TSTP Solution File: TOP018-1 by Prover9---1109a
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- Process Solution
%------------------------------------------------------------------------------
% File : Prover9---1109a
% Problem : TOP018-1 : TPTP v8.1.0. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : tptp2X_and_run_prover9 %d %s
% Computer : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 600s
% DateTime : Thu Jul 21 21:33:50 EDT 2022
% Result : Timeout 300.05s 300.37s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.11 % Problem : TOP018-1 : TPTP v8.1.0. Released v1.0.0.
% 0.03/0.12 % Command : tptp2X_and_run_prover9 %d %s
% 0.11/0.32 % Computer : n025.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 600
% 0.11/0.32 % DateTime : Sun May 29 08:24:41 EDT 2022
% 0.11/0.33 % CPUTime :
% 0.38/0.91 ============================== Prover9 ===============================
% 0.38/0.91 Prover9 (32) version 2009-11A, November 2009.
% 0.38/0.91 Process 18395 was started by sandbox on n025.cluster.edu,
% 0.38/0.91 Sun May 29 08:24:41 2022
% 0.38/0.91 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 300 -f /tmp/Prover9_18242_n025.cluster.edu".
% 0.38/0.91 ============================== end of head ===========================
% 0.38/0.91
% 0.38/0.91 ============================== INPUT =================================
% 0.38/0.91
% 0.38/0.91 % Reading from file /tmp/Prover9_18242_n025.cluster.edu
% 0.38/0.91
% 0.38/0.91 set(prolog_style_variables).
% 0.38/0.91 set(auto2).
% 0.38/0.91 % set(auto2) -> set(auto).
% 0.38/0.91 % set(auto) -> set(auto_inference).
% 0.38/0.91 % set(auto) -> set(auto_setup).
% 0.38/0.91 % set(auto_setup) -> set(predicate_elim).
% 0.38/0.91 % set(auto_setup) -> assign(eq_defs, unfold).
% 0.38/0.91 % set(auto) -> set(auto_limits).
% 0.38/0.91 % set(auto_limits) -> assign(max_weight, "100.000").
% 0.38/0.91 % set(auto_limits) -> assign(sos_limit, 20000).
% 0.38/0.91 % set(auto) -> set(auto_denials).
% 0.38/0.91 % set(auto) -> set(auto_process).
% 0.38/0.91 % set(auto2) -> assign(new_constants, 1).
% 0.38/0.91 % set(auto2) -> assign(fold_denial_max, 3).
% 0.38/0.91 % set(auto2) -> assign(max_weight, "200.000").
% 0.38/0.91 % set(auto2) -> assign(max_hours, 1).
% 0.38/0.91 % assign(max_hours, 1) -> assign(max_seconds, 3600).
% 0.38/0.91 % set(auto2) -> assign(max_seconds, 0).
% 0.38/0.91 % set(auto2) -> assign(max_minutes, 5).
% 0.38/0.91 % assign(max_minutes, 5) -> assign(max_seconds, 300).
% 0.38/0.91 % set(auto2) -> set(sort_initial_sos).
% 0.38/0.91 % set(auto2) -> assign(sos_limit, -1).
% 0.38/0.91 % set(auto2) -> assign(lrs_ticks, 3000).
% 0.38/0.91 % set(auto2) -> assign(max_megs, 400).
% 0.38/0.91 % set(auto2) -> assign(stats, some).
% 0.38/0.91 % set(auto2) -> clear(echo_input).
% 0.38/0.91 % set(auto2) -> set(quiet).
% 0.38/0.91 % set(auto2) -> clear(print_initial_clauses).
% 0.38/0.91 % set(auto2) -> clear(print_given).
% 0.38/0.91 assign(lrs_ticks,-1).
% 0.38/0.91 assign(sos_limit,10000).
% 0.38/0.91 assign(order,kbo).
% 0.38/0.91 set(lex_order_vars).
% 0.38/0.91 clear(print_given).
% 0.38/0.91
% 0.38/0.91 % formulas(sos). % not echoed (112 formulas)
% 0.38/0.91
% 0.38/0.91 ============================== end of input ==========================
% 0.38/0.91
% 0.38/0.91 % From the command line: assign(max_seconds, 300).
% 0.38/0.91
% 0.38/0.91 ============================== PROCESS NON-CLAUSAL FORMULAS ==========
% 0.38/0.91
% 0.38/0.91 % Formulas that are not ordinary clauses:
% 0.38/0.91
% 0.38/0.91 ============================== end of process non-clausal formulas ===
% 0.38/0.91
% 0.38/0.91 ============================== PROCESS INITIAL CLAUSES ===============
% 0.38/0.91
% 0.38/0.91 ============================== PREDICATE ELIMINATION =================
% 0.38/0.91 1 topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f3(A,B),B) | subset_collections(f5(A,B),B) # label(topological_space_12) # label(axiom). [assumption].
% 0.38/0.91 2 -topological_space(A,B) | -subset_collections(C,B) | element_of_collection(union_of_members(C),B) # label(topological_space_11) # label(axiom). [assumption].
% 0.38/0.91 Derived: topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f3(A,B),B) | -topological_space(C,B) | element_of_collection(union_of_members(f5(A,B)),B). [resolve(1,f,2,b)].
% 0.38/0.91 3 topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f4(A,B),B) | subset_collections(f5(A,B),B) # label(topological_space_14) # label(axiom). [assumption].
% 0.38/0.91 Derived: topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f4(A,B),B) | -topological_space(C,B) | element_of_collection(union_of_members(f5(A,B)),B). [resolve(3,f,2,b)].
% 0.38/0.91 4 topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | -element_of_collection(intersection_of_sets(f3(A,B),f4(A,B)),B) | subset_collections(f5(A,B),B) # label(topological_space_16) # label(axiom). [assumption].
% 0.38/0.91 Derived: topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | -element_of_collection(intersection_of_sets(f3(A,B),f4(A,B)),B) | -topological_space(C,B) | element_of_collection(union_of_members(f5(A,B)),B). [resolve(4,f,2,b)].
% 0.38/0.92 5 -finer(A,B,C) | subset_collections(B,A) # label(finer_topology_26) # label(axiom). [assumption].
% 0.38/0.92 Derived: -finer(A,B,C) | -topological_space(D,A) | element_of_collection(union_of_members(B),A). [resolve(5,b,2,b)].
% 0.38/0.92 6 finer(A,B,C) | -topological_space(C,A) | -topological_space(C,B) | -subset_collections(B,A) # label(finer_topology_27) # label(axiom). [assumption].
% 0.38/0.92 Derived: finer(A,f5(B,A),C) | -topological_space(C,A) | -topological_space(C,f5(B,A)) | topological_space(B,A) | -equal_sets(union_of_members(A),B) | -element_of_collection(empty_set,A) | -element_of_collection(B,A) | element_of_collection(f3(B,A),A). [resolve(6,d,1,f)].
% 0.38/0.92 Derived: finer(A,f5(B,A),C) | -topological_space(C,A) | -topological_space(C,f5(B,A)) | topological_space(B,A) | -equal_sets(union_of_members(A),B) | -element_of_collection(empty_set,A) | -element_of_collection(B,A) | element_of_collection(f4(B,A),A). [resolve(6,d,3,f)].
% 0.38/0.92 Derived: finer(A,f5(B,A),C) | -topological_space(C,A) | -topological_space(C,f5(B,A)) | topological_space(B,A) | -equal_sets(union_of_members(A),B) | -element_of_collection(empty_set,A) | -element_of_collection(B,A) | -element_of_collection(intersection_of_sets(f3(B,A),f4(B,A)),A). [resolve(6,d,4,f)].
% 0.38/0.92 Derived: finer(A,B,C) | -topological_space(C,A) | -topological_space(C,B) | -finer(A,B,D). [resolve(6,d,5,b)].
% 0.38/0.92 7 -open_covering(A,B,C) | subset_collections(A,C) # label(open_covering_97) # label(axiom). [assumption].
% 0.38/0.92 Derived: -open_covering(A,B,C) | -topological_space(D,C) | element_of_collection(union_of_members(A),C). [resolve(7,b,2,b)].
% 0.38/0.92 Derived: -open_covering(A,B,C) | finer(C,A,D) | -topological_space(D,C) | -topological_space(D,A). [resolve(7,b,6,d)].
% 0.38/0.92 8 open_covering(A,B,C) | -topological_space(B,C) | -subset_collections(A,C) | -equal_sets(union_of_members(A),B) # label(open_covering_99) # label(axiom). [assumption].
% 0.38/0.92 Derived: open_covering(f5(A,B),C,B) | -topological_space(C,B) | -equal_sets(union_of_members(f5(A,B)),C) | topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f3(A,B),B). [resolve(8,c,1,f)].
% 0.38/0.92 Derived: open_covering(f5(A,B),C,B) | -topological_space(C,B) | -equal_sets(union_of_members(f5(A,B)),C) | topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | element_of_collection(f4(A,B),B). [resolve(8,c,3,f)].
% 0.38/0.92 Derived: open_covering(f5(A,B),C,B) | -topological_space(C,B) | -equal_sets(union_of_members(f5(A,B)),C) | topological_space(A,B) | -equal_sets(union_of_members(B),A) | -element_of_collection(empty_set,B) | -element_of_collection(A,B) | -element_of_collection(intersection_of_sets(f3(A,B),f4(A,B)),B). [resolve(8,c,4,f)].
% 0.38/0.92 Derived: open_covering(A,B,C) | -topological_space(B,C) | -equal_sets(union_of_members(A),B) | -finer(C,A,D). [resolve(8,c,5,b)].
% 0.38/0.92 Derived: open_covering(A,B,C) | -topological_space(B,C) | -equal_sets(union_of_members(A),B) | -open_covering(A,D,C). [resolve(8,c,7,b)].
% 0.38/0.92 9 -compact_space(A,B) | -open_covering(C,A,B) | subset_collections(f23(A,B,C),C) # label(compact_space_102) # label(axiom). [assumption].
% 0.38/0.92 Derived: -compact_space(A,B) | -open_covering(C,A,B) | -topological_space(D,C) | element_of_collection(union_of_members(f23(A,B,C)),C). [resolve(9,c,2,b)].
% 0.38/0.92 Derived: -compact_space(A,B) | -open_covering(C,A,B) | finer(C,f23(A,B,C),D) | -topological_space(D,C) | -topological_space(D,f23(A,B,C)). [resolve(9,c,6,d)].
% 0.38/0.92 Derived: -compact_space(A,B) | -open_covering(C,A,B) | open_covering(f23(A,B,C),D,C) | -topological_space(D,C) | -equal_sets(union_of_members(f23(A,B,C)),D). [resolve(9,c,8,c)].
% 0.38/0.92 10 compact_space(A,B) | -topological_space(A,B) | -finite(C) | -subset_collections(C,f24(A,B)) | -open_covering(C,A,B) # label(compact_space_105) # label(axiom). [assumption].
% 0.38/0.92 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | element_of_collection(f3(C,f24(A,B)),f24(A,B)). [resolve(10,d,1,f)].
% 0.38/0.92 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | element_of_collection(f4(C,f24(A,B)),f24(A,B)). [resolve(10,d,3,f)].
% 0.38/0.92 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | -element_of_collection(intersection_of_sets(f3(C,f24(A,B)),f4(C,f24(A,B))),f24(A,B)). [resolve(10,d,4,f)].
% 0.38/0.92 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(C) | -open_covering(C,A,B) | -finer(f24(A,B),C,D). [resolve(10,d,5,b)].
% 0.38/0.92 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(C) | -open_covering(C,A,B) | -open_covering(C,D,f24(A,B)). [resolve(10,d,7,b)].
% 0.38/0.92 Derived: compact_space(A,B) | -topological_space(A,B) | -finite(f23(C,D,f24(A,B))) | -open_covering(f23(C,D,f24(A,B)),A,B) | -compact_space(C,D) | -open_covering(f24(A,B),C,D). [resolve(10,d,9,c)].
% 0.38/0.92 11 open(A,B,C) | -topological_space(B,C) | -element_of_collection(A,C) # label(open_set_20) # label(axiom). [assumption].
% 0.38/0.92 12 -open(A,B,C) | topological_space(B,C) # label(open_set_18) # label(axiom). [assumption].
% 0.38/0.92 13 -open(A,B,C) | element_of_collection(A,C) # label(open_set_19) # label(axiom). [assumption].
% 0.38/0.92 14 -closed(A,B,C) | open(relative_complement_sets(A,B),B,C) # label(closed_set_22) # label(axiom). [assumption].
% 0.38/0.92 Derived: -closed(A,B,C) | topological_space(B,C). [resolve(14,b,12,a)].
% 0.38/0.92 Derived: -closed(A,B,C) | element_of_collection(relative_complement_sets(A,B),C). [resolve(14,b,13,a)].
% 0.38/0.92 15 closed(A,B,C) | -topological_space(B,C) | -open(relative_complement_sets(A,B),B,C) # label(closed_set_23) # label(axiom). [assumption].
% 0.38/0.92 Derived: closed(A,B,C) | -topological_space(B,C) | -topological_space(B,C) | -element_of_collection(relative_complement_sets(A,B),C). [resolve(15,c,11,a)].
% 0.38/0.92 16 -element_of_set(A,interior(B,C,D)) | open(f13(B,C,D,A),C,D) # label(interior_51) # label(axiom). [assumption].
% 0.38/0.92 Derived: -element_of_set(A,interior(B,C,D)) | topological_space(C,D). [resolve(16,b,12,a)].
% 0.38/0.92 Derived: -element_of_set(A,interior(B,C,D)) | element_of_collection(f13(B,C,D,A),D). [resolve(16,b,13,a)].
% 0.38/0.92 17 element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,E) | -subset_sets(E,B) | -open(E,C,D) # label(interior_52) # label(axiom). [assumption].
% 0.38/0.92 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,E) | -subset_sets(E,B) | -topological_space(C,D) | -element_of_collection(E,D). [resolve(17,f,11,a)].
% 0.38/0.92 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,relative_complement_sets(E,C)) | -subset_sets(relative_complement_sets(E,C),B) | -closed(E,C,D). [resolve(17,f,14,b)].
% 0.38/0.92 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,f13(E,C,D,F)) | -subset_sets(f13(E,C,D,F),B) | -element_of_set(F,interior(E,C,D)). [resolve(17,f,16,b)].
% 0.38/0.92 18 -neighborhood(A,B,C,D) | open(A,C,D) # label(neighborhood_60) # label(axiom). [assumption].
% 0.38/0.92 Derived: -neighborhood(A,B,C,D) | topological_space(C,D). [resolve(18,b,12,a)].
% 0.38/0.92 Derived: -neighborhood(A,B,C,D) | element_of_collection(A,D). [resolve(18,b,13,a)].
% 0.38/0.92 Derived: -neighborhood(relative_complement_sets(A,B),C,B,D) | closed(A,B,D) | -topological_space(B,D). [resolve(18,b,15,c)].
% 0.38/0.92 Derived: -neighborhood(A,B,C,D) | element_of_set(E,interior(F,C,D)) | -topological_space(C,D) | -subset_sets(F,C) | -element_of_set(E,A) | -subset_sets(A,F). [resolve(18,b,17,f)].
% 0.38/0.92 19 neighborhood(A,B,C,D) | -topological_space(C,D) | -open(A,C,D) | -element_of_set(B,A) # label(neighborhood_62) # label(axiom). [assumption].
% 0.38/0.92 Derived: neighborhood(A,B,C,D) | -topological_space(C,D) | -element_of_set(B,A) | -topological_space(C,D) | -element_of_collection(A,D). [resolve(19,c,11,a)].
% 0.38/0.92 Derived: neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | -element_of_set(C,relative_complement_sets(A,B)) | -closed(A,B,D). [resolve(19,c,14,b)].
% 0.38/0.92 Derived: neighborhood(f13(A,B,C,D),E,B,C) | -topological_space(B,C) | -element_of_set(E,f13(A,B,C,D)) | -element_of_set(D,interior(A,B,C)). [resolve(19,c,16,b)].
% 0.38/0.92 Derived: neighborhood(A,B,C,D) | -topological_space(C,D) | -element_of_set(B,A) | -neighborhood(A,E,C,D). [resolve(19,c,18,b)].
% 0.38/0.92 20 element_of_set(A,closure(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | closed(f14(B,C,D,A),C,D) # label(closure_57) # label(axiom). [assumption].
% 0.38/0.92 21 -closed(A,B,C) | topological_space(B,C) # label(closed_set_21) # label(axiom). [assumption].
% 0.38/0.92 22 -element_of_set(A,closure(B,C,D)) | -subset_sets(B,E) | -closed(E,C,D) | element_of_set(A,E) # label(closure_55) # label(axiom). [assumption].
% 0.38/0.92 Derived: element_of_set(A,closure(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(E,closure(F,C,D)) | -subset_sets(F,f14(B,C,D,A)) | element_of_set(E,f14(B,C,D,A)). [resolve(20,d,22,c)].
% 0.38/0.92 23 -closed(A,B,C) | topological_space(B,C). [resolve(14,b,12,a)].
% 0.38/0.92 24 -closed(A,B,C) | element_of_collection(relative_complement_sets(A,B),C). [resolve(14,b,13,a)].
% 0.38/0.92 Derived: element_of_collection(relative_complement_sets(f14(A,B,C,D),B),C) | element_of_set(D,closure(A,B,C)) | -topological_space(B,C) | -subset_sets(A,B). [resolve(24,a,20,d)].
% 0.38/0.92 25 closed(A,B,C) | -topological_space(B,C) | -topological_space(B,C) | -element_of_collection(relative_complement_sets(A,B),C). [resolve(15,c,11,a)].
% 0.38/0.92 Derived: -topological_space(A,B) | -topological_space(A,B) | -element_of_collection(relative_complement_sets(C,A),B) | -element_of_set(D,closure(E,A,B)) | -subset_sets(E,C) | element_of_set(D,C). [resolve(25,a,22,c)].
% 0.38/0.92 26 element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,relative_complement_sets(E,C)) | -subset_sets(relative_complement_sets(E,C),B) | -closed(E,C,D). [resolve(17,f,14,b)].
% 0.38/0.92 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,relative_complement_sets(f14(E,C,D,F),C)) | -subset_sets(relative_complement_sets(f14(E,C,D,F),C),B) | element_of_set(F,closure(E,C,D)) | -topological_space(C,D) | -subset_sets(E,C). [resolve(26,f,20,d)].
% 0.38/0.92 Derived: element_of_set(A,interior(B,C,D)) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(A,relative_complement_sets(E,C)) | -subset_sets(relative_complement_sets(E,C),B) | -topological_space(C,D) | -topological_space(C,D) | -element_of_collection(relative_complement_sets(E,C),D). [resolve(26,f,25,a)].
% 0.38/0.92 27 -neighborhood(relative_complement_sets(A,B),C,B,D) | closed(A,B,D) | -topological_space(B,D). [resolve(18,b,15,c)].
% 0.38/0.92 Derived: -neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | -element_of_set(E,closure(F,B,D)) | -subset_sets(F,A) | element_of_set(E,A). [resolve(27,b,22,c)].
% 0.38/0.92 Derived: -neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | element_of_collection(relative_complement_sets(A,B),D). [resolve(27,b,24,a)].
% 0.38/0.92 28 neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | -element_of_set(C,relative_complement_sets(A,B)) | -closed(A,B,D). [resolve(19,c,14,b)].
% 0.38/0.92 Derived: neighborhood(relative_complement_sets(f14(A,B,C,D),B),E,B,C) | -topological_space(B,C) | -element_of_set(E,relative_complement_sets(f14(A,B,C,D),B)) | element_of_set(D,closure(A,B,C)) | -topological_space(B,C) | -subset_sets(A,B). [resolve(28,d,20,d)].
% 0.38/0.92 Derived: neighborhood(relative_complement_sets(A,B),C,B,D) | -topological_space(B,D) | -element_of_set(C,relative_complement_sets(A,B)) | -topological_space(B,D) | -topological_space(B,D) | -element_of_collection(relative_complement_sets(A,B),D). [resolve(28,d,25,a)].
% 0.38/0.92 29 basis(A,B) | -equal_sets(union_of_members(B),A) | element_of_set(f7(A,B),A) # label(basis_for_topology_32) # label(axiom). [assumption].
% 0.38/0.93 30 -basis(A,B) | equal_sets(union_of_members(B),A) # label(basis_for_topology_28) # label(axiom). [assumption].
% 0.38/0.93 31 -basis(A,B) | -element_of_set(C,A) | -element_of_collection(D,B) | -element_of_collection(E,B) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(A,B,C,D,E)) # label(basis_for_topology_29) # label(axiom). [assumption].
% 0.38/0.93 32 -basis(A,B) | -element_of_set(C,A) | -element_of_collection(D,B) | -element_of_collection(E,B) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(A,B,C,D,E),B) # label(basis_for_topology_30) # label(axiom). [assumption].
% 0.38/0.93 33 -basis(A,B) | -element_of_set(C,A) | -element_of_collection(D,B) | -element_of_collection(E,B) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(A,B,C,D,E),intersection_of_sets(D,E)) # label(basis_for_topology_31) # label(axiom). [assumption].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),B) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(B,A,C,D,E)). [resolve(29,a,31,a)].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),B) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(B,A,C,D,E),A). [resolve(29,a,32,a)].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),B) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(B,A,C,D,E),intersection_of_sets(D,E)). [resolve(29,a,33,a)].
% 0.38/0.93 34 basis(A,B) | -equal_sets(union_of_members(B),A) | element_of_collection(f8(A,B),B) # label(basis_for_topology_33) # label(axiom). [assumption].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f8(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(B,A,C,D,E)). [resolve(34,a,31,a)].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f8(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(B,A,C,D,E),A). [resolve(34,a,32,a)].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f8(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(B,A,C,D,E),intersection_of_sets(D,E)). [resolve(34,a,33,a)].
% 0.38/0.93 35 basis(A,B) | -equal_sets(union_of_members(B),A) | element_of_collection(f9(A,B),B) # label(basis_for_topology_34) # label(axiom). [assumption].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f9(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(B,A,C,D,E)). [resolve(35,a,31,a)].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f9(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(B,A,C,D,E),A). [resolve(35,a,32,a)].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_collection(f9(B,A),A) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(B,A,C,D,E),intersection_of_sets(D,E)). [resolve(35,a,33,a)].
% 0.38/0.93 36 basis(A,B) | -equal_sets(union_of_members(B),A) | element_of_set(f7(A,B),intersection_of_sets(f8(A,B),f9(A,B))) # label(basis_for_topology_35) # label(axiom). [assumption].
% 0.38/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_set(C,f6(B,A,C,D,E)). [resolve(36,a,31,a)].
% 0.63/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | element_of_collection(f6(B,A,C,D,E),A). [resolve(36,a,32,a)].
% 0.63/0.93 Derived: -equal_sets(union_of_members(A),B) | element_of_set(f7(B,A),intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(C,B) | -element_of_collection(D,A) | -element_of_collection(E,A) | -element_of_set(C,intersection_of_sets(D,E)) | subset_sets(f6(B,A,C,D,E),intersection_of_sets(D,E)). [resolve(36,a,33,a)].
% 0.63/0.93 37 basis(A,B) | -equal_sets(union_of_members(B),A) | -element_of_set(f7(A,B),C) | -element_of_collection(C,B) | -subset_sets(C,intersection_of_sets(f8(A,B),f9(A,B))) # label(basis_for_topology_36) # label(axiom). [assumption].
% 0.63/0.93 Derived: -equal_sets(union_of_members(A),B) | -element_of_set(f7(B,A),C) | -element_of_collection(C,A) | -subset_sets(C,intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(D,B) | -element_of_collection(E,A) | -element_of_collection(F,A) | -element_of_set(D,intersection_of_sets(E,F)) | element_of_set(D,f6(B,A,D,E,F)). [resolve(37,a,31,a)].
% 0.63/0.93 Derived: -equal_sets(union_of_members(A),B) | -element_of_set(f7(B,A),C) | -element_of_collection(C,A) | -subset_sets(C,intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(D,B) | -element_of_collection(E,A) | -element_of_collection(F,A) | -element_of_set(D,intersection_of_sets(E,F)) | element_of_collection(f6(B,A,D,E,F),A). [resolve(37,a,32,a)].
% 0.63/0.93 Derived: -equal_sets(union_of_members(A),B) | -element_of_set(f7(B,A),C) | -element_of_collection(C,A) | -subset_sets(C,intersection_of_sets(f8(B,A),f9(B,A))) | -element_of_set(D,B) | -element_of_collection(E,A) | -element_of_collection(F,A) | -element_of_set(D,intersection_of_sets(E,F)) | subset_sets(f6(B,A,D,E,F),intersection_of_sets(E,F)). [resolve(37,a,33,a)].
% 0.63/0.93 38 limit_point(A,B,C,D) | -topological_space(C,D) | -subset_sets(B,C) | neighborhood(f16(A,B,C,D),A,C,D) # label(limit_point_67) # label(axiom). [assumption].
% 0.63/0.93 39 -limit_point(A,B,C,D) | topological_space(C,D) # label(limit_point_63) # label(axiom). [assumption].
% 0.63/0.93 40 -limit_point(A,B,C,D) | subset_sets(B,C) # label(limit_point_64) # label(axiom). [assumption].
% 0.63/0.93 41 -limit_point(A,B,C,D) | -neighborhood(E,A,C,D) | element_of_set(f15(A,B,C,D,E),intersection_of_sets(E,B)) # label(limit_point_65) # label(axiom). [assumption].
% 0.63/0.93 42 -limit_point(A,B,C,D) | -neighborhood(E,A,C,D) | -eq_p(f15(A,B,C,D,E),A) # label(limit_point_66) # label(axiom). [assumption].
% 0.63/0.93 Derived: -topological_space(A,B) | -subset_sets(C,A) | neighborhood(f16(D,C,A,B),D,A,B) | -neighborhood(E,D,A,B) | element_of_set(f15(D,C,A,B,E),intersection_of_sets(E,C)). [resolve(38,a,41,a)].
% 0.63/0.93 Derived: -topological_space(A,B) | -subset_sets(C,A) | neighborhood(f16(D,C,A,B),D,A,B) | -neighborhood(E,D,A,B) | -eq_p(f15(D,C,A,B,E),D). [resolve(38,a,42,a)].
% 0.63/0.93 43 limit_point(A,B,C,D) | -topological_space(C,D) | -subset_sets(B,C) | -element_of_set(E,intersection_of_sets(f16(A,B,C,D),B)) | eq_p(E,A) # label(limit_point_68) # label(axiom). [assumption].
% 0.63/0.93 Derived: -topological_space(A,B) | -subset_sets(C,A) | -element_of_set(D,intersection_of_sets(f16(E,C,A,B),C)) | eq_p(D,E) | -neighborhood(F,E,A,B) | element_of_set(f15(E,C,A,B,F),intersection_of_sets(F,C)). [resolve(43,a,41,a)].
% 0.63/0.93 Derived: -topological_space(A,B) | -subset_sets(C,A) | -element_of_set(D,intersection_of_sets(f16(E,C,A,B),C)) | eq_p(D,E) | -neighborhood(F,E,A,B) | -eq_p(f15(E,C,A,B,F),E). [resolve(43,a,42,a)].
% 0.63/0.93 44 -element_of_set(A,b) | limit_point(A,a,cx,ct) # label(problem_13_160) # label(negated_conjecture). [assumption].
% 0.63/0.93 Derived: -element_of_set(A,b) | topological_space(cx,ct). [resolve(44,b,39,a)].
% 0.63/0.93 Derived: -element_of_set(A,b) | subset_sets(a,cx). [resolve(44,b,40,a)].
% 0.63/0.93 Derived: -element_of_set(A,b) | -neighborhood(B,A,cx,ct) | element_of_set(f15(A,a,cx,ct,B),intersection_of_sets(B,a)). [resolve(44,b,41,a)].
% 0.63/0.93 Derived: -element_of_set(A,b) | -neighborhood(B,A,cx,ct) | -eq_p(f15(A,a,cx,ct,B),A). [resolve(44,b,42,a)].
% 0.63/0.94 45 hausdorff(A,B) | -topological_space(A,B) | element_of_set(f19(A,B),A) # label(hausdorff_77) # label(axiom). [assumption].
% 0.63/0.94 46 -hausdorff(A,B) | topological_space(A,B) # label(hausdorff_73) # label(axiom). [assumption].
% 0.63/0.94 47 -hausdorff(A,B) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f17(A,B,C,D),C,A,B) # label(hausdorff_74) # label(axiom). [assumption].
% 0.63/0.94 48 -hausdorff(A,B) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f18(A,B,C,D),D,A,B) # label(hausdorff_75) # label(axiom). [assumption].
% 0.63/0.94 49 -hausdorff(A,B) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | disjoint_s(f17(A,B,C,D),f18(A,B,C,D)) # label(hausdorff_76) # label(axiom). [assumption].
% 0.63/0.94 Derived: -topological_space(A,B) | element_of_set(f19(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f17(A,B,C,D),C,A,B). [resolve(45,a,47,a)].
% 0.63/0.94 Derived: -topological_space(A,B) | element_of_set(f19(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f18(A,B,C,D),D,A,B). [resolve(45,a,48,a)].
% 0.63/0.94 Derived: -topological_space(A,B) | element_of_set(f19(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | disjoint_s(f17(A,B,C,D),f18(A,B,C,D)). [resolve(45,a,49,a)].
% 0.63/0.94 50 hausdorff(A,B) | -topological_space(A,B) | element_of_set(f20(A,B),A) # label(hausdorff_78) # label(axiom). [assumption].
% 0.63/0.94 Derived: -topological_space(A,B) | element_of_set(f20(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f17(A,B,C,D),C,A,B). [resolve(50,a,47,a)].
% 0.63/0.94 Derived: -topological_space(A,B) | element_of_set(f20(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f18(A,B,C,D),D,A,B). [resolve(50,a,48,a)].
% 0.63/0.94 Derived: -topological_space(A,B) | element_of_set(f20(A,B),A) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | disjoint_s(f17(A,B,C,D),f18(A,B,C,D)). [resolve(50,a,49,a)].
% 0.63/0.94 51 hausdorff(A,B) | -topological_space(A,B) | -eq_p(f19(A,B),f20(A,B)) # label(hausdorff_79) # label(axiom). [assumption].
% 0.63/0.94 Derived: -topological_space(A,B) | -eq_p(f19(A,B),f20(A,B)) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f17(A,B,C,D),C,A,B). [resolve(51,a,47,a)].
% 0.63/0.94 Derived: -topological_space(A,B) | -eq_p(f19(A,B),f20(A,B)) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | neighborhood(f18(A,B,C,D),D,A,B). [resolve(51,a,48,a)].
% 0.63/0.94 Derived: -topological_space(A,B) | -eq_p(f19(A,B),f20(A,B)) | -element_of_set(C,A) | -element_of_set(D,A) | eq_p(C,D) | disjoint_s(f17(A,B,C,D),f18(A,B,C,D)). [resolve(51,a,49,a)].
% 0.63/0.94 52 hausdorff(A,B) | -topological_space(A,B) | -neighborhood(C,f19(A,B),A,B) | -neighborhood(D,f20(A,B),A,B) | -disjoint_s(C,D) # label(hausdorff_80) # label(axiom). [assumption].
% 0.63/0.94 Derived: -topological_space(A,B) | -neighborhood(C,f19(A,B),A,B) | -neighborhood(D,f20(A,B),A,B) | -disjoint_s(C,D) | -element_of_set(E,A) | -element_of_set(F,A) | eq_p(E,F) | neighborhood(f17(A,B,E,F),E,A,B). [resolve(52,a,47,a)].
% 0.63/0.94 Derived: -topological_space(A,B) | -neighborhood(C,f19(A,B),A,B) | -neighborhood(D,f20(A,B),A,B) | -disjoint_s(C,D) | -element_of_set(E,A) | -element_of_set(F,A) | eq_p(E,F) | neighborhood(f18(A,B,E,F),F,A,B). [resolve(52,a,48,a)].
% 0.63/0.94 Derived: -topological_space(A,B) | -neighborhood(C,f19(A,B),A,B) | -neighborhood(D,f20(A,B),A,B) | -disjoint_s(C,D) | -element_of_set(E,A) | -element_of_set(F,A) | eq_p(E,F) | disjoint_s(f17(A,B,E,F),f18(A,B,E,F)). [resolve(52,a,49,a)].
% 0.63/0.94 53 separation(A,B,C,D) | -topological_space(C,D) | equal_sets(A,empty_set) | equal_sets(B,empty_set) | -element_of_collection(A,D) | -element_of_collection(B,D) | -equal_sets(union_of_sets(A,B),C) | -disjoint_s(A,B) # label(separation_88) # label(axiom). [assumption].
% 0.63/0.94 54 -separation(A,B,C,D) | topological_space(C,D) # label(separation_81) # label(axiom). [assumption].
% 0.63/0.94 55 -separation(A,B,C,D) | -equal_sets(A,empty_set) # label(separation_82) # label(axiom). [assumption].
% 0.63/0.94 56 -separation(A,B,C,D) | -equal_sets(B,empty_set) # label(separation_83) # label(axiom). [assumption].
% 0.63/0.94 57 -separation(A,B,C,D) | element_of_collection(A,D) # label(separation_84) # label(axiom). [assumption].
% 0.63/0.94 58 -separation(A,B,C,D) | element_of_collection(B,D) # label(separation_85) # label(axiom). [assumption].
% 0.63/0.94 59 -separation(A,B,C,D) | equal_sets(union_of_sets(A,B),C) # label(separation_86) # label(axiom). [assumption].
% 0.63/0.94 60 -separation(A,B,C,D) | disjoint_s(A,B) # label(separation_87) # label(axiom). [assumption].
% 0.63/0.94 61 -connected_space(A,B) | -separation(C,D,A,B) # label(connected_space_90) # label(axiom). [assumption].
% 0.63/0.94 Derived: -connected_space(A,B) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(61,b,53,a)].
% 0.63/0.94 62 connected_space(A,B) | -topological_space(A,B) | separation(f21(A,B),f22(A,B),A,B) # label(connected_space_91) # label(axiom). [assumption].
% 0.63/0.94 Derived: connected_space(A,B) | -topological_space(A,B) | -equal_sets(f21(A,B),empty_set). [resolve(62,c,55,a)].
% 0.63/0.94 Derived: connected_space(A,B) | -topological_space(A,B) | -equal_sets(f22(A,B),empty_set). [resolve(62,c,56,a)].
% 0.63/0.94 Derived: connected_space(A,B) | -topological_space(A,B) | element_of_collection(f21(A,B),B). [resolve(62,c,57,a)].
% 0.63/0.94 Derived: connected_space(A,B) | -topological_space(A,B) | element_of_collection(f22(A,B),B). [resolve(62,c,58,a)].
% 0.63/0.94 Derived: connected_space(A,B) | -topological_space(A,B) | equal_sets(union_of_sets(f21(A,B),f22(A,B)),A). [resolve(62,c,59,a)].
% 0.63/0.94 Derived: connected_space(A,B) | -topological_space(A,B) | disjoint_s(f21(A,B),f22(A,B)). [resolve(62,c,60,a)].
% 0.63/0.94 63 -connected_set(A,B,C) | connected_space(A,subspace_topology(B,C,A)) # label(connected_set_94) # label(axiom). [assumption].
% 0.63/0.94 64 -connected_space(A,B) | topological_space(A,B) # label(connected_space_89) # label(axiom). [assumption].
% 0.63/0.94 Derived: -connected_set(A,B,C) | topological_space(A,subspace_topology(B,C,A)). [resolve(63,b,64,a)].
% 0.63/0.94 65 connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B) | -connected_space(A,subspace_topology(B,C,A)) # label(connected_set_95) # label(axiom). [assumption].
% 0.63/0.94 66 -connected_space(A,B) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(61,b,53,a)].
% 0.63/0.94 Derived: -topological_space(A,subspace_topology(B,C,A)) | equal_sets(D,empty_set) | equal_sets(E,empty_set) | -element_of_collection(D,subspace_topology(B,C,A)) | -element_of_collection(E,subspace_topology(B,C,A)) | -equal_sets(union_of_sets(D,E),A) | -disjoint_s(D,E) | -connected_set(A,B,C). [resolve(66,a,63,b)].
% 0.63/0.94 67 connected_space(A,B) | -topological_space(A,B) | -equal_sets(f21(A,B),empty_set). [resolve(62,c,55,a)].
% 0.63/0.94 Derived: -topological_space(A,subspace_topology(B,C,A)) | -equal_sets(f21(A,subspace_topology(B,C,A)),empty_set) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(67,a,65,d)].
% 0.63/0.94 Derived: -topological_space(A,B) | -equal_sets(f21(A,B),empty_set) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(67,a,66,a)].
% 0.63/0.94 68 connected_space(A,B) | -topological_space(A,B) | -equal_sets(f22(A,B),empty_set). [resolve(62,c,56,a)].
% 0.63/0.94 Derived: -topological_space(A,subspace_topology(B,C,A)) | -equal_sets(f22(A,subspace_topology(B,C,A)),empty_set) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(68,a,65,d)].
% 0.63/0.94 Derived: -topological_space(A,B) | -equal_sets(f22(A,B),empty_set) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(68,a,66,a)].
% 0.63/0.94 69 connected_space(A,B) | -topological_space(A,B) | element_of_collection(f21(A,B),B). [resolve(62,c,57,a)].
% 0.63/0.94 Derived: -topological_space(A,subspace_topology(B,C,A)) | element_of_collection(f21(A,subspace_topology(B,C,A)),subspace_topology(B,C,A)) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(69,a,65,d)].
% 0.63/0.95 Derived: -topological_space(A,B) | element_of_collection(f21(A,B),B) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(69,a,66,a)].
% 0.63/0.95 70 connected_space(A,B) | -topological_space(A,B) | element_of_collection(f22(A,B),B). [resolve(62,c,58,a)].
% 0.63/0.95 Derived: -topological_space(A,subspace_topology(B,C,A)) | element_of_collection(f22(A,subspace_topology(B,C,A)),subspace_topology(B,C,A)) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(70,a,65,d)].
% 0.63/0.95 Derived: -topological_space(A,B) | element_of_collection(f22(A,B),B) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(70,a,66,a)].
% 0.63/0.95 71 connected_space(A,B) | -topological_space(A,B) | equal_sets(union_of_sets(f21(A,B),f22(A,B)),A). [resolve(62,c,59,a)].
% 0.63/0.95 Derived: -topological_space(A,subspace_topology(B,C,A)) | equal_sets(union_of_sets(f21(A,subspace_topology(B,C,A)),f22(A,subspace_topology(B,C,A))),A) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(71,a,65,d)].
% 0.63/0.95 Derived: -topological_space(A,B) | equal_sets(union_of_sets(f21(A,B),f22(A,B)),A) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(71,a,66,a)].
% 0.63/0.95 72 connected_space(A,B) | -topological_space(A,B) | disjoint_s(f21(A,B),f22(A,B)). [resolve(62,c,60,a)].
% 0.63/0.95 Derived: -topological_space(A,subspace_topology(B,C,A)) | disjoint_s(f21(A,subspace_topology(B,C,A)),f22(A,subspace_topology(B,C,A))) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(72,a,65,d)].
% 0.63/0.95 Derived: -topological_space(A,B) | disjoint_s(f21(A,B),f22(A,B)) | -topological_space(A,B) | equal_sets(C,empty_set) | equal_sets(D,empty_set) | -element_of_collection(C,B) | -element_of_collection(D,B) | -equal_sets(union_of_sets(C,D),A) | -disjoint_s(C,D). [resolve(72,a,66,a)].
% 0.63/0.95 73 connected_set(a,cx,ct) # label(problem_13_159) # label(negated_conjecture). [assumption].
% 0.63/0.95 74 -connected_set(A,B,C) | topological_space(B,C) # label(connected_set_92) # label(axiom). [assumption].
% 0.63/0.95 75 -connected_set(A,B,C) | subset_sets(A,B) # label(connected_set_93) # label(axiom). [assumption].
% 0.63/0.95 Derived: topological_space(cx,ct). [resolve(73,a,74,a)].
% 0.63/0.95 Derived: subset_sets(a,cx). [resolve(73,a,75,a)].
% 0.63/0.95 76 -connected_set(union_of_sets(a,b),cx,ct) # label(problem_13_161) # label(negated_conjecture). [assumption].
% 0.63/0.95 77 -connected_set(A,B,C) | topological_space(A,subspace_topology(B,C,A)). [resolve(63,b,64,a)].
% 0.63/0.95 Derived: topological_space(a,subspace_topology(cx,ct,a)). [resolve(77,a,73,a)].
% 0.63/0.95 78 -topological_space(A,subspace_topology(B,C,A)) | equal_sets(D,empty_set) | equal_sets(E,empty_set) | -element_of_collection(D,subspace_topology(B,C,A)) | -element_of_collection(E,subspace_topology(B,C,A)) | -equal_sets(union_of_sets(D,E),A) | -disjoint_s(D,E) | -connected_set(A,B,C). [resolve(66,a,63,b)].
% 0.63/0.95 Derived: -topological_space(a,subspace_topology(cx,ct,a)) | equal_sets(A,empty_set) | equal_sets(B,empty_set) | -element_of_collection(A,subspace_topology(cx,ct,a)) | -element_of_collection(B,subspace_topology(cx,ct,a)) | -equal_sets(union_of_sets(A,B),a) | -disjoint_s(A,B). [resolve(78,h,73,a)].
% 0.63/0.95 79 -topological_space(A,subspace_topology(B,C,A)) | -equal_sets(f21(A,subspace_topology(B,C,A)),empty_set) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(67,a,65,d)].
% 0.63/0.95 Derived: -topological_space(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))) | -equal_sets(f21(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))),empty_set) | -topological_space(cx,ct) | -subset_sets(union_of_sets(a,b),cx). [resolve(79,c,76,a)].
% 0.63/0.95 80 -topological_space(A,subspace_topology(B,C,A)) | -equal_sets(f22(A,subspace_topology(B,C,A)),empty_set) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(68,a,65,d)].
% 0.63/0.95 Derived: -topological_space(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))) | -equal_sets(f22(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))),empty_set) | -topological_space(cx,ct) | -subset_sets(union_of_sets(a,b),cx). [resolve(80,c,76,a)].
% 0.63/0.95 81 -topological_space(A,subspace_topology(B,C,A)) | element_of_collection(f21(A,subspace_topology(B,C,A)),subspace_topology(B,C,A)) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(69,a,65,d)].
% 0.63/0.95 Derived: -topological_space(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))) | element_of_collection(f21(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))),subspace_topology(cx,ct,union_of_sets(a,b))) | -topological_space(cx,ct) | -subset_sets(union_of_sets(a,b),cx). [resolve(81,c,76,a)].
% 0.63/0.95 82 -topological_space(A,subspace_topology(B,C,A)) | element_of_collection(f22(A,subspace_topology(B,C,A)),subspace_topology(B,C,A)) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(70,a,65,d)].
% 0.63/0.95 Derived: -topological_space(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))) | element_of_collection(f22(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))),subspace_topology(cx,ct,union_of_sets(a,b))) | -topological_space(cx,ct) | -subset_sets(union_of_sets(a,b),cx). [resolve(82,c,76,a)].
% 0.63/0.95 83 -topological_space(A,subspace_topology(B,C,A)) | equal_sets(union_of_sets(f21(A,subspace_topology(B,C,A)),f22(A,subspace_topology(B,C,A))),A) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(71,a,65,d)].
% 0.63/0.95 Derived: -topological_space(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))) | equal_sets(union_of_sets(f21(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))),f22(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b)))),union_of_sets(a,b)) | -topological_space(cx,ct) | -subset_sets(union_of_sets(a,b),cx). [resolve(83,c,76,a)].
% 0.63/0.95 84 -topological_space(A,subspace_topology(B,C,A)) | disjoint_s(f21(A,subspace_topology(B,C,A)),f22(A,subspace_topology(B,C,A))) | connected_set(A,B,C) | -topological_space(B,C) | -subset_sets(A,B). [resolve(72,a,65,d)].
% 0.63/0.95 Derived: -topological_space(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))) | disjoint_s(f21(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b))),f22(union_of_sets(a,b),subspace_topology(cx,ct,union_of_sets(a,b)))) | -topological_space(cx,ct) | -subset_sets(union_of_sets(a,b),cx). [resolve(84,c,76,a)].
% 0.63/0.95 85 compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | element_of_collection(f3(C,f24(A,B)),f24(A,B)). [resolve(10,d,1,f)].
% 0.63/0.95 86 -compact_space(A,B) | -open_covering(C,A,B) | finite(f23(A,B,C)) # label(compact_space_101) # label(axiom). [assumption].
% 0.63/0.95 87 compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | element_of_collection(f4(C,f24(A,B)),f24(A,B)). [resolve(10,d,3,f)].
% 0.63/0.95 88 compact_space(A,B) | -topological_space(A,B) | -finite(f5(C,f24(A,B))) | -open_covering(f5(C,f24(A,B)),A,B) | topological_space(C,f24(A,B)) | -equal_sets(union_of_members(f24(A,B)),C) | -element_of_collection(empty_set,f24(A,B)) | -element_of_collection(C,f24(A,B)) | -element_of_collection(intersection_of_sets(f3(C,f24(A,B)),f4(C,f24(A,B))),f24(A,B)). [resolve(10,d,4,f)].
% 0.63/0.95 89 compact_space(A,B) | -topological_space(A,B) | -finite(C) | -open_covering(C,A,B) | -finer(f24(A,B),C,Cputime limit exceeded (core dumped)
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