%------------------------------------------------------------------------------
% File     : cvc5---1.0.5
% Problem  : SEU324+2 : TPTP v8.1.2. Released v3.3.0.
% Transfm  : none
% Format   : tptp
% Command  : do_cvc5 %s %d

% Computer : n010.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  : 300s
% DateTime : Thu Aug 31 16:30:55 EDT 2023

% Result   : Theorem 6.14s 6.43s
% Output   : Proof 6.14s
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13  % Problem    : SEU324+2 : TPTP v8.1.2. Released v3.3.0.
% 0.10/0.14  % Command    : do_cvc5 %s %d
% 0.10/0.34  % Computer : n010.cluster.edu
% 0.10/0.34  % Model    : x86_64 x86_64
% 0.10/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.34  % Memory   : 8042.1875MB
% 0.10/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.10/0.34  % CPULimit   : 300
% 0.10/0.34  % WCLimit    : 300
% 0.10/0.34  % DateTime   : Wed Aug 23 18:48:51 EDT 2023
% 0.10/0.34  % CPUTime    : 
% 0.15/0.51  %----Proving TF0_NAR, FOF, or CNF
% 6.14/6.43  ------- convert to smt2 : /export/starexec/sandbox2/tmp/tmp.a5th9xsyI6/cvc5---1.0.5_19196.p...
% 6.14/6.43  ------- get file name : TPTP file name is SEU324+2
% 6.14/6.43  ------- cvc5-fof : /export/starexec/sandbox2/solver/bin/cvc5---1.0.5_19196.smt2...
% 6.14/6.43  --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10...
% 6.14/6.43  % SZS status Theorem for SEU324+2
% 6.14/6.43  % SZS output start Proof for SEU324+2
% 6.14/6.43  (
% 6.14/6.43  (let ((_let_1 (not (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.top_str B) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.the_carrier A))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier B))) (and (=> (tptp.open_subset D B) (= (tptp.interior B D) D)) (=> (= (tptp.interior A C) C) (tptp.open_subset C A)))))))))))))) (let ((_let_2 (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.closed_subset B A))) (let ((_let_2 (= (tptp.topstr_closure A B) B))) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (and (=> _let_1 _let_2) (=> (and (tptp.topological_space A) _let_2) _let_1)))))))))) (let ((_let_3 (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.open_subset B A) (tptp.closed_subset (tptp.subset_complement _let_1 B) A))))))))) (let ((_let_4 (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.subset_complement _let_1 B) (tptp.subset_difference _let_1 (tptp.cast_as_carrier_subset A) B))))))))) (let ((_let_5 (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (= (tptp.subset_complement A (tptp.subset_complement A B)) B))))) (let ((_let_6 (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (tptp.open_subset (tptp.interior A B) A))))) (let ((_let_7 (tptp.empty tptp.empty_set))) (let ((_let_8 (tptp.relation tptp.empty_set))) (let ((_let_9 (tptp.relation_empty_yielding tptp.empty_set))) (let ((_let_10 (forall ((A $$unsorted)) (=> (tptp.top_str A) (tptp.one_sorted_str A))))) (let ((_let_11 (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset (tptp.the_carrier A)))) (=> (and (tptp.top_str A) (tptp.element B _let_1)) (tptp.element (tptp.topstr_closure A B) _let_1)))))) (let ((_let_12 (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B _let_1) (tptp.element (tptp.subset_complement A B) _let_1)))))) (let ((_let_13 (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (= (tptp.subset_complement A B) (tptp.set_difference A B)))))) (let ((_let_14 (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (= (tptp.cast_as_carrier_subset A) (tptp.the_carrier A)))))) (let ((_let_15 (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.interior A B) (tptp.subset_complement _let_1 (tptp.topstr_closure A (tptp.subset_complement _let_1 B))))))))))) (let ((_let_16 (tptp.the_carrier SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28))) (let ((_let_17 (tptp.subset_complement _let_16 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_27))) (let ((_let_18 (tptp.topstr_closure SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28 _let_17))) (let ((_let_19 (tptp.subset_complement _let_16 _let_18))) (let ((_let_20 (tptp.interior SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_27))) (let ((_let_21 (= _let_20 _let_19))) (let ((_let_22 (tptp.subset_complement _let_16 _let_17))) (let ((_let_23 (= SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_27 _let_22))) (let ((_let_24 (tptp.cast_as_carrier_subset SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28))) (let ((_let_25 (= _let_16 _let_24))) (let ((_let_26 (= _let_17 (tptp.subset_difference _let_16 _let_24 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_27)))) (let ((_let_27 (= _let_22 (tptp.set_difference _let_16 _let_17)))) (let ((_let_28 (= _let_19 (tptp.set_difference _let_16 _let_18)))) (let ((_let_29 (= SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_27 _let_20))) (let ((_let_30 (= _let_17 _let_18))) (let ((_let_31 (tptp.powerset _let_16))) (let ((_let_32 (tptp.element SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_27 _let_31))) (let ((_let_33 (not _let_32))) (let ((_let_34 (tptp.top_str SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28))) (let ((_let_35 (not _let_34))) (let ((_let_36 (or _let_35 _let_33 _let_21))) (let ((_let_37 (forall ((A $$unsorted) (BOUND_VARIABLE_9525 $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (or (not (tptp.top_str A)) (not (tptp.element BOUND_VARIABLE_9525 (tptp.powerset _let_1))) (= (tptp.interior A BOUND_VARIABLE_9525) (tptp.subset_complement _let_1 (tptp.topstr_closure A (tptp.subset_complement _let_1 BOUND_VARIABLE_9525))))))))) (let ((_let_38 (EQ_RESOLVE (ASSUME :args (_let_15)) (MACRO_SR_EQ_INTRO :args (_let_15 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_39 (tptp.open_subset SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_26 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_25))) (let ((_let_40 (tptp.interior SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_25 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_26))) (let ((_let_41 (= SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_26 _let_40))) (let ((_let_42 (not _let_41))) (let ((_let_43 (or _let_42 _let_39))) (let ((_let_44 (tptp.open_subset SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_27 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28))) (let ((_let_45 (not _let_44))) (let ((_let_46 (or _let_45 _let_29))) (let ((_let_47 (and _let_46 _let_43))) (let ((_let_48 (tptp.element SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_26 (tptp.powerset (tptp.the_carrier SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_25))))) (let ((_let_49 (not _let_48))) (let ((_let_50 (tptp.top_str SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_25))) (let ((_let_51 (not _let_50))) (let ((_let_52 (tptp.topological_space SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_25))) (let ((_let_53 (not _let_52))) (let ((_let_54 (or _let_53 _let_51 _let_35 _let_49 _let_33 _let_47))) (let ((_let_55 (forall ((A $$unsorted) (BOUND_VARIABLE_16686 $$unsorted) (BOUND_VARIABLE_16684 $$unsorted) (BOUND_VARIABLE_16682 $$unsorted)) (or (not (tptp.topological_space A)) (not (tptp.top_str A)) (not (tptp.top_str BOUND_VARIABLE_16682)) (not (tptp.element BOUND_VARIABLE_16686 (tptp.powerset (tptp.the_carrier A)))) (not (tptp.element BOUND_VARIABLE_16684 (tptp.powerset (tptp.the_carrier BOUND_VARIABLE_16682)))) (and (or (not (tptp.open_subset BOUND_VARIABLE_16684 BOUND_VARIABLE_16682)) (= BOUND_VARIABLE_16684 (tptp.interior BOUND_VARIABLE_16682 BOUND_VARIABLE_16684))) (or (not (= BOUND_VARIABLE_16686 (tptp.interior A BOUND_VARIABLE_16686))) (tptp.open_subset BOUND_VARIABLE_16686 A))))))) (let ((_let_56 (not _let_54))) (let ((_let_57 (EQ_RESOLVE (ASSUME :args (_let_1)) (MACRO_SR_EQ_INTRO :args (_let_1 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_58 (or))) (let ((_let_59 (not _let_55))) (let ((_let_60 (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (IMPLIES_ELIM (SCOPE (SKOLEMIZE _let_57) :args (_let_59))) (CONG (MACRO_SR_PRED_INTRO :args ((= (not _let_59) _let_55))) (REFL :args (_let_56)) :args _let_58)) _let_57 :args (_let_56 true _let_55)))) (let ((_let_61 (REFL :args (_let_54)))) (let ((_let_62 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_54 4)) (CONG _let_61 (MACRO_SR_PRED_INTRO :args ((= (not _let_33) _let_32))) :args _let_58)) :args ((or _let_32 _let_54))) _let_60 :args (_let_32 true _let_54)))) (let ((_let_63 (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_54 2)) (CONG _let_61 (MACRO_SR_PRED_INTRO :args ((= (not _let_35) _let_34))) :args _let_58)) :args ((or _let_34 _let_54))) _let_60 :args (_let_34 true _let_54)))) (let ((_let_64 (or _let_33 _let_23))) (let ((_let_65 (forall ((A $$unsorted) (B $$unsorted)) (or (not (tptp.element B (tptp.powerset A))) (= B (tptp.subset_complement A (tptp.subset_complement A B))))))) (let ((_let_66 (EQ_RESOLVE (ASSUME :args (_let_5)) (MACRO_SR_EQ_INTRO :args (_let_5 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_67 (tptp.one_sorted_str SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28))) (let ((_let_68 (not _let_67))) (let ((_let_69 (or _let_68 _let_25))) (let ((_let_70 (forall ((A $$unsorted)) (or (not (tptp.one_sorted_str A)) (= (tptp.cast_as_carrier_subset A) (tptp.the_carrier A)))))) (let ((_let_71 (EQ_RESOLVE (ASSUME :args (_let_14)) (MACRO_SR_EQ_INTRO :args (_let_14 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_72 (or _let_35 _let_67))) (let ((_let_73 (forall ((A $$unsorted)) (or (not (tptp.top_str A)) (tptp.one_sorted_str A))))) (let ((_let_74 (EQ_RESOLVE (ASSUME :args (_let_10)) (MACRO_SR_EQ_INTRO :args (_let_10 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_75 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_72)) :args ((or _let_35 _let_67 (not _let_72)))) _let_63 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_74 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((not (= (tptp.top_str A) false))))) :args (_let_73))) _let_74 :args (_let_72 false _let_73)) :args (_let_67 false _let_34 false _let_72)))) (let ((_let_76 (or _let_68 _let_33 _let_26))) (let ((_let_77 (forall ((A $$unsorted) (BOUND_VARIABLE_14765 $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (or (not (tptp.one_sorted_str A)) (not (tptp.element BOUND_VARIABLE_14765 (tptp.powerset _let_1))) (= (tptp.subset_complement _let_1 BOUND_VARIABLE_14765) (tptp.subset_difference _let_1 (tptp.cast_as_carrier_subset A) BOUND_VARIABLE_14765))))))) (let ((_let_78 (EQ_RESOLVE (ASSUME :args (_let_4)) (MACRO_SR_EQ_INTRO :args (_let_4 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_79 (tptp.element _let_17 _let_31))) (let ((_let_80 (not _let_79))) (let ((_let_81 (or _let_80 _let_27))) (let ((_let_82 (forall ((A $$unsorted) (B $$unsorted)) (or (not (tptp.element B (tptp.powerset A))) (= (tptp.subset_complement A B) (tptp.set_difference A B)))))) (let ((_let_83 (EQ_RESOLVE (ASSUME :args (_let_13)) (MACRO_SR_EQ_INTRO :args (_let_13 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_84 (_let_82))) (let ((_let_85 ((tptp.subset_complement A B)))) (let ((_let_86 (or _let_33 _let_79))) (let ((_let_87 (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (or (not (tptp.element B _let_1)) (tptp.element (tptp.subset_complement A B) _let_1)))))) (let ((_let_88 (EQ_RESOLVE (ASSUME :args (_let_12)) (MACRO_SR_EQ_INTRO :args (_let_12 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_89 (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_86)) :args ((or _let_33 _let_79 (not _let_86)))) _let_62 (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_88 :args (_let_16 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_27 QUANTIFIERS_INST_E_MATCHING_SIMPLE ((tptp.subset_complement A B)))) :args (_let_87))) _let_88 :args (_let_86 false _let_87)) :args (_let_79 false _let_32 false _let_86)))) (let ((_let_90 (tptp.element _let_18 _let_31))) (let ((_let_91 (not _let_90))) (let ((_let_92 (or _let_91 _let_28))) (let ((_let_93 (or _let_35 _let_80 _let_90))) (let ((_let_94 (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset (tptp.the_carrier A)))) (or (not (tptp.top_str A)) (not (tptp.element B _let_1)) (tptp.element (tptp.topstr_closure A B) _let_1)))))) (let ((_let_95 (EQ_RESOLVE (ASSUME :args (_let_11)) (MACRO_SR_EQ_INTRO :args (_let_11 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_96 (tptp.open_subset _let_40 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_25))) (let ((_let_97 (or _let_53 _let_51 _let_49 _let_96))) (let ((_let_98 (forall ((A $$unsorted) (B $$unsorted)) (or (not (tptp.topological_space A)) (not (tptp.top_str A)) (not (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (tptp.open_subset (tptp.interior A B) A))))) (let ((_let_99 (EQ_RESOLVE (ASSUME :args (_let_6)) (MACRO_SR_EQ_INTRO :args (_let_6 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_100 (not _let_39))) (let ((_let_101 (ASSUME :args (_let_96)))) (let ((_let_102 (ASSUME :args (_let_41)))) (let ((_let_103 (ASSUME :args (_let_100)))) (let ((_let_104 (MACRO_RESOLUTION_TRUST (CNF_AND_NEG :args (_let_47)) (MACRO_RESOLUTION_TRUST (CNF_OR_NEG :args (_let_54 5)) _let_60 :args ((not _let_47) true _let_54)) (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (NOT_AND (MACRO_SR_PRED_TRANSFORM (SCOPE (AND_INTRO _let_101 _let_102 _let_103) :args (_let_41 _let_100 _let_96)) (SCOPE (MACRO_SR_PRED_ELIM (TRANS (SYMM (FALSE_INTRO _let_103)) (CONG (SYMM (SYMM _let_102)) (REFL :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_25)) :args (APPLY_UF tptp.open_subset)) (TRUE_INTRO _let_101))) :args (_let_96 _let_41 _let_100)) :args ((not (and _let_41 _let_100 _let_96)) SB_LITERAL))) (CONG (REFL :args (_let_42)) (MACRO_SR_PRED_INTRO :args ((= (not _let_100) _let_39))) (REFL :args ((not _let_96))) :args _let_58)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_97)) :args ((or _let_53 _let_51 _let_49 _let_96 (not _let_97)))) (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_54 0)) (CONG _let_61 (MACRO_SR_PRED_INTRO :args ((= (not _let_53) _let_52))) :args _let_58)) :args ((or _let_52 _let_54))) _let_60 :args (_let_52 true _let_54)) (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_54 1)) (CONG _let_61 (MACRO_SR_PRED_INTRO :args ((= (not _let_51) _let_50))) :args _let_58)) :args ((or _let_50 _let_54))) _let_60 :args (_let_50 true _let_54)) (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_54 3)) (CONG _let_61 (MACRO_SR_PRED_INTRO :args ((= (not _let_49) _let_48))) :args _let_58)) :args ((or _let_48 _let_54))) _let_60 :args (_let_48 true _let_54)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_99 :args (SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_25 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_26 QUANTIFIERS_INST_E_MATCHING ((not (= (tptp.element B (tptp.powerset (tptp.the_carrier A))) false))))) :args (_let_98))) _let_99 :args (_let_97 false _let_98)) :args (_let_96 false _let_52 false _let_50 false _let_48 false _let_97)) (CNF_OR_NEG :args (_let_43 1)) (REORDERING (EQ_RESOLVE (CNF_OR_NEG :args (_let_43 0)) (CONG (REFL :args (_let_43)) (MACRO_SR_PRED_INTRO :args ((= (not _let_42) _let_41))) :args _let_58)) :args ((or _let_41 _let_43))) :args (_let_43 false _let_96 true _let_39 false _let_41)) :args ((not _let_46) true _let_47 false _let_43)))) (let ((_let_105 (tptp.closed_subset _let_17 SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28))) (let ((_let_106 (not _let_105))) (let ((_let_107 (or _let_106 _let_30))) (let ((_let_108 (not _let_30))) (let ((_let_109 (and _let_107 (or (not (tptp.topological_space SKOLEM_FUN_QUANTIFIERS_SKOLEMIZE_28)) _let_108 _let_105)))) (let ((_let_110 (or _let_35 _let_80 _let_109))) (let ((_let_111 (forall ((A $$unsorted) (BOUND_VARIABLE_16308 $$unsorted)) (let ((_let_1 (tptp.closed_subset BOUND_VARIABLE_16308 A))) (let ((_let_2 (= BOUND_VARIABLE_16308 (tptp.topstr_closure A BOUND_VARIABLE_16308)))) (or (not (tptp.top_str A)) (not (tptp.element BOUND_VARIABLE_16308 (tptp.powerset (tptp.the_carrier A)))) (and (or (not _let_1) _let_2) (or (not (tptp.topological_space A)) (not _let_2) _let_1)))))))) (let ((_let_112 (EQ_RESOLVE (ASSUME :args (_let_2)) (MACRO_SR_EQ_INTRO :args (_let_2 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_113 (= _let_44 _let_105))) (let ((_let_114 (or _let_35 _let_33 _let_113))) (let ((_let_115 (forall ((A $$unsorted) (BOUND_VARIABLE_15420 $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (or (not (tptp.top_str A)) (not (tptp.element BOUND_VARIABLE_15420 (tptp.powerset _let_1))) (= (tptp.open_subset BOUND_VARIABLE_15420 A) (tptp.closed_subset (tptp.subset_complement _let_1 BOUND_VARIABLE_15420) A))))))) (let ((_let_116 (EQ_RESOLVE (ASSUME :args (_let_3)) (MACRO_SR_EQ_INTRO :args (_let_3 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_117 (and _let_21 _let_25 _let_23 _let_26 _let_27 _let_28 _let_30))) (let ((_let_118 (ASSUME :args (_let_21)))) (let ((_let_119 (ASSUME :args (_let_28)))) (let ((_let_120 (APPLY_UF tptp.set_difference))) (let ((_let_121 (ASSUME :args (_let_30)))) (let ((_let_122 (ASSUME :args (_let_26)))) (let ((_let_123 (ASSUME :args (_let_25)))) (let ((_let_124 (ASSUME :args (_let_27)))) (let ((_let_125 (ASSUME :args (_let_23)))) (SCOPE (SCOPE (MACRO_RESOLUTION_TRUST (REORDERING (RESOLUTION (CNF_AND_NEG :args (_let_117)) (IMPLIES_ELIM (SCOPE (MODUS_PONENS (AND_INTRO _let_118 _let_119 _let_123 _let_121 _let_122 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(tptp.relation_rng B) A) (tptp.in (tptp.relation_dom B) tptp.omega))))) (forall ((A $$unsorted)) (= (tptp.function A) (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.in (tptp.ordered_pair B C) A) (tptp.in (tptp.ordered_pair B D) A)) (= C D))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.quasi_total C A B))) (let ((_let_2 (= A tptp.empty_set))) (let ((_let_3 (= B tptp.empty_set))) (=> (tptp.relation_of2_as_subset C A B) (and (=> (=> _let_3 _let_2) (= _let_1 (= A (tptp.relation_dom_as_subset A B C)))) (=> _let_3 (or _let_2 (= _let_1 (= C tptp.empty_set)))))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.join_semilatt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element C _let_1) (= (tptp.join A B C) (tptp.apply_binary_as_element _let_1 _let_1 _let_1 (tptp.the_L_join A) B C))))))))) (forall ((A $$unsorted)) (=> (exists ((B $$unsorted) (C $$unsorted)) (= A (tptp.ordered_pair B C))) (forall ((B $$unsorted)) (= (= B (tptp.pair_first A)) (forall ((C $$unsorted) (D $$unsorted)) (=> (= A (tptp.ordered_pair C D)) (= B C))))))) (forall ((A $$unsorted)) (= (tptp.succ A) (tptp.set_union2 A (tptp.singleton A)))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (= (tptp.topological_space A) (and (tptp.in (tptp.the_carrier A) (tptp.the_topology A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_topology A))) (let ((_let_2 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset (tptp.powerset _let_2))) (=> (tptp.subset B _let_1) (tptp.in (tptp.union_of_subsets _let_2 B) _let_1)))))) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (forall ((C $$unsorted)) (let ((_let_1 (tptp.the_topology A))) (let ((_let_2 (tptp.the_carrier A))) (=> (tptp.element C (tptp.powerset _let_2)) (=> (and (tptp.in B _let_1) (tptp.in C _let_1)) (tptp.in (tptp.subset_intersection2 _let_2 B C) _let_1)))))))))))) (forall ((A $$unsorted)) (= (tptp.relation A) (forall ((B $$unsorted)) (not (and (tptp.in B A) (forall ((C $$unsorted) (D $$unsorted)) (not (= B (tptp.ordered_pair C D))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_reflexive_in A B) (forall ((C $$unsorted)) (=> (tptp.in C B) (tptp.in (tptp.ordered_pair C C) A))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (tptp.relation_of2 C A B) (tptp.subset C (tptp.cartesian_product2 A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (= B (tptp.set_meet A)))) (let ((_let_2 (= A tptp.empty_set))) (and (=> (not _let_2) (= _let_1 (forall ((C $$unsorted)) (= (tptp.in C B) (forall ((D $$unsorted)) (=> (tptp.in D A) (tptp.in C D))))))) (=> _let_2 (= _let_1 (= B tptp.empty_set))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (= (tptp.empty_carrier A) (tptp.empty (tptp.the_carrier A))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= B (tptp.singleton A)) (forall ((C $$unsorted)) (= (tptp.in C B) (= C A))))) _let_15 (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted) (C $$unsorted)) (= (= C (tptp.fiber A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (not (= D B)) (tptp.in (tptp.ordered_pair D B) A)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (= B (tptp.inclusion_relation A)) (and (= (tptp.relation_field B) A) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C A) (tptp.in D A)) (= (tptp.in (tptp.ordered_pair C D) B) (tptp.subset C D)))))))) (forall ((A $$unsorted)) (= (= A tptp.empty_set) (forall ((B $$unsorted)) (not (tptp.in B A))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= B (tptp.powerset A)) (forall ((C $$unsorted)) (= (tptp.in C B) (tptp.subset C A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_semilatt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element C _let_1) (= (tptp.meet A B C) (tptp.apply_binary_as_element _let_1 _let_1 _let_1 (tptp.the_L_meet A) B C))))))))) (forall ((A $$unsorted)) (=> (exists ((B $$unsorted) (C $$unsorted)) (= A (tptp.ordered_pair B C))) (forall ((B $$unsorted)) (= (= B (tptp.pair_second A)) (forall ((C $$unsorted) (D $$unsorted)) (=> (= A (tptp.ordered_pair C D)) (= B D))))))) (forall ((A $$unsorted)) (= (tptp.epsilon_transitive A) (forall ((B $$unsorted)) (=> (tptp.in B A) (tptp.subset B A))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (= (tptp.empty_carrier_subset A) tptp.empty_set))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (= A B) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.ordered_pair C D))) (= (tptp.in _let_1 A) (tptp.in _let_1 B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.element B A))) (let ((_let_2 (tptp.empty A))) (and (=> (not _let_2) (= _let_1 (tptp.in B A))) (=> _let_2 (= _let_1 (tptp.empty B))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.unordered_pair A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (or (= D A) (= D B)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.well_founded_relation A) (forall ((B $$unsorted)) (not (and (tptp.subset B (tptp.relation_field A)) (not (= B tptp.empty_set)) (forall ((C $$unsorted)) (not (and (tptp.in C B) (tptp.disjoint (tptp.fiber A C) B)))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.set_union2 A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (or (tptp.in D A) (tptp.in D B)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.cartesian_product2 A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted) (F $$unsorted)) (and (tptp.in E A) (tptp.in F B) (= D (tptp.ordered_pair E F)))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.join_semilatt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.below A B C) (= (tptp.join A B C) C)))))))) (forall ((A $$unsorted)) (= (tptp.epsilon_connected A) (forall ((B $$unsorted) (C $$unsorted)) (not (and (tptp.in B A) (tptp.in C A) (not (tptp.in B C)) (not (= B C)) (not (tptp.in C B))))))) _let_14 (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (tptp.subset A B) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.ordered_pair C D))) (=> (tptp.in _let_1 A) (tptp.in _let_1 B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.subset A B) (forall ((C $$unsorted)) (=> (tptp.in C A) (tptp.in C B))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_well_founded_in A B) (forall ((C $$unsorted)) (not (and (tptp.subset C B) (not (= C tptp.empty_set)) (forall ((D $$unsorted)) (not (and (tptp.in D C) (tptp.disjoint (tptp.fiber A D) C))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.set_intersection2 A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D A) (tptp.in D B)))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (forall ((B $$unsorted) (C $$unsorted)) (let ((_let_1 (= C (tptp.apply A B)))) (let ((_let_2 (tptp.in B (tptp.relation_dom A)))) (and (=> _let_2 (= _let_1 (tptp.in (tptp.ordered_pair B C) A))) (=> (not _let_2) (= _let_1 (= C tptp.empty_set))))))))) (forall ((A $$unsorted)) (= (tptp.ordinal A) (and (tptp.epsilon_transitive A) (tptp.epsilon_connected A)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (= B (tptp.relation_dom A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (tptp.in (tptp.ordered_pair C D) A)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_antisymmetric_in A B) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C B) (tptp.in D B) (tptp.in (tptp.ordered_pair C D) A) (tptp.in (tptp.ordered_pair D C) A)) (= C D))))))) (forall ((A $$unsorted)) (= (tptp.cast_to_subset A) A)) (forall ((A $$unsorted) (B $$unsorted)) (= (= B (tptp.union A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (and (tptp.in C D) (tptp.in D A))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.well_ordering A) (and (tptp.reflexive A) (tptp.transitive A) (tptp.antisymmetric A) (tptp.connected A) (tptp.well_founded_relation A))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.equipotent A B) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (tptp.one_to_one C) (= (tptp.relation_dom C) A) (= (tptp.relation_rng C) B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (= C (tptp.set_difference A B)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D A) (not (tptp.in D B))))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (forall ((B $$unsorted)) (= (= B (tptp.relation_rng A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (and (tptp.in D (tptp.relation_dom A)) (= C (tptp.apply A D)))))))))) (forall ((A $$unsorted)) (= (= A tptp.omega) (and (tptp.in tptp.empty_set A) (tptp.being_limit_ordinal A) (tptp.ordinal A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (=> (and (tptp.in tptp.empty_set B) (tptp.being_limit_ordinal B)) (tptp.subset A B))))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (= (tptp.open_subset B A) (tptp.in B (tptp.the_topology A))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (= B (tptp.relation_rng A)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (tptp.in (tptp.ordered_pair D C) A)))))))) _let_13 (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.ordered_pair A B) (tptp.unordered_pair (tptp.unordered_pair A B) (tptp.singleton A)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.well_orders A B) (and (tptp.is_reflexive_in A B) (tptp.is_transitive_in A B) (tptp.is_antisymmetric_in A B) (tptp.is_connected_in A B) (tptp.is_well_founded_in A B)))))) (forall ((A $$unsorted)) (= (tptp.being_limit_ordinal A) (= A (tptp.union A)))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.closed_subset B A) (tptp.open_subset (tptp.subset_difference _let_1 (tptp.cast_as_carrier_subset A) B) A))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_field A) (tptp.set_union2 (tptp.relation_dom A) (tptp.relation_rng A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_connected_in A B) (forall ((C $$unsorted) (D $$unsorted)) (not (and (tptp.in C B) (tptp.in D B) (not (= C D)) (not (tptp.in (tptp.ordered_pair C D) A)) (not (tptp.in (tptp.ordered_pair D C) A))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.relation_restriction A B) (tptp.set_intersection2 A (tptp.cartesian_product2 B B)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (= B (tptp.relation_inverse A)) (forall ((C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair C D) B) (tptp.in (tptp.ordered_pair D C) A)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.relation_isomorphism A B C) (and (= (tptp.relation_dom C) (tptp.relation_field A)) (= (tptp.relation_rng C) (tptp.relation_field B)) (tptp.one_to_one C) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.relation_field A))) (= (tptp.in (tptp.ordered_pair D E) A) (and (tptp.in D _let_1) (tptp.in E _let_1) (tptp.in (tptp.ordered_pair (tptp.apply C D) (tptp.apply C E)) B))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.disjoint A B) (= (tptp.set_intersection2 A B) tptp.empty_set))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (= (tptp.one_to_one A) (forall ((B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_dom A))) (=> (and (tptp.in B _let_1) (tptp.in C _let_1) (= (tptp.apply A B) (tptp.apply A C))) (= B C))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.latt_str A)) (= (tptp.meet_absorbing A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (= (tptp.join A (tptp.meet A B C) C) C)))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (tptp.relation C) (= (= C (tptp.relation_composition A B)) (forall ((D $$unsorted) (E $$unsorted)) (= (tptp.in (tptp.ordered_pair D E) C) (exists ((F $$unsorted)) (and (tptp.in (tptp.ordered_pair D F) A) (tptp.in (tptp.ordered_pair F E) B)))))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (= (tptp.is_transitive_in A B) (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (tptp.in C B) (tptp.in D B) (tptp.in E B) (tptp.in (tptp.ordered_pair C D) A) (tptp.in (tptp.ordered_pair D E) A)) (tptp.in (tptp.ordered_pair C E) A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.powerset A))) (= (= C (tptp.complements_of_subsets A B)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset A)) (= (tptp.in D C) (tptp.in (tptp.subset_complement A D) B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.proper_subset A B) (and (tptp.subset A B) (not (= A B))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (= (tptp.function_inverse A) (tptp.relation_inverse A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.reflexive A) (tptp.is_reflexive_in A (tptp.relation_field A))))) true true true true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.join_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.join A B C) _let_1)))) true true (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (tptp.element (tptp.empty_carrier_subset A) (tptp.powerset (tptp.the_carrier A))))) true true true (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset (tptp.the_carrier A)))) (=> (and (tptp.top_str A) (tptp.element B _let_1)) (tptp.element (tptp.interior A B) _let_1)))) true (forall ((A $$unsorted)) (tptp.relation (tptp.inclusion_relation A))) true true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted) (E $$unsorted) (F $$unsorted)) (let ((_let_1 (tptp.cartesian_product2 A B))) (=> (and (not (tptp.empty A)) (not (tptp.empty B)) (tptp.function D) (tptp.quasi_total D _let_1 C) (tptp.relation_of2 D _let_1 C) (tptp.element E A) (tptp.element F B)) (tptp.element (tptp.apply_binary_as_element A B C D E F) C)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.function_inverse A))) (=> (and (tptp.relation A) (tptp.function A)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.meet A B C) _let_1)))) true (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (tptp.element (tptp.cast_as_carrier_subset A) (tptp.powerset (tptp.the_carrier A))))) true (forall ((A $$unsorted)) (tptp.element (tptp.cast_to_subset A) (tptp.powerset A))) true (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation A) (tptp.relation (tptp.relation_restriction A B)))) true true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.join_commutative A) (tptp.join_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.join_commut A B C) _let_1)))) true _let_12 true true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.meet_commut A B C) _let_1)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (tptp.relation (tptp.relation_inverse A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2 C A B) (tptp.element (tptp.relation_dom_as_subset A B C) (tptp.powerset A)))) true true true (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.relation_composition A B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2 C A B) (tptp.element (tptp.relation_rng_as_subset A B C) (tptp.powerset B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B (tptp.powerset _let_1)) (tptp.element (tptp.union_of_subsets A B) _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.subset_intersection2 A B C) _let_1)))) _let_11 (forall ((A $$unsorted)) (tptp.relation (tptp.identity_relation A))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (tptp.element B (tptp.powerset _let_1)) (tptp.element (tptp.meet_of_subsets A B) _let_1)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (tptp.element (tptp.subset_difference A B C) _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation A) (tptp.relation (tptp.relation_dom_restriction A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.powerset (tptp.powerset A)))) (=> (tptp.element B _let_1) (tptp.element (tptp.complements_of_subsets A B) _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.relation (tptp.relation_rng_restriction A B)))) true (forall ((A $$unsorted)) (=> (tptp.meet_semilatt_str A) (tptp.one_sorted_str A))) _let_10 true (forall ((A $$unsorted)) (=> (tptp.join_semilatt_str A) (tptp.one_sorted_str A))) (forall ((A $$unsorted)) (=> (tptp.latt_str A) (and (tptp.meet_semilatt_str A) (tptp.join_semilatt_str A)))) true true (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2_as_subset C A B) (tptp.element C (tptp.powerset (tptp.cartesian_product2 A B))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.cartesian_product2 _let_1 _let_1))) (let ((_let_3 (tptp.the_L_meet A))) (=> (tptp.meet_semilatt_str A) (and (tptp.function _let_3) (tptp.quasi_total _let_3 _let_2 _let_1) (tptp.relation_of2_as_subset _let_3 _let_2 _let_1))))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (tptp.element (tptp.the_topology A) (tptp.powerset (tptp.powerset (tptp.the_carrier A)))))) true (forall ((A $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (let ((_let_2 (tptp.cartesian_product2 _let_1 _let_1))) (let ((_let_3 (tptp.the_L_join A))) (=> (tptp.join_semilatt_str A) (and (tptp.function _let_3) (tptp.quasi_total _let_3 _let_2 _let_1) (tptp.relation_of2_as_subset _let_3 _let_2 _let_1))))))) (exists ((A $$unsorted)) (tptp.meet_semilatt_str A)) (exists ((A $$unsorted)) (tptp.top_str A)) (exists ((A $$unsorted)) (tptp.one_sorted_str A)) (exists ((A $$unsorted)) (tptp.join_semilatt_str A)) (exists ((A $$unsorted)) (tptp.latt_str A)) (forall ((A $$unsorted) (B $$unsorted)) (exists ((C $$unsorted)) (tptp.relation_of2 C A B))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (tptp.element B A))) (forall ((A $$unsorted) (B $$unsorted)) (exists ((C $$unsorted)) (tptp.relation_of2_as_subset C A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.finite B) (tptp.finite (tptp.set_intersection2 A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_composition B A))) (=> (and (tptp.empty A) (tptp.relation B)) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.finite A) (tptp.finite (tptp.set_intersection2 A B)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_inverse A))) (=> (tptp.empty A) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.finite A) (tptp.finite (tptp.set_difference A B)))) (and _let_7 _let_8 _let_9) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.function A) (tptp.finite B)) (tptp.finite (tptp.relation_image A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_dom_restriction A B))) (=> (and (tptp.relation A) (tptp.relation_empty_yielding A)) (and (tptp.relation _let_1) (tptp.relation_empty_yielding _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.singleton A))) (and (not (tptp.empty _let_1)) (tptp.finite _let_1)))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.powerset A))) (and (not (tptp.empty _let_1)) (tptp.cup_closed _let_1) (tptp.diff_closed _let_1) (tptp.preboolean _let_1)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_composition A B))) (=> (and (tptp.relation A) (tptp.function A) (tptp.relation B) (tptp.function B)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted)) (not (tptp.empty (tptp.succ A)))) (and (tptp.epsilon_transitive tptp.omega) (tptp.epsilon_connected tptp.omega) (tptp.ordinal tptp.omega) (not (tptp.empty tptp.omega))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.empty_carrier_subset A))) (=> (tptp.one_sorted_str A) (and (tptp.empty _let_1) (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1) (tptp.v5_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.set_intersection2 A B)))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (not (tptp.empty (tptp.the_carrier A))))) (forall ((A $$unsorted)) (not (tptp.empty (tptp.powerset A)))) _let_7 (forall ((A $$unsorted) (B $$unsorted)) (not (tptp.empty (tptp.ordered_pair A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.v1_membered A) (tptp.v1_membered (tptp.set_intersection2 A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.v1_membered A) (tptp.v1_membered (tptp.set_intersection2 B A)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 A B))) (=> (tptp.v2_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.succ A))) (=> (and (tptp.ordinal A) (tptp.natural A)) (and (not (tptp.empty _let_1)) (tptp.epsilon_transitive _let_1) (tptp.epsilon_connected _let_1) (tptp.ordinal _let_1) (tptp.natural _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.identity_relation A))) (and (tptp.relation _let_1) (tptp.function _let_1)))) (and _let_8 _let_9 (tptp.function tptp.empty_set) (tptp.one_to_one tptp.empty_set) _let_7 (tptp.epsilon_transitive tptp.empty_set) (tptp.epsilon_connected tptp.empty_set) (tptp.ordinal tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.set_union2 A B)))) (forall ((A $$unsorted)) (not (tptp.empty (tptp.singleton A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (tptp.closed_subset (tptp.topstr_closure A B) A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.empty A)) (not (tptp.empty (tptp.set_union2 A B))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 B A))) (=> (tptp.v2_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 A B))) (=> (tptp.v3_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 B A))) (=> (tptp.v3_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 A B))) (=> (tptp.v4_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 B A))) (=> (tptp.v4_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 A B))) (=> (tptp.v5_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1) (tptp.v5_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_intersection2 B A))) (=> (tptp.v5_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1) (tptp.v5_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.v1_membered A) (tptp.v1_membered (tptp.set_difference A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_difference A B))) (=> (tptp.v2_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_difference A B))) (=> (tptp.v3_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_inverse A))) (=> (and (tptp.relation A) (tptp.function A) (tptp.one_to_one A)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.succ A))) (=> (tptp.ordinal A) (and (not (tptp.empty _let_1)) (tptp.epsilon_transitive _let_1) (tptp.epsilon_connected _let_1) (tptp.ordinal _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation A) (tptp.relation B)) (tptp.relation (tptp.set_difference A B)))) (forall ((A $$unsorted) (B $$unsorted)) (not (tptp.empty (tptp.unordered_pair A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.closed_subset B A) (tptp.element B (tptp.powerset _let_1))) (tptp.open_subset (tptp.subset_complement _let_1 B) A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.empty A)) (not (tptp.empty (tptp.set_union2 B A))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_difference A B))) (=> (tptp.v4_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.set_difference A B))) (=> (tptp.v5_membered A) (and (tptp.v1_membered _let_1) (tptp.v2_membered _let_1) (tptp.v3_membered _let_1) (tptp.v4_membered _let_1) (tptp.v5_membered _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_dom_restriction A B))) (=> (and (tptp.relation A) (tptp.function A)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.union A))) (=> (tptp.ordinal A) (and (tptp.epsilon_transitive _let_1) (tptp.epsilon_connected _let_1) (tptp.ordinal _let_1))))) (and _let_7 _let_8) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (not (tptp.empty B))) (not (tptp.empty (tptp.cartesian_product2 A B))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.open_subset B A) (tptp.element B (tptp.powerset _let_1))) (tptp.closed_subset (tptp.subset_complement _let_1 B) A)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_rng_restriction A B))) (=> (and (tptp.relation B) (tptp.function B)) (and (tptp.relation _let_1) (tptp.function _let_1))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (tptp.closed_subset (tptp.cast_as_carrier_subset A) A))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation A)) (not (tptp.empty (tptp.relation_dom A))))) (and _let_7 (tptp.v1_membered tptp.empty_set) (tptp.v2_membered tptp.empty_set) (tptp.v3_membered tptp.empty_set) (tptp.v4_membered tptp.empty_set) (tptp.v5_membered tptp.empty_set)) (forall ((A $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation A)) (not (tptp.empty (tptp.relation_rng A))))) _let_6 (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_dom A))) (=> (tptp.empty A) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_rng A))) (=> (tptp.empty A) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.finite A) (tptp.finite B)) (tptp.finite (tptp.set_union2 A B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_composition A B))) (=> (and (tptp.empty A) (tptp.relation B)) (and (tptp.empty _let_1) (tptp.relation _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_union2 A A) A)) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_intersection2 A A) A)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_intersection2 A B B) B)))) _let_5 (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_inverse (tptp.relation_inverse A)) A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (= (tptp.complements_of_subsets A (tptp.complements_of_subsets A B)) B))) (forall ((A $$unsorted) (B $$unsorted)) (not (tptp.proper_subset A A))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.reflexive A) (forall ((B $$unsorted)) (=> (tptp.in B (tptp.relation_field A)) (tptp.in (tptp.ordered_pair B B) A)))))) (forall ((A $$unsorted)) (not (= (tptp.singleton A) tptp.empty_set))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (= (tptp.set_union2 (tptp.singleton A) B) B))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.disjoint (tptp.singleton A) B) (tptp.in A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (not (tptp.in A B)) (tptp.disjoint (tptp.singleton A) B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_dom (tptp.relation_rng_restriction A B)) (tptp.relation_dom B)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.transitive A) (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.in (tptp.ordered_pair B C) A) (tptp.in (tptp.ordered_pair C D) A)) (tptp.in (tptp.ordered_pair B D) A)))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.subset (tptp.singleton A) B) (tptp.in A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (not (and (tptp.well_ordering B) (tptp.equipotent A (tptp.relation_field B)) (forall ((C $$unsorted)) (=> (tptp.relation C) (not (tptp.well_orders C A)))))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= (tptp.set_difference A B) tptp.empty_set) (tptp.subset A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.in C B) (tptp.in C A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.antisymmetric A) (forall ((B $$unsorted) (C $$unsorted)) (=> (and (tptp.in (tptp.ordered_pair B C) A) (tptp.in (tptp.ordered_pair C B) A)) (= B C)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (or (tptp.in C A) (tptp.subset A (tptp.set_difference B (tptp.singleton C)))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (forall ((C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element C _let_1) (= (tptp.in C (tptp.subset_complement _let_1 B)) (not (tptp.in C B)))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.connected A) (forall ((B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_field A))) (not (and (tptp.in B _let_1) (tptp.in C _let_1) (not (= B C)) (not (tptp.in (tptp.ordered_pair B C) A)) (not (tptp.in (tptp.ordered_pair C B) A))))))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.singleton B))) (= (tptp.subset A _let_1) (or (= A tptp.empty_set) (= A _let_1))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (tptp.subset A (tptp.union B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair A B) (tptp.cartesian_product2 C D)) (and (tptp.in A C) (tptp.in B D)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (forall ((C $$unsorted)) (=> (tptp.in C A) (tptp.in C B))) (tptp.element A (tptp.powerset B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.in B (tptp.relation_dom (tptp.relation_dom_restriction C A))) (and (tptp.in B (tptp.relation_dom C)) (tptp.in B A))))) (exists ((A $$unsorted)) (and (not (tptp.empty A)) (tptp.epsilon_transitive A) (tptp.epsilon_connected A) (tptp.ordinal A) (tptp.natural A))) (exists ((A $$unsorted)) (and (not (tptp.empty A)) (tptp.finite A))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.function A))) (forall ((A $$unsorted) (B $$unsorted)) (exists ((C $$unsorted)) (and (tptp.relation_of2 C A B) (tptp.relation C) (tptp.function C) (tptp.quasi_total C A B)))) (exists ((A $$unsorted)) (and (not (tptp.empty A)) (tptp.v1_membered A) (tptp.v2_membered A) (tptp.v3_membered A) (tptp.v4_membered A) (tptp.v5_membered A))) (exists ((A $$unsorted)) (and (tptp.epsilon_transitive A) (tptp.epsilon_connected A) (tptp.ordinal A))) (exists ((A $$unsorted)) (and (tptp.epsilon_transitive A) (tptp.epsilon_connected A) (tptp.ordinal A) (tptp.being_limit_ordinal A))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.function A) (tptp.one_to_one A) (tptp.empty A))) (exists ((A $$unsorted)) (and (tptp.empty A) (tptp.relation A))) (forall ((A $$unsorted)) (=> (not (tptp.empty A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset A)) (not (tptp.empty B)))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.open_subset B A))))) (exists ((A $$unsorted)) (tptp.empty A)) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset A)) (tptp.empty B) (tptp.relation B) (tptp.function B) (tptp.one_to_one B) (tptp.epsilon_transitive B) (tptp.epsilon_connected B) (tptp.ordinal B) (tptp.natural B) (tptp.finite B)))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.empty A) (tptp.function A))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.function A) (tptp.one_to_one A) (tptp.empty A) (tptp.epsilon_transitive A) (tptp.epsilon_connected A) (tptp.ordinal A))) (forall ((A $$unsorted) (B $$unsorted)) (exists ((C $$unsorted)) (and (tptp.relation_of2 C A B) (tptp.relation C) (tptp.function C)))) (exists ((A $$unsorted)) (and (not (tptp.empty A)) (tptp.relation A))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset A)) (tptp.empty B)))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.open_subset B A) (tptp.closed_subset B A))))) (exists ((A $$unsorted)) (not (tptp.empty A))) (forall ((A $$unsorted)) (=> (not (tptp.empty A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset A)) (not (tptp.empty B)) (tptp.finite B))))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.function A) (tptp.one_to_one A))) (exists ((A $$unsorted)) (and (not (tptp.empty A)) (tptp.epsilon_transitive A) (tptp.epsilon_connected A) (tptp.ordinal A))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.relation_empty_yielding A))) (exists ((A $$unsorted)) (and (tptp.one_sorted_str A) (not (tptp.empty_carrier A)))) (exists ((A $$unsorted)) (and (tptp.relation A) (tptp.relation_empty_yielding A) (tptp.function A))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.one_sorted_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (not (tptp.empty B)))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (exists ((B $$unsorted)) (and (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.closed_subset B A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted) (E $$unsorted) (F $$unsorted)) (let ((_let_1 (tptp.cartesian_product2 A B))) (=> (and (not (tptp.empty A)) (not (tptp.empty B)) (tptp.function D) (tptp.quasi_total D _let_1 C) (tptp.relation_of2 D _let_1 C) (tptp.element E A) (tptp.element F B)) (= (tptp.apply_binary_as_element A B C D E F) (tptp.apply_binary D E F))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.join_commutative A) (tptp.join_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.join_commut A B C) (tptp.join A B C))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_semilatt_str A) (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.meet_commut A B C) (tptp.meet A B C))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2 C A B) (= (tptp.relation_dom_as_subset A B C) (tptp.relation_dom C)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2 C A B) (= (tptp.relation_rng_as_subset A B C) (tptp.relation_rng C)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (= (tptp.union_of_subsets A B) (tptp.union B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_intersection2 A B C) (tptp.set_intersection2 B C))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (= (tptp.meet_of_subsets A B) (tptp.set_meet B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.powerset A))) (=> (and (tptp.element B _let_1) (tptp.element C _let_1)) (= (tptp.subset_difference A B C) (tptp.set_difference B C))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (tptp.relation_of2_as_subset C A B) (tptp.relation_of2 C A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.ordinal A) (tptp.ordinal B)) (= (tptp.ordinal_subset A B) (tptp.subset A B)))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.equipotent A B) (tptp.are_equipotent A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.ordinal A) (tptp.ordinal B)) (tptp.ordinal_subset A A))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset A A)) (forall ((A $$unsorted) (B $$unsorted)) (tptp.equipotent A A)) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in C A))) (=> (and _let_1 (exists ((F $$unsorted)) (and (= C F) (tptp.in D F) (forall ((G $$unsorted)) (=> (tptp.in G F) (tptp.in (tptp.ordered_pair D G) B))))) _let_1 (exists ((H $$unsorted)) (and (= C H) (tptp.in E H) (forall ((I $$unsorted)) (=> (tptp.in I H) (tptp.in (tptp.ordered_pair E I) B)))))) (= D E)))) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (forall ((D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in D A))) (= (tptp.in (tptp.ordered_pair D E) C) (and _let_1 _let_1 (exists ((J $$unsorted)) (and (= D J) (tptp.in E J) (forall ((K $$unsorted)) (=> (tptp.in K J) (tptp.in (tptp.ordered_pair E K) B)))))))))))))) (forall ((A $$unsorted)) (=> (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.singleton B))) (let ((_let_2 (tptp.in B A))) (=> (and _let_2 (= C _let_1) _let_2 (= D _let_1)) (= C D))))) (exists ((B $$unsorted)) (and (tptp.relation B) (tptp.function B) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.in C A))) (= (tptp.in (tptp.ordered_pair C D) B) (and _let_1 _let_1 (= D (tptp.singleton C)))))))))) (forall ((A $$unsorted)) (=> (exists ((B $$unsorted)) (and (tptp.ordinal B) (tptp.in B A))) (exists ((B $$unsorted)) (and (tptp.ordinal B) (tptp.in B A) (forall ((C $$unsorted)) (=> (tptp.ordinal C) (=> (tptp.in C A) (tptp.ordinal_subset B C)))))))) (=> (and (=> (tptp.in tptp.empty_set tptp.omega) (forall ((A $$unsorted)) (=> (tptp.element A (tptp.powerset (tptp.powerset tptp.empty_set))) (not (and (not (= A tptp.empty_set)) (forall ((B $$unsorted)) (not (and (tptp.in B A) (forall ((C $$unsorted)) (=> (and (tptp.in C A) (tptp.subset B C)) (= C B))))))))))) (forall ((D $$unsorted)) (=> (tptp.ordinal D) (=> (=> (tptp.in D tptp.omega) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.powerset (tptp.powerset D))) (not (and (not (= E tptp.empty_set)) (forall ((F $$unsorted)) (not (and (tptp.in F E) (forall ((G $$unsorted)) (=> (and (tptp.in G E) (tptp.subset F G)) (= G F))))))))))) (=> (tptp.in (tptp.succ D) tptp.omega) (forall ((H $$unsorted)) (=> (tptp.element H (tptp.powerset (tptp.powerset (tptp.succ D)))) (not (and (not (= H tptp.empty_set)) (forall ((I $$unsorted)) (not (and (tptp.in I H) (forall ((J $$unsorted)) (=> (and (tptp.in J H) (tptp.subset I J)) (= J I)))))))))))))) (forall ((D $$unsorted)) (=> (tptp.ordinal D) (=> (and (tptp.being_limit_ordinal D) (forall ((K $$unsorted)) (=> (tptp.ordinal K) (=> (tptp.in K D) (=> (tptp.in K tptp.omega) (forall ((L $$unsorted)) (=> (tptp.element L (tptp.powerset (tptp.powerset K))) (not (and (not (= L tptp.empty_set)) (forall ((M $$unsorted)) (not (and (tptp.in M L) (forall ((N $$unsorted)) (=> (and (tptp.in N L) (tptp.subset M N)) (= N M))))))))))))))) (or (= D tptp.empty_set) (=> (tptp.in D tptp.omega) (forall ((O $$unsorted)) (=> (tptp.element O (tptp.powerset (tptp.powerset D))) (not (and (not (= O tptp.empty_set)) (forall ((P $$unsorted)) (not (and (tptp.in P O) (forall ((Q $$unsorted)) (=> (and (tptp.in Q O) (tptp.subset P Q)) (= Q P)))))))))))))))) (forall ((D $$unsorted)) (=> (tptp.ordinal D) (=> (tptp.in D tptp.omega) (forall ((R $$unsorted)) (=> (tptp.element R (tptp.powerset (tptp.powerset D))) (not (and (not (= R tptp.empty_set)) (forall ((S $$unsorted)) (not (and (tptp.in S R) (forall ((T $$unsorted)) (=> (and (tptp.in T R) (tptp.subset S T)) (= T S)))))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation B) (tptp.relation C) (tptp.function C)) (exists ((D $$unsorted)) (and (tptp.relation D) (forall ((E $$unsorted) (F $$unsorted)) (= (tptp.in (tptp.ordered_pair E F) D) (and (tptp.in E A) (tptp.in F A) (tptp.in (tptp.ordered_pair (tptp.apply C E) (tptp.apply C F)) B)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.in C A))) (=> (and _let_1 (exists ((F $$unsorted)) (and (= C F) (tptp.in D F) (forall ((G $$unsorted)) (=> (tptp.in G F) (tptp.in (tptp.ordered_pair D G) B))))) _let_1 (exists ((H $$unsorted)) (and (= C H) (tptp.in E H) (forall ((I $$unsorted)) (=> (tptp.in I H) (tptp.in (tptp.ordered_pair E I) B)))))) (= D E)))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (let ((_let_1 (tptp.in E A))) (and _let_1 _let_1 (exists ((J $$unsorted)) (and (= E J) (tptp.in D J) (forall ((K $$unsorted)) (=> (tptp.in K J) (tptp.in (tptp.ordered_pair D K) B)))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (forall ((C $$unsorted)) (=> (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (and (= D E) (exists ((G $$unsorted) (H $$unsorted)) (and (= (tptp.ordered_pair G H) E) (tptp.in G A) (exists ((I $$unsorted)) (and (= G I) (tptp.in H I) (forall ((J $$unsorted)) (=> (tptp.in J I) (tptp.in (tptp.ordered_pair H J) B))))))) (= D F) (exists ((K $$unsorted) (L $$unsorted)) (and (= (tptp.ordered_pair K L) F) (tptp.in K A) (exists ((M $$unsorted)) (and (= K M) (tptp.in L M) (forall ((N $$unsorted)) (=> (tptp.in N M) (tptp.in (tptp.ordered_pair L N) B)))))))) (= E F))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (exists ((F $$unsorted)) (and (tptp.in F (tptp.cartesian_product2 A C)) (= F E) (exists ((O $$unsorted) (P $$unsorted)) (and (= (tptp.ordered_pair O P) E) (tptp.in O A) (exists ((Q $$unsorted)) (and (= O Q) (tptp.in P Q) (forall ((R $$unsorted)) (=> (tptp.in R Q) (tptp.in (tptp.ordered_pair P R) B)))))))))))))))) (forall ((A $$unsorted)) (=> (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.singleton B))) (let ((_let_2 (tptp.in B A))) (=> (and _let_2 (= C _let_1) _let_2 (= D _let_1)) (= C D))))) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (let ((_let_1 (tptp.in D A))) (and _let_1 _let_1 (= C (tptp.singleton D)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (= C D) (exists ((F $$unsorted) (G $$unsorted)) (and (= (tptp.ordered_pair F G) D) (tptp.in F A) (= G (tptp.singleton F)))) (= C E) (exists ((H $$unsorted) (I $$unsorted)) (and (= (tptp.ordered_pair H I) E) (tptp.in H A) (= I (tptp.singleton H))))) (= D E))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.cartesian_product2 A B)) (= E D) (exists ((J $$unsorted) (K $$unsorted)) (and (= (tptp.ordered_pair J K) D) (tptp.in J A) (= K (tptp.singleton J))))))))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (=> (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (= B C) (exists ((E $$unsorted)) (and (tptp.ordinal E) (= C E) (=> (tptp.in E tptp.omega) (forall ((F $$unsorted)) (=> (tptp.element F (tptp.powerset (tptp.powerset E))) (not (and (not (= F tptp.empty_set)) (forall ((G $$unsorted)) (not (and (tptp.in G F) (forall ((H $$unsorted)) (=> (and (tptp.in H F) (tptp.subset G H)) (= H G))))))))))))) (= B D) (exists ((I $$unsorted)) (and (tptp.ordinal I) (= D I) (=> (tptp.in I tptp.omega) (forall ((J $$unsorted)) (=> (tptp.element J (tptp.powerset (tptp.powerset I))) (not (and (not (= J tptp.empty_set)) (forall ((K $$unsorted)) (not (and (tptp.in K J) (forall ((L $$unsorted)) (=> (and (tptp.in L J) (tptp.subset K L)) (= L K)))))))))))))) (= C D))) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (and (tptp.in D (tptp.succ A)) (= D C) (exists ((M $$unsorted)) (and (tptp.ordinal M) (= C M) (=> (tptp.in M tptp.omega) (forall ((N $$unsorted)) (=> (tptp.element N (tptp.powerset (tptp.powerset M))) (not (and (not (= N tptp.empty_set)) (forall ((O $$unsorted)) (not (and (tptp.in O N) (forall ((P $$unsorted)) (=> (and (tptp.in P N) (tptp.subset O P)) (= P O))))))))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (= C D) (exists ((F $$unsorted)) (and (tptp.element F (tptp.powerset (tptp.the_carrier A))) (= F D) (tptp.closed_subset F A) (tptp.subset B D))) (= C E) (exists ((G $$unsorted)) (and (tptp.element G (tptp.powerset (tptp.the_carrier A))) (= G E) (tptp.closed_subset G A) (tptp.subset B E)))) (= D E))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.powerset (tptp.the_carrier A))) (= E D) (exists ((H $$unsorted)) (and (tptp.element H (tptp.powerset (tptp.the_carrier A))) (= H D) (tptp.closed_subset H A) (tptp.subset B D))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (=> (and (= C D) (tptp.in (tptp.set_difference _let_1 D) B) (= C E) (tptp.in (tptp.set_difference _let_1 E) B)) (= D E)))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.powerset (tptp.the_carrier A))) (= E D) (tptp.in (tptp.set_difference (tptp.cast_as_carrier_subset A) D) B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.ordinal A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.succ A))))) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (= C D) (exists ((F $$unsorted)) (and (tptp.in F B) (= D (tptp.set_difference F (tptp.singleton A))))) (= C E) (exists ((G $$unsorted)) (and (tptp.in G B) (= E (tptp.set_difference G (tptp.singleton A)))))) (= D E))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.powerset A)) (= E D) (exists ((H $$unsorted)) (and (tptp.in H B) (= D (tptp.set_difference H (tptp.singleton A))))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation B) (tptp.relation C) (tptp.function C)) (=> (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (and (= D E) (exists ((G $$unsorted) (H $$unsorted)) (and (= E (tptp.ordered_pair G H)) (tptp.in (tptp.ordered_pair (tptp.apply C G) (tptp.apply C H)) B))) (= D F) (exists ((I $$unsorted) (J $$unsorted)) (and (= F (tptp.ordered_pair I J)) (tptp.in (tptp.ordered_pair (tptp.apply C I) (tptp.apply C J)) B)))) (= E F))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (exists ((F $$unsorted)) (and (tptp.in F (tptp.cartesian_product2 A A)) (= F E) (exists ((K $$unsorted) (L $$unsorted)) (and (= E (tptp.ordered_pair K L)) (tptp.in (tptp.ordered_pair (tptp.apply C K) (tptp.apply C L)) B))))))))))) (forall ((A $$unsorted)) (=> (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (= B C) (tptp.ordinal C) (= B D) (tptp.ordinal D)) (= C D))) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (exists ((D $$unsorted)) (and (tptp.in D A) (= D C) (tptp.ordinal C)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.element B (tptp.powerset (tptp.powerset A))) (tptp.relation C) (tptp.function C)) (=> (forall ((D $$unsorted) (E $$unsorted) (F $$unsorted)) (=> (and (= D E) (tptp.in (tptp.relation_image C E) B) (= D F) (tptp.in (tptp.relation_image C F) B)) (= E F))) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (exists ((F $$unsorted)) (and (tptp.in F (tptp.powerset (tptp.relation_dom C))) (= F E) (tptp.in (tptp.relation_image C E) B))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.ordinal B) (=> (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (= C D) (exists ((F $$unsorted)) (and (tptp.ordinal F) (= D F) (tptp.in F A))) (= C E) (exists ((G $$unsorted)) (and (tptp.ordinal G) (= E G) (tptp.in G A)))) (= D E))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.in E (tptp.succ B)) (= E D) (exists ((H $$unsorted)) (and (tptp.ordinal H) (= D H) (tptp.in H A))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (forall ((C $$unsorted)) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (and (tptp.in E (tptp.cartesian_product2 A C)) (exists ((F $$unsorted) (G $$unsorted)) (and (= (tptp.ordered_pair F G) E) (tptp.in F A) (exists ((H $$unsorted)) (and (= F H) (tptp.in G H) (forall ((I $$unsorted)) (=> (tptp.in I H) (tptp.in (tptp.ordered_pair G I) B)))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.cartesian_product2 A B)) (exists ((E $$unsorted) (F $$unsorted)) (and (= (tptp.ordered_pair E F) D) (tptp.in E A) (= F (tptp.singleton E))))))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (and (tptp.in C (tptp.succ A)) (exists ((D $$unsorted)) (and (tptp.ordinal D) (= C D) (=> (tptp.in D tptp.omega) (forall ((E $$unsorted)) (=> (tptp.element E (tptp.powerset (tptp.powerset D))) (not (and (not (= E tptp.empty_set)) (forall ((F $$unsorted)) (not (and (tptp.in F E) (forall ((G $$unsorted)) (=> (and (tptp.in G E) (tptp.subset F G)) (= G F))))))))))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.powerset (tptp.the_carrier A))) (exists ((E $$unsorted)) (and (tptp.element E (tptp.powerset (tptp.the_carrier A))) (= E D) (tptp.closed_subset E A) (tptp.subset B D))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.powerset (tptp.the_carrier A))) (tptp.in (tptp.set_difference (tptp.cast_as_carrier_subset A) D) B))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.ordinal A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.succ A))))) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.powerset A)) (exists ((E $$unsorted)) (and (tptp.in E B) (= D (tptp.set_difference E (tptp.singleton A))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation B) (tptp.relation C) (tptp.function C)) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (and (tptp.in E (tptp.cartesian_product2 A A)) (exists ((F $$unsorted) (G $$unsorted)) (and (= E (tptp.ordered_pair F G)) (tptp.in (tptp.ordered_pair (tptp.apply C F) (tptp.apply C G)) B))))))))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (forall ((C $$unsorted)) (= (tptp.in C B) (and (tptp.in C A) (tptp.ordinal C)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.element B (tptp.powerset (tptp.powerset A))) (tptp.relation C) (tptp.function C)) (exists ((D $$unsorted)) (forall ((E $$unsorted)) (= (tptp.in E D) (and (tptp.in E (tptp.powerset (tptp.relation_dom C))) (tptp.in (tptp.relation_image C E) B))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.ordinal B) (exists ((C $$unsorted)) (forall ((D $$unsorted)) (= (tptp.in D C) (and (tptp.in D (tptp.succ B)) (exists ((E $$unsorted)) (and (tptp.ordinal E) (= D E) (tptp.in E A))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (not (tptp.empty A)) (tptp.relation B)) (=> (and (forall ((C $$unsorted) (D $$unsorted) (E $$unsorted)) (=> (and (tptp.in C A) (exists ((F $$unsorted)) (and (= C F) (tptp.in D F) (forall ((G $$unsorted)) (=> (tptp.in G F) (tptp.in (tptp.ordered_pair D G) B))))) (exists ((H $$unsorted)) (and (= C H) (tptp.in E H) (forall ((I $$unsorted)) (=> (tptp.in I H) (tptp.in (tptp.ordered_pair E I) B)))))) (= D E))) (forall ((C $$unsorted)) (not (and (tptp.in C A) (forall ((D $$unsorted)) (not (exists ((J $$unsorted)) (and (= C J) (tptp.in D J) (forall ((K $$unsorted)) (=> (tptp.in K J) (tptp.in (tptp.ordered_pair D K) B))))))))))) (exists ((C $$unsorted)) (and (tptp.relation C) (tptp.function C) (= (tptp.relation_dom C) A) (forall ((D $$unsorted)) (=> (tptp.in D A) (exists ((L $$unsorted)) (and (= D L) (tptp.in (tptp.apply C D) L) (forall ((M $$unsorted)) (=> (tptp.in M L) (tptp.in (tptp.ordered_pair (tptp.apply C D) M) B)))))))))))) (forall ((A $$unsorted)) (=> (and (forall ((B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.singleton B))) (=> (and (tptp.in B A) (= C _let_1) (= D _let_1)) (= C D)))) (forall ((B $$unsorted)) (not (and (tptp.in B A) (forall ((C $$unsorted)) (not (= C (tptp.singleton B)))))))) (exists ((B $$unsorted)) (and (tptp.relation B) (tptp.function B) (= (tptp.relation_dom B) A) (forall ((C $$unsorted)) (=> (tptp.in C A) (= (tptp.apply B C) (tptp.singleton C)))))))) (=> (forall ((A $$unsorted)) (=> (tptp.ordinal A) (=> (forall ((B $$unsorted)) (=> (tptp.ordinal B) (=> (tptp.in B A) (=> (tptp.in B tptp.omega) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.powerset B))) (not (and (not (= C tptp.empty_set)) (forall ((D $$unsorted)) (not (and (tptp.in D C) (forall ((E $$unsorted)) (=> (and (tptp.in E C) (tptp.subset D E)) (= E D)))))))))))))) (=> (tptp.in A tptp.omega) (forall ((F $$unsorted)) (=> (tptp.element F (tptp.powerset (tptp.powerset A))) (not (and (not (= F tptp.empty_set)) (forall ((G $$unsorted)) (not (and (tptp.in G F) (forall ((H $$unsorted)) (=> (and (tptp.in H F) (tptp.subset G H)) (= H G)))))))))))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (=> (tptp.in A tptp.omega) (forall ((I $$unsorted)) (=> (tptp.element I (tptp.powerset (tptp.powerset A))) (not (and (not (= I tptp.empty_set)) (forall ((J $$unsorted)) (not (and (tptp.in J I) (forall ((K $$unsorted)) (=> (and (tptp.in K I) (tptp.subset J K)) (= K J)))))))))))))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.relation B) (tptp.function B) (= (tptp.relation_dom B) A) (forall ((C $$unsorted)) (=> (tptp.in C A) (= (tptp.apply B C) (tptp.singleton C))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.the_carrier A)))) (exists ((C $$unsorted)) (and (tptp.element C (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier A))) (= (tptp.in D C) (exists ((E $$unsorted)) (and (tptp.element E (tptp.powerset (tptp.the_carrier A))) (= E D) (tptp.closed_subset E A) (tptp.subset B D)))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A) (tptp.element B (tptp.powerset (tptp.powerset (tptp.the_carrier A))))) (exists ((C $$unsorted)) (and (tptp.element C (tptp.powerset (tptp.powerset (tptp.the_carrier A)))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier A))) (= (tptp.in D C) (tptp.in (tptp.set_difference (tptp.cast_as_carrier_subset A) D) B)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.disjoint A B) (tptp.disjoint B A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.equipotent A B) (tptp.equipotent B A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (= (tptp.in (tptp.ordered_pair A B) (tptp.cartesian_product2 C D)) (and (tptp.in A C) (tptp.in B D)))) (forall ((A $$unsorted)) (tptp.in A (tptp.succ A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (not (and (= (tptp.unordered_pair A B) (tptp.unordered_pair C D)) (not (= A C)) (not (= A D))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_rng (tptp.relation_rng_restriction B C))) (and (tptp.in A B) (tptp.in A (tptp.relation_rng C)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_rng_restriction A B)) A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng_restriction A B) B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_rng_restriction A B)) (tptp.relation_rng B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (and (tptp.subset (tptp.cartesian_product2 A C) (tptp.cartesian_product2 B C)) (tptp.subset (tptp.cartesian_product2 C A) (tptp.cartesian_product2 C B))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_rng (tptp.relation_rng_restriction A B)) (tptp.set_intersection2 (tptp.relation_rng B) A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset C D)) (tptp.subset (tptp.cartesian_product2 A C) (tptp.cartesian_product2 B D)))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (= (tptp.cast_as_carrier_subset A) (tptp.the_carrier A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2_as_subset C A B) (and (tptp.subset (tptp.relation_dom C) A) (tptp.subset (tptp.relation_rng C) B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset A B) (= (tptp.set_union2 A B) B))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.in A B) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C B) (tptp.subset D C)) (tptp.in D B))) (forall ((C $$unsorted)) (=> (tptp.in C B) (tptp.in (tptp.powerset C) B))) (forall ((C $$unsorted)) (not (and (tptp.subset C B) (not (tptp.are_equipotent C B)) (not (tptp.in C B)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.subset A B) (tptp.finite B)) (tptp.finite A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.relation_dom_restriction (tptp.relation_rng_restriction A C) B) (tptp.relation_rng_restriction A (tptp.relation_dom_restriction C B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_image C B)) (exists ((D $$unsorted)) (and (tptp.in D (tptp.relation_dom C)) (tptp.in (tptp.ordered_pair D A) C) (tptp.in D B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_image B A) (tptp.relation_rng B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (tptp.subset (tptp.relation_image B (tptp.relation_inverse_image B A)) A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_image B A) (tptp.relation_image B (tptp.set_intersection2 (tptp.relation_dom B) A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset A (tptp.relation_dom B)) (tptp.subset A (tptp.relation_inverse_image B (tptp.relation_image B A)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.relation_image A (tptp.relation_dom A)) (tptp.relation_rng A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (=> (tptp.subset A (tptp.relation_rng B)) (= (tptp.relation_image B (tptp.relation_inverse_image B A)) A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (tptp.relation_of2_as_subset D C A) (=> (tptp.subset (tptp.relation_rng D) B) (tptp.relation_of2_as_subset D C B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.finite A) (tptp.finite (tptp.set_intersection2 A B)))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.subset_intersection2 _let_1 B (tptp.cast_as_carrier_subset A)) B)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_rng (tptp.relation_composition A B)) (tptp.relation_image B (tptp.relation_rng A))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_inverse_image C B)) (exists ((D $$unsorted)) (and (tptp.in D (tptp.relation_rng C)) (tptp.in (tptp.ordered_pair A D) C) (tptp.in D B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_inverse_image B A) (tptp.relation_dom B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (tptp.relation_of2_as_subset D C A) (=> (tptp.subset A B) (tptp.relation_of2_as_subset D C B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_restriction C B)) (and (tptp.in A C) (tptp.in A (tptp.cartesian_product2 B B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (not (and (not (= A tptp.empty_set)) (tptp.subset A (tptp.relation_rng B)) (= (tptp.relation_inverse_image B A) tptp.empty_set))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (=> (tptp.subset A B) (tptp.subset (tptp.relation_inverse_image C A) (tptp.relation_inverse_image C B))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (=> (tptp.finite A) (tptp.finite (tptp.relation_image B A))))) _let_4 (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_restriction B A) (tptp.relation_dom_restriction (tptp.relation_rng_restriction A B) A)))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset (tptp.set_intersection2 A B) A)) (forall ((A $$unsorted)) (=> (tptp.finite A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (not (and (not (= B tptp.empty_set)) (forall ((C $$unsorted)) (not (and (tptp.in C B) (forall ((D $$unsorted)) (=> (and (tptp.in D B) (tptp.subset C D)) (= D C)))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_restriction B A) (tptp.relation_rng_restriction A (tptp.relation_dom_restriction B A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (=> (tptp.in A (tptp.relation_field (tptp.relation_restriction C B))) (and (tptp.in A (tptp.relation_field C)) (tptp.in A B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset A C)) (tptp.subset A (tptp.set_intersection2 B C)))) (forall ((A $$unsorted)) (= (tptp.set_union2 A tptp.empty_set) A)) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (tptp.element A B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset B C)) (tptp.subset A C))) (= (tptp.powerset tptp.empty_set) (tptp.singleton tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (=> (tptp.in (tptp.ordered_pair A B) C) (and (tptp.in A (tptp.relation_dom C)) (tptp.in B (tptp.relation_rng C)))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_field (tptp.relation_restriction B A)))) (=> (tptp.relation B) (and (tptp.subset _let_1 (tptp.relation_field B)) (tptp.subset _let_1 A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.in A (tptp.relation_dom (tptp.relation_composition C B))) (and (tptp.in A (tptp.relation_dom C)) (tptp.in (tptp.apply C A) (tptp.relation_dom B)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.function D) (tptp.quasi_total D A B) (tptp.relation_of2_as_subset D A B)) (forall ((E $$unsorted)) (=> (and (tptp.relation E) (tptp.function E)) (=> (tptp.in C A) (or (= B tptp.empty_set) (= (tptp.apply (tptp.relation_composition D E) C) (tptp.apply E (tptp.apply D C))))))))) (forall ((A $$unsorted)) (=> (tptp.epsilon_transitive A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (=> (tptp.proper_subset A B) (tptp.in A B)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (tptp.subset A (tptp.cartesian_product2 (tptp.relation_dom A) (tptp.relation_rng A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (tptp.subset (tptp.fiber (tptp.relation_restriction C A) B) (tptp.fiber C B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (let ((_let_1 (tptp.relation_composition C B))) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in A (tptp.relation_dom _let_1)) (= (tptp.apply _let_1 A) (tptp.apply B (tptp.apply C A))))))))) (forall ((A $$unsorted)) (=> (tptp.one_sorted_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.cast_as_carrier_subset A))) (let ((_let_2 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_2)) (= (tptp.subset_difference _let_2 _let_1 (tptp.subset_difference _let_2 _let_1 B)) B))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2_as_subset C B A) (= (forall ((D $$unsorted)) (not (and (tptp.in D B) (forall ((E $$unsorted)) (not (tptp.in (tptp.ordered_pair D E) C)))))) (= (tptp.relation_dom_as_subset B A C) B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.reflexive B) (tptp.reflexive (tptp.relation_restriction B A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in A (tptp.relation_dom B)) (= (tptp.apply (tptp.relation_composition B C) A) (tptp.apply C (tptp.apply B A)))))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.meet_commutative A) (tptp.meet_absorbing A) (tptp.latt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (tptp.below A (tptp.meet_commut A B C) B))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.ordinal B) (=> (tptp.in A B) (tptp.ordinal A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation_of2_as_subset C A B) (= (forall ((D $$unsorted)) (not (and (tptp.in D B) (forall ((E $$unsorted)) (not (tptp.in (tptp.ordered_pair E D) C)))))) (= (tptp.relation_rng_as_subset A B C) B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.connected B) (tptp.connected (tptp.relation_restriction B A))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (not (and (not (tptp.in A B)) (not (= A B)) (not (tptp.in B A)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.transitive B) (tptp.transitive (tptp.relation_restriction B A))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset A B) (and (tptp.subset (tptp.relation_dom A) (tptp.relation_dom B)) (tptp.subset (tptp.relation_rng A) (tptp.relation_rng B)))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.antisymmetric B) (tptp.antisymmetric (tptp.relation_restriction B A))))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.relation_restriction B A))) (=> (tptp.relation B) (=> (tptp.well_orders B A) (and (= (tptp.relation_field _let_1) A) (tptp.well_ordering _let_1)))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.finite (tptp.relation_dom A)) (tptp.finite (tptp.relation_rng A))))) (forall ((A $$unsorted)) (=> (and (not (tptp.empty_carrier A)) (tptp.join_commutative A) (tptp.join_semilatt_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.the_carrier A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.the_carrier A)) (=> (and (tptp.below A B C) (tptp.below A C B)) (= B C)))))))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.relation B) (tptp.well_orders B A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (tptp.subset (tptp.set_intersection2 A C) (tptp.set_intersection2 B C)))) (forall ((A $$unsorted)) (=> (not (tptp.empty A)) (not (and (forall ((B $$unsorted)) (not (and (tptp.in B A) (= B tptp.empty_set)))) (forall ((B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (not (and (= (tptp.relation_dom B) A) (forall ((C $$unsorted)) (=> (tptp.in C A) (tptp.in (tptp.apply B C) C))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset A B) (= (tptp.set_intersection2 A B) A))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset _let_1)) (= (tptp.closed_subset B A) (tptp.open_subset (tptp.subset_complement _let_1 B) A))))))) (forall ((A $$unsorted)) (= (tptp.set_intersection2 A tptp.empty_set) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element A B) (or (tptp.empty B) (tptp.in A B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (forall ((C $$unsorted)) (= (tptp.in C A) (tptp.in C B))) (= A B))) (forall ((A $$unsorted)) (tptp.reflexive (tptp.inclusion_relation A))) (forall ((A $$unsorted)) (tptp.subset tptp.empty_set A)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_field C))) (=> (tptp.relation C) (=> (tptp.in (tptp.ordered_pair A B) C) (and (tptp.in A _let_1) (tptp.in B _let_1)))))) _let_3 (forall ((A $$unsorted)) (=> (forall ((B $$unsorted)) (=> (tptp.in B A) (and (tptp.ordinal B) (tptp.subset B A)))) (tptp.ordinal A))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.well_founded_relation B) (tptp.well_founded_relation (tptp.relation_restriction B A))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.ordinal B) (not (and (tptp.subset A B) (not (= A tptp.empty_set)) (forall ((C $$unsorted)) (=> (tptp.ordinal C) (not (and (tptp.in C A) (forall ((D $$unsorted)) (=> (tptp.ordinal D) (=> (tptp.in D A) (tptp.ordinal_subset C D)))))))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (tptp.well_ordering B) (tptp.well_ordering (tptp.relation_restriction B A))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (= (tptp.in A B) (tptp.ordinal_subset (tptp.succ A) B)))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.subset A B) (tptp.subset (tptp.set_difference A C) (tptp.set_difference B C)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (= (tptp.ordered_pair A B) (tptp.ordered_pair C D)) (and (= A C) (= B D)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (= (= B (tptp.identity_relation A)) (and (= (tptp.relation_dom B) A) (forall ((C $$unsorted)) (=> (tptp.in C A) (= (tptp.apply B C) C))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in B A) (= (tptp.apply (tptp.identity_relation A) B) B))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset (tptp.set_difference A B) A)) (forall ((A $$unsorted)) (let ((_let_1 (tptp.relation_inverse A))) (=> (tptp.relation A) (and (= (tptp.relation_rng A) (tptp.relation_dom _let_1)) (= (tptp.relation_dom A) (tptp.relation_rng _let_1)))))) (forall ((A $$unsorted) (B $$unsorted)) (= (= (tptp.set_difference A B) tptp.empty_set) (tptp.subset A B))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.subset (tptp.singleton A) B) (tptp.in A B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (= (tptp.subset (tptp.unordered_pair A B) C) (and (tptp.in A C) (tptp.in B C)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (=> (and (tptp.well_ordering B) (tptp.subset A (tptp.relation_field B))) (= (tptp.relation_field (tptp.relation_restriction B A)) A)))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_union2 A (tptp.set_difference B A)) (tptp.set_union2 A B))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.singleton B))) (= (tptp.subset A _let_1) (or (= A tptp.empty_set) (= A _let_1))))) (forall ((A $$unsorted)) (= (tptp.set_difference A tptp.empty_set) A)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (not (and (tptp.in A B) (tptp.in B C) (tptp.in C A)))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.element A (tptp.powerset B)) (tptp.subset A B))) (forall ((A $$unsorted)) (tptp.transitive (tptp.inclusion_relation A))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.disjoint A B))) (and (not (and (not _let_1) (forall ((C $$unsorted)) (not (and (tptp.in C A) (tptp.in C B)))))) (not (and (exists ((C $$unsorted)) (and (tptp.in C A) (tptp.in C B))) _let_1))))) (forall ((A $$unsorted)) (=> (tptp.subset A tptp.empty_set) (= A tptp.empty_set))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_difference (tptp.set_union2 A B) B) (tptp.set_difference A B))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (= (tptp.being_limit_ordinal A) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (=> (tptp.in B A) (tptp.in (tptp.succ B) A))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.being_limit_ordinal A))) (=> (tptp.ordinal A) (and (not (and (not _let_1) (forall ((B $$unsorted)) (=> (tptp.ordinal B) (not (= A (tptp.succ B))))))) (not (and (exists ((B $$unsorted)) (and (tptp.ordinal B) (= A (tptp.succ B)))) _let_1)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset A)) (= (tptp.disjoint B C) (tptp.subset B (tptp.subset_complement A C))))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (=> (tptp.element B (tptp.powerset (tptp.powerset _let_1))) (=> (forall ((C $$unsorted)) (=> (tptp.element C (tptp.powerset (tptp.the_carrier A))) (=> (tptp.in C B) (tptp.closed_subset C A)))) (tptp.closed_subset (tptp.meet_of_subsets _let_1 B) A))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_dom (tptp.relation_composition A B)) (tptp.relation_dom A)))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.subset (tptp.interior A B) B))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (forall ((C $$unsorted)) (=> (tptp.in C (tptp.the_carrier A)) (= (tptp.in C (tptp.topstr_closure A B)) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier A))) (=> (and (tptp.closed_subset D A) (tptp.subset B D)) (tptp.in C D))))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_composition A B)) (tptp.relation_rng B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset A B) (= B (tptp.set_union2 A (tptp.set_difference B A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.function D) (tptp.quasi_total D A B) (tptp.relation_of2_as_subset D A B)) (=> (not (= B tptp.empty_set)) (forall ((E $$unsorted)) (= (tptp.in E (tptp.relation_inverse_image D C)) (and (tptp.in E A) (tptp.in (tptp.apply D E) C))))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (exists ((C $$unsorted)) (let ((_let_1 (tptp.the_carrier A))) (and (tptp.element C (tptp.powerset (tptp.powerset _let_1))) (forall ((D $$unsorted)) (=> (tptp.element D (tptp.powerset (tptp.the_carrier A))) (= (tptp.in D C) (and (tptp.closed_subset D A) (tptp.subset B D))))) (= (tptp.topstr_closure A B) (tptp.meet_of_subsets _let_1 C))))))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset (tptp.relation_rng A) (tptp.relation_dom B)) (= (tptp.relation_dom (tptp.relation_composition A B)) (tptp.relation_dom A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (not (and (not (= B tptp.empty_set)) (= (tptp.complements_of_subsets A B) tptp.empty_set))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (= (tptp.set_union2 (tptp.singleton A) B) B))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (=> (tptp.subset (tptp.relation_dom A) (tptp.relation_rng B)) (= (tptp.relation_rng (tptp.relation_composition B A)) (tptp.relation_rng A))))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (=> (not (= B tptp.empty_set)) (= (tptp.subset_difference A (tptp.cast_to_subset A) (tptp.union_of_subsets A B)) (tptp.meet_of_subsets A (tptp.complements_of_subsets A B)))))) (forall ((A $$unsorted)) (=> (tptp.top_str A) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.subset B (tptp.topstr_closure A B)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.powerset A))) (=> (not (= B tptp.empty_set)) (= (tptp.union_of_subsets A (tptp.complements_of_subsets A B)) (tptp.subset_difference A (tptp.cast_to_subset A) (tptp.meet_of_subsets A B)))))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.set_difference A (tptp.set_difference A B)) (tptp.set_intersection2 A B))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.relation_isomorphism A B C) (tptp.relation_isomorphism B A (tptp.function_inverse C))))))))) (forall ((A $$unsorted)) (= (tptp.set_difference tptp.empty_set A) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.in A B) (tptp.element B (tptp.powerset C))) (tptp.element A C))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (tptp.connected (tptp.inclusion_relation A)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.disjoint A B))) (and (not (and (not _let_1) (forall ((C $$unsorted)) (not (tptp.in C (tptp.set_intersection2 A B)))))) (not (and (exists ((C $$unsorted)) (tptp.in C (tptp.set_intersection2 A B))) _let_1))))) (forall ((A $$unsorted)) (=> (not (= A tptp.empty_set)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset A)) (forall ((C $$unsorted)) (=> (tptp.element C A) (=> (not (tptp.in C B)) (tptp.in C (tptp.subset_complement A B))))))))) (forall ((A $$unsorted)) (=> (and (tptp.topological_space A) (tptp.top_str A)) (forall ((B $$unsorted)) (=> (tptp.element B (tptp.powerset (tptp.the_carrier A))) (tptp.open_subset (tptp.interior A B) A))))) _let_2 (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.relation_isomorphism A B C) (and (=> (tptp.reflexive A) (tptp.reflexive B)) (=> (tptp.transitive A) (tptp.transitive B)) (=> (tptp.connected A) (tptp.connected B)) (=> (tptp.antisymmetric A) (tptp.antisymmetric B)) (=> (tptp.well_founded_relation A) (tptp.well_founded_relation B)))))))))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (forall ((B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (= (= B (tptp.function_inverse A)) (and (= (tptp.relation_dom B) (tptp.relation_rng A)) (forall ((C $$unsorted) (D $$unsorted)) (let ((_let_1 (and (tptp.in C (tptp.relation_rng A)) (= D (tptp.apply B C))))) (let ((_let_2 (and (tptp.in D (tptp.relation_dom A)) (= C (tptp.apply A D))))) (and (=> _let_1 _let_2) (=> _let_2 _let_1)))))))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.element C (tptp.powerset A)) (not (and (tptp.in B (tptp.subset_complement A C)) (tptp.in B C))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (forall ((B $$unsorted)) (=> (tptp.relation B) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (and (tptp.well_ordering A) (tptp.relation_isomorphism A B C)) (tptp.well_ordering B)))))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.function_inverse A))) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (and (= (tptp.relation_rng A) (tptp.relation_dom _let_1)) (= (tptp.relation_dom A) (tptp.relation_rng _let_1))))))) _let_1 (forall ((A $$unsorted)) (=> (tptp.relation A) (=> (forall ((B $$unsorted) (C $$unsorted)) (not (tptp.in (tptp.ordered_pair B C) A))) (= A tptp.empty_set)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.function_inverse B))) (=> (and (tptp.relation B) (tptp.function B)) (=> (and (tptp.one_to_one B) (tptp.in A (tptp.relation_rng B))) (and (= A (tptp.apply B (tptp.apply _let_1 A))) (= A (tptp.apply (tptp.relation_composition _let_1 B) A))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (not (and (tptp.in A B) (tptp.element B (tptp.powerset C)) (tptp.empty C)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.well_founded_relation A) (tptp.is_well_founded_in A (tptp.relation_field A))))) (forall ((A $$unsorted)) (tptp.antisymmetric (tptp.inclusion_relation A))) (and (= (tptp.relation_dom tptp.empty_set) tptp.empty_set) (= (tptp.relation_rng tptp.empty_set) tptp.empty_set)) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.subset A B) (tptp.proper_subset B A)))) (forall ((A $$unsorted)) (=> (and (tptp.relation A) (tptp.function A)) (=> (tptp.one_to_one A) (tptp.one_to_one (tptp.function_inverse A))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.disjoint B C)) (tptp.disjoint A C))) (forall ((A $$unsorted)) (=> (tptp.relation A) (=> (or (= (tptp.relation_dom A) tptp.empty_set) (= (tptp.relation_rng A) tptp.empty_set)) (= A tptp.empty_set)))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (= (tptp.relation_dom A) tptp.empty_set) (= (tptp.relation_rng A) tptp.empty_set)))) (forall ((A $$unsorted) (B $$unsorted)) (= (= (tptp.set_difference A (tptp.singleton B)) A) (not (tptp.in B A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (and (tptp.relation B) (tptp.function B)) (forall ((C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (= B (tptp.relation_dom_restriction C A)) (and (= (tptp.relation_dom B) (tptp.set_intersection2 (tptp.relation_dom C) A)) (forall ((D $$unsorted)) (=> (tptp.in D (tptp.relation_dom B)) (= (tptp.apply B D) (tptp.apply C D)))))))))) (forall ((A $$unsorted)) (= (tptp.unordered_pair A A) (tptp.singleton A))) (forall ((A $$unsorted)) (=> (tptp.empty A) (= A tptp.empty_set))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (=> (and (tptp.function D) (tptp.quasi_total D A B) (tptp.relation_of2_as_subset D A B)) (=> (tptp.in C A) (or (= B tptp.empty_set) (tptp.in (tptp.apply D C) (tptp.relation_rng D)))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (tptp.well_founded_relation (tptp.inclusion_relation A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.subset (tptp.singleton A) (tptp.singleton B)) (= A B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (tptp.relation_dom_restriction C A))) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in B (tptp.relation_dom _let_1)) (= (tptp.apply _let_1 B) (tptp.apply C B)))))) (forall ((A $$unsorted)) (let ((_let_1 (tptp.identity_relation A))) (and (= (tptp.relation_dom _let_1) A) (= (tptp.relation_rng _let_1) A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (=> (tptp.in B A) (= (tptp.apply (tptp.relation_dom_restriction C A) B) (tptp.apply C B))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.ordered_pair A B))) (=> (tptp.relation D) (= (tptp.in _let_1 (tptp.relation_composition (tptp.identity_relation C) D)) (and (tptp.in A C) (tptp.in _let_1 D)))))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.in A B) (tptp.empty B)))) (forall ((A $$unsorted) (B $$unsorted)) (let ((_let_1 (tptp.ordered_pair A B))) (and (= (tptp.pair_first _let_1) A) (= (tptp.pair_second _let_1) B)))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.in A B) (forall ((C $$unsorted)) (not (and (tptp.in C B) (forall ((D $$unsorted)) (not (and (tptp.in D B) (tptp.in D C)))))))))) (forall ((A $$unsorted)) (=> (tptp.ordinal A) (tptp.well_ordering (tptp.inclusion_relation A)))) (forall ((A $$unsorted) (B $$unsorted)) (tptp.subset A (tptp.set_union2 A B))) (forall ((A $$unsorted) (B $$unsorted)) (= (tptp.disjoint A B) (= (tptp.set_difference A B) A))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (tptp.relation C) (= (tptp.in A (tptp.relation_dom (tptp.relation_dom_restriction C B))) (and (tptp.in A B) (tptp.in A (tptp.relation_dom C)))))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_dom_restriction B A) B))) (forall ((A $$unsorted) (B $$unsorted)) (not (and (tptp.empty A) (not (= A B)) (tptp.empty B)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.relation C) (tptp.function C)) (= (tptp.in (tptp.ordered_pair A B) C) (and (tptp.in A (tptp.relation_dom C)) (= B (tptp.apply C A)))))) (forall ((A $$unsorted)) (=> (tptp.relation A) (= (tptp.well_orders A (tptp.relation_field A)) (tptp.well_ordering A)))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (and (tptp.subset A B) (tptp.subset C B)) (tptp.subset (tptp.set_union2 A C) B))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (= (tptp.singleton A) (tptp.unordered_pair B C)) (= A B))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_dom (tptp.relation_dom_restriction B A)) (tptp.set_intersection2 (tptp.relation_dom B) A)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.in A B) (tptp.subset A (tptp.union B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (= (tptp.relation_dom_restriction B A) (tptp.relation_composition (tptp.identity_relation A) B)))) (forall ((A $$unsorted) (B $$unsorted)) (=> (tptp.relation B) (tptp.subset (tptp.relation_rng (tptp.relation_dom_restriction B A)) (tptp.relation_rng B)))) (forall ((A $$unsorted)) (= (tptp.union (tptp.powerset A)) A)) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted) (D $$unsorted)) (let ((_let_1 (tptp.function D))) (=> (and _let_1 (tptp.quasi_total D A B) (tptp.relation_of2_as_subset D A B)) (=> (tptp.subset B C) (or (and (= B tptp.empty_set) (not (= A tptp.empty_set))) (and _let_1 (tptp.quasi_total D A C) (tptp.relation_of2_as_subset D A C))))))) (forall ((A $$unsorted)) (exists ((B $$unsorted)) (and (tptp.in A B) (forall ((C $$unsorted) (D $$unsorted)) (=> (and (tptp.in C B) (tptp.subset D C)) (tptp.in D B))) (forall ((C $$unsorted)) (not (and (tptp.in C B) (forall ((D $$unsorted)) (not (and (tptp.in D B) (forall ((E $$unsorted)) (=> (tptp.subset E C) (tptp.in E D))))))))) (forall ((C $$unsorted)) (not (and (tptp.subset C B) (not (tptp.are_equipotent C B)) (not (tptp.in C B)))))))) (forall ((A $$unsorted) (B $$unsorted) (C $$unsorted)) (=> (= (tptp.singleton A) (tptp.unordered_pair B C)) (= B C))) true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 6.14/6.46  )
% 6.14/6.46  % SZS output end Proof for SEU324+2
% 6.14/6.46  % cvc5---1.0.5 exiting
% 6.14/6.46  % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------