $false )
| ( ( topological_space @ SV16 @ SV126 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1080]) ).
thf(1115,plain,
! [SV127: $i,SV17: $i] :
( ( ( element_of_collection @ ( f4 @ SV17 @ SV127 ) @ SV127 )
= $true )
| ( ( subset_collections @ ( f5 @ SV17 @ SV127 ) @ SV127 )
= $true )
| ( ( element_of_collection @ SV17 @ SV127 )
= $false )
| ( ( element_of_collection @ empty_set @ SV127 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV127 ) @ SV17 )
= $false )
| ( ( topological_space @ SV17 @ SV127 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1081]) ).
thf(1116,plain,
! [SV128: $i,SV18: $i] :
( ( ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
= $true )
| ( ( ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 ) )
= $true )
| ( ( element_of_collection @ SV18 @ SV128 )
= $false )
| ( ( element_of_collection @ empty_set @ SV128 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 )
= $false )
| ( ( topological_space @ SV18 @ SV128 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1082]) ).
thf(1117,plain,
! [SV129: $i,SV19: $i] :
( ( ( ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 ) )
= $true )
| ( ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 )
= $true )
| ( ( element_of_collection @ SV19 @ SV129 )
= $false )
| ( ( element_of_collection @ empty_set @ SV129 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 )
= $false )
| ( ( topological_space @ SV19 @ SV129 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1083]) ).
thf(1118,plain,
! [SV130: $i,SV20: $i] :
( ( ( ~ ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true )
| ( ( ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true )
| ( ( element_of_collection @ SV20 @ SV130 )
= $false )
| ( ( element_of_collection @ empty_set @ SV130 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
= $false )
| ( ( topological_space @ SV20 @ SV130 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1084]) ).
thf(1119,plain,
! [SV32: $i,SV232: $i,SV142: $i,SV349: $i] :
( ( ( element_of_collection @ SV349 @ SV142 )
= $false )
| ( ( ! [SY812: $i] :
( ~ ( element_of_collection @ SY812 @ SV142 )
| ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SV349 @ SY812 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SV349 @ SY812 ) ) ) )
= $true )
| ( ( element_of_set @ SV232 @ SV32 )
= $false )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1086]) ).
thf(1120,plain,
! [SV33: $i,SV233: $i,SV143: $i,SV350: $i] :
( ( ( element_of_collection @ SV350 @ SV143 )
= $false )
| ( ( ! [SY813: $i] :
( ~ ( element_of_collection @ SY813 @ SV143 )
| ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SV350 @ SY813 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SV350 @ SY813 ) @ SV143 ) ) )
= $true )
| ( ( element_of_set @ SV233 @ SV33 )
= $false )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1087]) ).
thf(1121,plain,
! [SV34: $i,SV234: $i,SV144: $i,SV351: $i] :
( ( ( element_of_collection @ SV351 @ SV144 )
= $false )
| ( ( ! [SY814: $i] :
( ~ ( element_of_collection @ SY814 @ SV144 )
| ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SV351 @ SY814 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SV351 @ SY814 ) @ ( intersection_of_sets @ SV351 @ SY814 ) ) ) )
= $true )
| ( ( element_of_set @ SV234 @ SV34 )
= $false )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1088]) ).
thf(1122,plain,
! [SV39: $i,SV149: $i,SV341: $i] :
( ( ( element_of_collection @ SV341 @ SV149 )
= $false )
| ( ( ~ ( subset_sets @ SV341 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) ) )
= $true )
| ( ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SV341 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 )
= $false )
| ( ( basis @ SV39 @ SV149 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1089]) ).
thf(1123,plain,
! [SV159: $i,SV285: $i,SV49: $i,SV239: $i,SV356: $i] :
( ( ( ~ ( element_of_collection @ SV356 @ SV239 )
| ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SV356 ) ) )
= $true )
| ( ( subset_sets @ SV285 @ SV159 )
= $false )
| ( ( topological_space @ SV159 @ SV239 )
= $false )
| ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[1091]) ).
thf(1124,plain,
! [SV290: $i,SV245: $i,SV165: $i,SV357: $i,SV55: $i] :
( ( ( ~ ( element_of_set @ SV55 @ SV357 )
| ~ ( subset_sets @ SV357 @ SV165 )
| ~ ( open @ SV357 @ SV245 @ SV290 ) )
= $true )
| ( ( subset_sets @ SV165 @ SV245 )
= $false )
| ( ( topological_space @ SV245 @ SV290 )
= $false )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[1092]) ).
thf(1125,plain,
! [SV168: $i,SV58: $i,SV292: $i,SV248: $i,SV345: $i] :
( ( ( ~ ( closed @ SV345 @ SV248 @ SV292 ) )
= $true )
| ( ( element_of_set @ SV58 @ SV345 )
= $true )
| ( ( subset_sets @ SV168 @ SV345 )
= $false )
| ( ( element_of_set @ SV58 @ ( closure @ SV168 @ SV248 @ SV292 ) )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1093]) ).
thf(1126,plain,
! [SV295: $i,SV251: $i,SV171: $i,SV61: $i] :
( ( ( element_of_set @ SV61 @ ( f14 @ SV171 @ SV251 @ SV295 @ SV61 ) )
= $false )
| ( ( subset_sets @ SV171 @ SV251 )
= $false )
| ( ( topological_space @ SV251 @ SV295 )
= $false )
| ( ( element_of_set @ SV61 @ ( closure @ SV171 @ SV251 @ SV295 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1096]) ).
thf(1127,plain,
! [SV299: $i,SV254: $i,SV65: $i,SV175: $i] :
( ( ( element_of_set @ SV175 @ SV65 )
= $false )
| ( ( open @ SV65 @ SV254 @ SV299 )
= $false )
| ( ( topological_space @ SV254 @ SV299 )
= $false )
| ( ( neighborhood @ SV65 @ SV175 @ SV254 @ SV299 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1097]) ).
thf(1128,plain,
! [SV347: $i,SV302: $i,SV258: $i,SV179: $i,SV69: $i] :
( ( ( eq_p @ ( f15 @ SV69 @ SV179 @ SV258 @ SV302 @ SV347 ) @ SV69 )
= $false )
| ( ( neighborhood @ SV347 @ SV69 @ SV258 @ SV302 )
= $false )
| ( ( limit_point @ SV69 @ SV179 @ SV258 @ SV302 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1099]) ).
thf(1129,plain,
! [SV304: $i,SV260: $i,SV181: $i,SV71: $i,SV358: $i] :
( ( ( ~ ( element_of_set @ SV358 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SV304 ) @ SV181 ) )
| ( eq_p @ SV358 @ SV71 ) )
= $true )
| ( ( subset_sets @ SV181 @ SV260 )
= $false )
| ( ( topological_space @ SV260 @ SV304 )
= $false )
| ( ( limit_point @ SV71 @ SV181 @ SV260 @ SV304 )
= $true ) ),
inference(extcnf_forall_pos,[status(thm)],[1101]) ).
thf(1130,plain,
! [SV308: $i,SV264: $i,SV185: $i,SV75: $i] :
( ( ( element_of_set @ SV75 @ ( closure @ ( relative_complement_sets @ SV185 @ SV264 ) @ SV264 @ SV308 ) )
= $false )
| ( ( element_of_set @ SV75 @ ( closure @ SV185 @ SV264 @ SV308 ) )
= $false )
| ( ( topological_space @ SV264 @ SV308 )
= $false )
| ( ( element_of_set @ SV75 @ ( boundary @ SV185 @ SV264 @ SV308 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1102]) ).
thf(1131,plain,
! [SV187: $i,SV330: $i,SV77: $i,SV352: $i] :
( ( ( element_of_set @ SV352 @ SV77 )
= $false )
| ( ( ( eq_p @ SV330 @ SV352 )
| ( neighborhood @ ( f17 @ SV77 @ SV187 @ SV330 @ SV352 ) @ SV330 @ SV77 @ SV187 ) )
= $true )
| ( ( element_of_set @ SV330 @ SV77 )
= $false )
| ( ( hausdorff @ SV77 @ SV187 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1103]) ).
thf(1132,plain,
! [SV188: $i,SV331: $i,SV78: $i,SV353: $i] :
( ( ( element_of_set @ SV353 @ SV78 )
= $false )
| ( ( ( eq_p @ SV331 @ SV353 )
| ( neighborhood @ ( f18 @ SV78 @ SV188 @ SV331 @ SV353 ) @ SV353 @ SV78 @ SV188 ) )
= $true )
| ( ( element_of_set @ SV331 @ SV78 )
= $false )
| ( ( hausdorff @ SV78 @ SV188 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1104]) ).
thf(1133,plain,
! [SV332: $i,SV189: $i,SV79: $i,SV354: $i] :
( ( ( element_of_set @ SV354 @ SV79 )
= $false )
| ( ( ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SV332 @ SV354 ) @ ( f18 @ SV79 @ SV189 @ SV332 @ SV354 ) )
| ( eq_p @ SV332 @ SV354 ) )
= $true )
| ( ( element_of_set @ SV332 @ SV79 )
= $false )
| ( ( hausdorff @ SV79 @ SV189 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1105]) ).
thf(1134,plain,
! [SV342: $i,SV193: $i,SV83: $i,SV355: $i] :
( ( ( ~ ( neighborhood @ SV355 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 ) )
= $true )
| ( ( ~ ( disjoint_s @ SV342 @ SV355 ) )
= $true )
| ( ( neighborhood @ SV342 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
= $false )
| ( ( topological_space @ SV83 @ SV193 )
= $false )
| ( ( hausdorff @ SV83 @ SV193 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1106]) ).
thf(1135,plain,
! [SV269: $i,SV201: $i,SV315: $i,SV91: $i] :
( ( ( ~ ( element_of_collection @ SV91 @ SV315 ) )
= $true )
| ( ( ~ ( element_of_collection @ SV201 @ SV315 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( equal_sets @ SV201 @ empty_set )
= $true )
| ( ( equal_sets @ SV91 @ empty_set )
= $true )
| ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1107]) ).
thf(1136,plain,
! [SV218: $i,SV108: $i,SV344: $i] :
( ( ( subset_collections @ SV344 @ ( f24 @ SV108 @ SV218 ) )
= $false )
| ( ( ~ ( open_covering @ SV344 @ SV108 @ SV218 ) )
= $true )
| ( ( finite @ SV344 )
= $false )
| ( ( topological_space @ SV108 @ SV218 )
= $false )
| ( ( compact_space @ SV108 @ SV218 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1110]) ).
thf(1137,plain,
! [SV126: $i,SV16: $i] :
( ( ( element_of_collection @ ( union_of_members @ ( f5 @ SV16 @ SV126 ) ) @ SV126 )
= $false )
| ( ( element_of_collection @ ( f3 @ SV16 @ SV126 ) @ SV126 )
= $true )
| ( ( element_of_collection @ SV16 @ SV126 )
= $false )
| ( ( element_of_collection @ empty_set @ SV126 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV126 ) @ SV16 )
= $false )
| ( ( topological_space @ SV16 @ SV126 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1114]) ).
thf(1138,plain,
! [SV128: $i,SV18: $i] :
( ( ( element_of_collection @ ( union_of_members @ ( f5 @ SV18 @ SV128 ) ) @ SV128 )
= $false )
| ( ( element_of_collection @ ( f4 @ SV18 @ SV128 ) @ SV128 )
= $true )
| ( ( element_of_collection @ SV18 @ SV128 )
= $false )
| ( ( element_of_collection @ empty_set @ SV128 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV128 ) @ SV18 )
= $false )
| ( ( topological_space @ SV18 @ SV128 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1116]) ).
thf(1139,plain,
! [SV129: $i,SV19: $i] :
( ( ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV19 @ SV129 ) @ ( f4 @ SV19 @ SV129 ) ) @ SV129 )
= $false )
| ( ( subset_collections @ ( f5 @ SV19 @ SV129 ) @ SV129 )
= $true )
| ( ( element_of_collection @ SV19 @ SV129 )
= $false )
| ( ( element_of_collection @ empty_set @ SV129 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV129 ) @ SV19 )
= $false )
| ( ( topological_space @ SV19 @ SV129 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1117]) ).
thf(1140,plain,
! [SV130: $i,SV20: $i] :
( ( ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
= $false )
| ( ( ~ ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 ) )
= $true )
| ( ( element_of_collection @ SV20 @ SV130 )
= $false )
| ( ( element_of_collection @ empty_set @ SV130 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
= $false )
| ( ( topological_space @ SV20 @ SV130 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1118]) ).
thf(1141,plain,
! [SV32: $i,SV349: $i,SV232: $i,SV142: $i,SV359: $i] :
( ( ( ~ ( element_of_collection @ SV359 @ SV142 )
| ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SV349 @ SV359 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SV349 @ SV359 ) ) )
= $true )
| ( ( element_of_collection @ SV349 @ SV142 )
= $false )
| ( ( element_of_set @ SV232 @ SV32 )
= $false )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[1119]) ).
thf(1142,plain,
! [SV33: $i,SV350: $i,SV233: $i,SV143: $i,SV360: $i] :
( ( ( ~ ( element_of_collection @ SV360 @ SV143 )
| ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SV350 @ SV360 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SV350 @ SV360 ) @ SV143 ) )
= $true )
| ( ( element_of_collection @ SV350 @ SV143 )
= $false )
| ( ( element_of_set @ SV233 @ SV33 )
= $false )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[1120]) ).
thf(1143,plain,
! [SV34: $i,SV351: $i,SV234: $i,SV144: $i,SV361: $i] :
( ( ( ~ ( element_of_collection @ SV361 @ SV144 )
| ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SV351 @ SV361 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SV351 @ SV361 ) @ ( intersection_of_sets @ SV351 @ SV361 ) ) )
= $true )
| ( ( element_of_collection @ SV351 @ SV144 )
= $false )
| ( ( element_of_set @ SV234 @ SV34 )
= $false )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_forall_pos,[status(thm)],[1121]) ).
thf(1144,plain,
! [SV149: $i,SV39: $i,SV341: $i] :
( ( ( subset_sets @ SV341 @ ( intersection_of_sets @ ( f8 @ SV39 @ SV149 ) @ ( f9 @ SV39 @ SV149 ) ) )
= $false )
| ( ( element_of_collection @ SV341 @ SV149 )
= $false )
| ( ( element_of_set @ ( f7 @ SV39 @ SV149 ) @ SV341 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV149 ) @ SV39 )
= $false )
| ( ( basis @ SV39 @ SV149 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1122]) ).
thf(1145,plain,
! [SV159: $i,SV285: $i,SV49: $i,SV239: $i,SV356: $i] :
( ( ( ~ ( element_of_collection @ SV356 @ SV239 ) )
= $true )
| ( ( ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SV356 ) ) )
= $true )
| ( ( subset_sets @ SV285 @ SV159 )
= $false )
| ( ( topological_space @ SV159 @ SV239 )
= $false )
| ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1123]) ).
thf(1146,plain,
! [SV290: $i,SV245: $i,SV165: $i,SV357: $i,SV55: $i] :
( ( ( ~ ( element_of_set @ SV55 @ SV357 ) )
= $true )
| ( ( ~ ( subset_sets @ SV357 @ SV165 )
| ~ ( open @ SV357 @ SV245 @ SV290 ) )
= $true )
| ( ( subset_sets @ SV165 @ SV245 )
= $false )
| ( ( topological_space @ SV245 @ SV290 )
= $false )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1124]) ).
thf(1147,plain,
! [SV168: $i,SV58: $i,SV292: $i,SV248: $i,SV345: $i] :
( ( ( closed @ SV345 @ SV248 @ SV292 )
= $false )
| ( ( element_of_set @ SV58 @ SV345 )
= $true )
| ( ( subset_sets @ SV168 @ SV345 )
= $false )
| ( ( element_of_set @ SV58 @ ( closure @ SV168 @ SV248 @ SV292 ) )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1125]) ).
thf(1148,plain,
! [SV304: $i,SV260: $i,SV181: $i,SV71: $i,SV358: $i] :
( ( ( ~ ( element_of_set @ SV358 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SV304 ) @ SV181 ) ) )
= $true )
| ( ( eq_p @ SV358 @ SV71 )
= $true )
| ( ( subset_sets @ SV181 @ SV260 )
= $false )
| ( ( topological_space @ SV260 @ SV304 )
= $false )
| ( ( limit_point @ SV71 @ SV181 @ SV260 @ SV304 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1129]) ).
thf(1149,plain,
! [SV187: $i,SV77: $i,SV352: $i,SV330: $i] :
( ( ( eq_p @ SV330 @ SV352 )
= $true )
| ( ( neighborhood @ ( f17 @ SV77 @ SV187 @ SV330 @ SV352 ) @ SV330 @ SV77 @ SV187 )
= $true )
| ( ( element_of_set @ SV352 @ SV77 )
= $false )
| ( ( element_of_set @ SV330 @ SV77 )
= $false )
| ( ( hausdorff @ SV77 @ SV187 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1131]) ).
thf(1150,plain,
! [SV188: $i,SV78: $i,SV353: $i,SV331: $i] :
( ( ( eq_p @ SV331 @ SV353 )
= $true )
| ( ( neighborhood @ ( f18 @ SV78 @ SV188 @ SV331 @ SV353 ) @ SV353 @ SV78 @ SV188 )
= $true )
| ( ( element_of_set @ SV353 @ SV78 )
= $false )
| ( ( element_of_set @ SV331 @ SV78 )
= $false )
| ( ( hausdorff @ SV78 @ SV188 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1132]) ).
thf(1151,plain,
! [SV354: $i,SV332: $i,SV189: $i,SV79: $i] :
( ( ( disjoint_s @ ( f17 @ SV79 @ SV189 @ SV332 @ SV354 ) @ ( f18 @ SV79 @ SV189 @ SV332 @ SV354 ) )
= $true )
| ( ( eq_p @ SV332 @ SV354 )
= $true )
| ( ( element_of_set @ SV354 @ SV79 )
= $false )
| ( ( element_of_set @ SV332 @ SV79 )
= $false )
| ( ( hausdorff @ SV79 @ SV189 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1133]) ).
thf(1152,plain,
! [SV342: $i,SV193: $i,SV83: $i,SV355: $i] :
( ( ( neighborhood @ SV355 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
= $false )
| ( ( ~ ( disjoint_s @ SV342 @ SV355 ) )
= $true )
| ( ( neighborhood @ SV342 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
= $false )
| ( ( topological_space @ SV83 @ SV193 )
= $false )
| ( ( hausdorff @ SV83 @ SV193 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1134]) ).
thf(1153,plain,
! [SV269: $i,SV201: $i,SV315: $i,SV91: $i] :
( ( ( element_of_collection @ SV91 @ SV315 )
= $false )
| ( ( ~ ( element_of_collection @ SV201 @ SV315 )
| ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( equal_sets @ SV201 @ empty_set )
= $true )
| ( ( equal_sets @ SV91 @ empty_set )
= $true )
| ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1135]) ).
thf(1154,plain,
! [SV218: $i,SV108: $i,SV344: $i] :
( ( ( open_covering @ SV344 @ SV108 @ SV218 )
= $false )
| ( ( subset_collections @ SV344 @ ( f24 @ SV108 @ SV218 ) )
= $false )
| ( ( finite @ SV344 )
= $false )
| ( ( topological_space @ SV108 @ SV218 )
= $false )
| ( ( compact_space @ SV108 @ SV218 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1136]) ).
thf(1155,plain,
! [SV130: $i,SV20: $i] :
( ( ( element_of_collection @ ( union_of_members @ ( f5 @ SV20 @ SV130 ) ) @ SV130 )
= $false )
| ( ( element_of_collection @ ( intersection_of_sets @ ( f3 @ SV20 @ SV130 ) @ ( f4 @ SV20 @ SV130 ) ) @ SV130 )
= $false )
| ( ( element_of_collection @ SV20 @ SV130 )
= $false )
| ( ( element_of_collection @ empty_set @ SV130 )
= $false )
| ( ( equal_sets @ ( union_of_members @ SV130 ) @ SV20 )
= $false )
| ( ( topological_space @ SV20 @ SV130 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1140]) ).
thf(1156,plain,
! [SV32: $i,SV349: $i,SV232: $i,SV142: $i,SV359: $i] :
( ( ( ~ ( element_of_collection @ SV359 @ SV142 ) )
= $true )
| ( ( ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SV349 @ SV359 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SV349 @ SV359 ) ) )
= $true )
| ( ( element_of_collection @ SV349 @ SV142 )
= $false )
| ( ( element_of_set @ SV232 @ SV32 )
= $false )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1141]) ).
thf(1157,plain,
! [SV33: $i,SV350: $i,SV233: $i,SV143: $i,SV360: $i] :
( ( ( ~ ( element_of_collection @ SV360 @ SV143 ) )
= $true )
| ( ( ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SV350 @ SV360 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SV350 @ SV360 ) @ SV143 ) )
= $true )
| ( ( element_of_collection @ SV350 @ SV143 )
= $false )
| ( ( element_of_set @ SV233 @ SV33 )
= $false )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1142]) ).
thf(1158,plain,
! [SV34: $i,SV351: $i,SV234: $i,SV144: $i,SV361: $i] :
( ( ( ~ ( element_of_collection @ SV361 @ SV144 ) )
= $true )
| ( ( ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SV351 @ SV361 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SV351 @ SV361 ) @ ( intersection_of_sets @ SV351 @ SV361 ) ) )
= $true )
| ( ( element_of_collection @ SV351 @ SV144 )
= $false )
| ( ( element_of_set @ SV234 @ SV34 )
= $false )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1143]) ).
thf(1159,plain,
! [SV159: $i,SV285: $i,SV49: $i,SV239: $i,SV356: $i] :
( ( ( element_of_collection @ SV356 @ SV239 )
= $false )
| ( ( ~ ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SV356 ) ) )
= $true )
| ( ( subset_sets @ SV285 @ SV159 )
= $false )
| ( ( topological_space @ SV159 @ SV239 )
= $false )
| ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1145]) ).
thf(1160,plain,
! [SV290: $i,SV245: $i,SV165: $i,SV357: $i,SV55: $i] :
( ( ( element_of_set @ SV55 @ SV357 )
= $false )
| ( ( ~ ( subset_sets @ SV357 @ SV165 )
| ~ ( open @ SV357 @ SV245 @ SV290 ) )
= $true )
| ( ( subset_sets @ SV165 @ SV245 )
= $false )
| ( ( topological_space @ SV245 @ SV290 )
= $false )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1146]) ).
thf(1161,plain,
! [SV304: $i,SV260: $i,SV181: $i,SV71: $i,SV358: $i] :
( ( ( element_of_set @ SV358 @ ( intersection_of_sets @ ( f16 @ SV71 @ SV181 @ SV260 @ SV304 ) @ SV181 ) )
= $false )
| ( ( eq_p @ SV358 @ SV71 )
= $true )
| ( ( subset_sets @ SV181 @ SV260 )
= $false )
| ( ( topological_space @ SV260 @ SV304 )
= $false )
| ( ( limit_point @ SV71 @ SV181 @ SV260 @ SV304 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1148]) ).
thf(1162,plain,
! [SV193: $i,SV83: $i,SV355: $i,SV342: $i] :
( ( ( disjoint_s @ SV342 @ SV355 )
= $false )
| ( ( neighborhood @ SV355 @ ( f20 @ SV83 @ SV193 ) @ SV83 @ SV193 )
= $false )
| ( ( neighborhood @ SV342 @ ( f19 @ SV83 @ SV193 ) @ SV83 @ SV193 )
= $false )
| ( ( topological_space @ SV83 @ SV193 )
= $false )
| ( ( hausdorff @ SV83 @ SV193 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1152]) ).
thf(1163,plain,
! [SV269: $i,SV91: $i,SV315: $i,SV201: $i] :
( ( ( ~ ( element_of_collection @ SV201 @ SV315 ) )
= $true )
| ( ( ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( element_of_collection @ SV91 @ SV315 )
= $false )
| ( ( equal_sets @ SV201 @ empty_set )
= $true )
| ( ( equal_sets @ SV91 @ empty_set )
= $true )
| ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1153]) ).
thf(1164,plain,
! [SV32: $i,SV349: $i,SV232: $i,SV142: $i,SV359: $i] :
( ( ( element_of_collection @ SV359 @ SV142 )
= $false )
| ( ( ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SV349 @ SV359 ) )
| ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SV349 @ SV359 ) ) )
= $true )
| ( ( element_of_collection @ SV349 @ SV142 )
= $false )
| ( ( element_of_set @ SV232 @ SV32 )
= $false )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1156]) ).
thf(1165,plain,
! [SV33: $i,SV350: $i,SV233: $i,SV143: $i,SV360: $i] :
( ( ( element_of_collection @ SV360 @ SV143 )
= $false )
| ( ( ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SV350 @ SV360 ) )
| ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SV350 @ SV360 ) @ SV143 ) )
= $true )
| ( ( element_of_collection @ SV350 @ SV143 )
= $false )
| ( ( element_of_set @ SV233 @ SV33 )
= $false )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1157]) ).
thf(1166,plain,
! [SV34: $i,SV351: $i,SV234: $i,SV144: $i,SV361: $i] :
( ( ( element_of_collection @ SV361 @ SV144 )
= $false )
| ( ( ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SV351 @ SV361 ) )
| ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SV351 @ SV361 ) @ ( intersection_of_sets @ SV351 @ SV361 ) ) )
= $true )
| ( ( element_of_collection @ SV351 @ SV144 )
= $false )
| ( ( element_of_set @ SV234 @ SV34 )
= $false )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1158]) ).
thf(1167,plain,
! [SV159: $i,SV239: $i,SV356: $i,SV285: $i,SV49: $i] :
( ( ( equal_sets @ SV49 @ ( intersection_of_sets @ SV285 @ SV356 ) )
= $false )
| ( ( element_of_collection @ SV356 @ SV239 )
= $false )
| ( ( subset_sets @ SV285 @ SV159 )
= $false )
| ( ( topological_space @ SV159 @ SV239 )
= $false )
| ( ( element_of_collection @ SV49 @ ( subspace_topology @ SV159 @ SV239 @ SV285 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1159]) ).
thf(1168,plain,
! [SV55: $i,SV290: $i,SV245: $i,SV165: $i,SV357: $i] :
( ( ( ~ ( subset_sets @ SV357 @ SV165 ) )
= $true )
| ( ( ~ ( open @ SV357 @ SV245 @ SV290 ) )
= $true )
| ( ( element_of_set @ SV55 @ SV357 )
= $false )
| ( ( subset_sets @ SV165 @ SV245 )
= $false )
| ( ( topological_space @ SV245 @ SV290 )
= $false )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1160]) ).
thf(1169,plain,
! [SV269: $i,SV91: $i,SV315: $i,SV201: $i] :
( ( ( element_of_collection @ SV201 @ SV315 )
= $false )
| ( ( ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
| ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( element_of_collection @ SV91 @ SV315 )
= $false )
| ( ( equal_sets @ SV201 @ empty_set )
= $true )
| ( ( equal_sets @ SV91 @ empty_set )
= $true )
| ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1163]) ).
thf(1170,plain,
! [SV142: $i,SV32: $i,SV359: $i,SV349: $i,SV232: $i] :
( ( ( ~ ( element_of_set @ SV232 @ ( intersection_of_sets @ SV349 @ SV359 ) ) )
= $true )
| ( ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SV349 @ SV359 ) )
= $true )
| ( ( element_of_collection @ SV359 @ SV142 )
= $false )
| ( ( element_of_collection @ SV349 @ SV142 )
= $false )
| ( ( element_of_set @ SV232 @ SV32 )
= $false )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1164]) ).
thf(1171,plain,
! [SV143: $i,SV33: $i,SV360: $i,SV350: $i,SV233: $i] :
( ( ( ~ ( element_of_set @ SV233 @ ( intersection_of_sets @ SV350 @ SV360 ) ) )
= $true )
| ( ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SV350 @ SV360 ) @ SV143 )
= $true )
| ( ( element_of_collection @ SV360 @ SV143 )
= $false )
| ( ( element_of_collection @ SV350 @ SV143 )
= $false )
| ( ( element_of_set @ SV233 @ SV33 )
= $false )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1165]) ).
thf(1172,plain,
! [SV144: $i,SV34: $i,SV361: $i,SV351: $i,SV234: $i] :
( ( ( ~ ( element_of_set @ SV234 @ ( intersection_of_sets @ SV351 @ SV361 ) ) )
= $true )
| ( ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SV351 @ SV361 ) @ ( intersection_of_sets @ SV351 @ SV361 ) )
= $true )
| ( ( element_of_collection @ SV361 @ SV144 )
= $false )
| ( ( element_of_collection @ SV351 @ SV144 )
= $false )
| ( ( element_of_set @ SV234 @ SV34 )
= $false )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_or_pos,[status(thm)],[1166]) ).
thf(1173,plain,
! [SV55: $i,SV290: $i,SV245: $i,SV165: $i,SV357: $i] :
( ( ( subset_sets @ SV357 @ SV165 )
= $false )
| ( ( ~ ( open @ SV357 @ SV245 @ SV290 ) )
= $true )
| ( ( element_of_set @ SV55 @ SV357 )
= $false )
| ( ( subset_sets @ SV165 @ SV245 )
= $false )
| ( ( topological_space @ SV245 @ SV290 )
= $false )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1168]) ).
thf(1174,plain,
! [SV315: $i,SV269: $i,SV201: $i,SV91: $i] :
( ( ( ~ ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 ) )
= $true )
| ( ( ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( element_of_collection @ SV201 @ SV315 )
= $false )
| ( ( element_of_collection @ SV91 @ SV315 )
= $false )
| ( ( equal_sets @ SV201 @ empty_set )
= $true )
| ( ( equal_sets @ SV91 @ empty_set )
= $true )
| ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_or_pos,[status(thm)],[1169]) ).
thf(1175,plain,
! [SV142: $i,SV32: $i,SV359: $i,SV349: $i,SV232: $i] :
( ( ( element_of_set @ SV232 @ ( intersection_of_sets @ SV349 @ SV359 ) )
= $false )
| ( ( element_of_set @ SV232 @ ( f6 @ SV32 @ SV142 @ SV232 @ SV349 @ SV359 ) )
= $true )
| ( ( element_of_collection @ SV359 @ SV142 )
= $false )
| ( ( element_of_collection @ SV349 @ SV142 )
= $false )
| ( ( element_of_set @ SV232 @ SV32 )
= $false )
| ( ( basis @ SV32 @ SV142 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1170]) ).
thf(1176,plain,
! [SV143: $i,SV33: $i,SV360: $i,SV350: $i,SV233: $i] :
( ( ( element_of_set @ SV233 @ ( intersection_of_sets @ SV350 @ SV360 ) )
= $false )
| ( ( element_of_collection @ ( f6 @ SV33 @ SV143 @ SV233 @ SV350 @ SV360 ) @ SV143 )
= $true )
| ( ( element_of_collection @ SV360 @ SV143 )
= $false )
| ( ( element_of_collection @ SV350 @ SV143 )
= $false )
| ( ( element_of_set @ SV233 @ SV33 )
= $false )
| ( ( basis @ SV33 @ SV143 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1171]) ).
thf(1177,plain,
! [SV144: $i,SV34: $i,SV361: $i,SV351: $i,SV234: $i] :
( ( ( element_of_set @ SV234 @ ( intersection_of_sets @ SV351 @ SV361 ) )
= $false )
| ( ( subset_sets @ ( f6 @ SV34 @ SV144 @ SV234 @ SV351 @ SV361 ) @ ( intersection_of_sets @ SV351 @ SV361 ) )
= $true )
| ( ( element_of_collection @ SV361 @ SV144 )
= $false )
| ( ( element_of_collection @ SV351 @ SV144 )
= $false )
| ( ( element_of_set @ SV234 @ SV34 )
= $false )
| ( ( basis @ SV34 @ SV144 )
= $false ) ),
inference(extcnf_not_pos,[status(thm)],[1172]) ).
thf(1178,plain,
! [SV55: $i,SV165: $i,SV290: $i,SV245: $i,SV357: $i] :
( ( ( open @ SV357 @ SV245 @ SV290 )
= $false )
| ( ( subset_sets @ SV357 @ SV165 )
= $false )
| ( ( element_of_set @ SV55 @ SV357 )
= $false )
| ( ( subset_sets @ SV165 @ SV245 )
= $false )
| ( ( topological_space @ SV245 @ SV290 )
= $false )
| ( ( element_of_set @ SV55 @ ( interior @ SV165 @ SV245 @ SV290 ) )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1173]) ).
thf(1179,plain,
! [SV315: $i,SV269: $i,SV201: $i,SV91: $i] :
( ( ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
= $false )
| ( ( ~ ( disjoint_s @ SV91 @ SV201 ) )
= $true )
| ( ( element_of_collection @ SV201 @ SV315 )
= $false )
| ( ( element_of_collection @ SV91 @ SV315 )
= $false )
| ( ( equal_sets @ SV201 @ empty_set )
= $true )
| ( ( equal_sets @ SV91 @ empty_set )
= $true )
| ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1174]) ).
thf(1180,plain,
! [SV315: $i,SV269: $i,SV201: $i,SV91: $i] :
( ( ( disjoint_s @ SV91 @ SV201 )
= $false )
| ( ( equal_sets @ ( union_of_sets @ SV91 @ SV201 ) @ SV269 )
= $false )
| ( ( element_of_collection @ SV201 @ SV315 )
= $false )
| ( ( element_of_collection @ SV91 @ SV315 )
= $false )
| ( ( equal_sets @ SV201 @ empty_set )
= $true )
| ( ( equal_sets @ SV91 @ empty_set )
= $true )
| ( ( topological_space @ SV269 @ SV315 )
= $false )
| ( ( separation @ SV91 @ SV201 @ SV269 @ SV315 )
= $true ) ),
inference(extcnf_not_pos,[status(thm)],[1179]) ).
thf(1181,plain,
$false = $true,
inference(fo_atp_e,[status(thm)],[272,1180,1178,1177,1176,1175,1167,1162,1161,1155,1154,1151,1150,1149,1147,1144,1139,1138,1137,1130,1128,1127,1126,1115,1113,1112,1111,1109,1108,1100,1098,1095,1094,1090,1085,1041,1040,1039,1036,1034,1033,1015,1014,1013,1007,1006,999,997,996,986,985,984,983,980,978,973,972,971,966,965,963,962,961,956,955,953,952,951,950,949,947,946,945,944,921,920,919,917,912,911,910,908,907,906,905,893,892,852,851,850,849,844,843,842,840,839,837,836,816,805,789,745,726,725,724,723,720,719,649,614,609,498,387,386]) ).
thf(1182,plain,
$false,
inference(solved_all_splits,[solved_all_splits(join,[])],[1181]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : TOP004-1 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.06/0.17 % Computer : n018.cluster.edu
% 0.06/0.17 % Model : x86_64 x86_64
% 0.06/0.17 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.06/0.17 % Memory : 8046.5625MB
% 0.06/0.17 % OS : Linux 6.8.0-71-generic
% 0.06/0.17 % CPULimit : 300
% 0.06/0.17 % WCLimit : 300
% 0.06/0.17 % DateTime : Tue Oct 6 17:57:22 UTC 2026
% 0.06/0.17 % CPUTime :
% 0.06/0.17 Running leo --cores 7 --timeout 300 --proofoutput 1 --foatp e --atp e=/export/starexec/sandbox2/solver/bin/eprover /export/starexec/sandbox2/benchmark/theBenchmark.p
% 5.14/1.07
% 5.14/1.07 No.of.Axioms: 113
% 5.14/1.07
% 5.14/1.07 Length.of.Defs: 0
% 5.14/1.07
% 5.14/1.07 Contains.Choice.Funs: false
% 5.14/1.07 ........................................
% 5.14/1.07
% 5.14/1.07 ********************************
% 5.14/1.07 * All subproblems solved! *
% 5.14/1.07 ********************************
% 5.14/1.07 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:0,axioms:113,ps:3,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:true,expand_extuni:false,foatp:e,atp_timeout:299,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:1181,loop_count:0,foatp_calls:1,translation:fof_full)
% 5.14/1.07
% 5.14/1.07 %**** Beginning of derivation protocol ****
% 5.14/1.07 % SZS output start CNFRefutation
% See solution above
% 5.14/1.08
% 5.14/1.08 %**** End of derivation protocol ****
% 5.14/1.08 %**** no. of clauses in derivation: 1182 ****
% 5.14/1.08 %**** clause counter: 1181 ****
% 5.14/1.08
% 5.14/1.08 % SZS status Unsatisfiable for /export/starexec/sandbox2/benchmark/theBenchmark.p : (rf:0,axioms:113,ps:3,u:6,ude:true,rLeibEQ:true,rAndEQ:true,use_choice:true,use_extuni:true,use_extcnf_combined:true,expand_extuni:false,foatp:e,atp_timeout:299,atp_calls_frequency:5,ordering:none,proof_output:1,protocol_output:false,clause_count:1181,loop_count:0,foatp_calls:1,translation:fof_full)
%------------------------------------------------------------------------------