%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Fishers_Inequality/0035_Dual_Systems/prob_00501_023078__28112356_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1496 ( 561 unt; 216 typ;   0 def)
%            Number of atoms       : 3259 (1358 equ;   0 cnn)
%            Maximal formula atoms :    7 (   2 avg)
%            Number of connectives : 11763 ( 448   ~;  46   |; 213   &;9694   @)
%                                         (   0 <=>;1362  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   18 (   6 avg)
%            Number of types       :   20 (  19 usr)
%            Number of type conns  :  410 ( 410   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  200 ( 197 usr;  20 con; 0-4 aty)
%            Number of variables   : 2720 (  72   ^;2571   !;  77   ?;2720   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 15:49:12.156
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    multiset_set_set_nat: $tType ).

thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    multiset_set_set_a: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    set_set_set_nat: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    list_set_set_a: $tType ).

thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
    multiset_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    set_set_set_a: $tType ).

thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
    multiset_set_a: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    list_set_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_set_nat: $tType ).

thf(ty_n_t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
    list_set_a: $tType ).

thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
    multiset_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    set_set_a: $tType ).

thf(ty_n_t__Multiset__Omultiset_Itf__a_J,type,
    multiset_a: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
    set_nat: $tType ).

thf(ty_n_t__List__Olist_Itf__a_J,type,
    list_a: $tType ).

thf(ty_n_t__Set__Oset_Itf__a_J,type,
    set_a: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (197)
thf(sy_c_Block__Designs_Opairwise__balance_001t__Nat__Onat,type,
    block_1456364645985477531ce_nat: set_nat > multiset_set_nat > nat > $o ).

thf(sy_c_Block__Designs_Opairwise__balance_001t__Set__Oset_It__Nat__Onat_J,type,
    block_5429691286618170193et_nat: set_set_nat > multiset_set_set_nat > nat > $o ).

thf(sy_c_Block__Designs_Opairwise__balance_001t__Set__Oset_Itf__a_J,type,
    block_6207159848980890963_set_a: set_set_a > multiset_set_set_a > nat > $o ).

thf(sy_c_Block__Designs_Opairwise__balance_001tf__a,type,
    block_5355636846524985331ance_a: set_a > multiset_set_a > nat > $o ).

thf(sy_c_Design__Basics_Odesign_001t__Nat__Onat,type,
    design_design_nat: set_nat > multiset_set_nat > $o ).

thf(sy_c_Design__Basics_Odesign_001t__Set__Oset_Itf__a_J,type,
    design_design_set_a: set_set_a > multiset_set_set_a > $o ).

thf(sy_c_Design__Basics_Odesign_001tf__a,type,
    design_design_a: set_a > multiset_set_a > $o ).

thf(sy_c_Design__Basics_Ofinite__incidence__system_001t__Nat__Onat,type,
    design5426232790142929158em_nat: set_nat > multiset_set_nat > $o ).

thf(sy_c_Design__Basics_Ofinite__incidence__system_001t__Set__Oset_It__Nat__Onat_J,type,
    design4015805878629997756et_nat: set_set_nat > multiset_set_set_nat > $o ).

thf(sy_c_Design__Basics_Ofinite__incidence__system_001t__Set__Oset_Itf__a_J,type,
    design1749870844763721896_set_a: set_set_a > multiset_set_set_a > $o ).

thf(sy_c_Design__Basics_Ofinite__incidence__system_001tf__a,type,
    design9187838744727572296stem_a: set_a > multiset_set_a > $o ).

thf(sy_c_Design__Basics_Oincidence__system_001t__Nat__Onat,type,
    design3753904077504641269em_nat: set_nat > multiset_set_nat > $o ).

thf(sy_c_Design__Basics_Oincidence__system_Oblock__complement_001t__Nat__Onat,type,
    design2875492832550762736nt_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Oblock__complement_001tf__a,type,
    design6447616907850319326ment_a: set_a > set_a > set_a ).

thf(sy_c_Design__Basics_Oincidence__system_Ocomplement__blocks_001t__Nat__Onat,type,
    design5569578106646884273ks_nat: set_nat > multiset_set_nat > multiset_set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Ocomplement__blocks_001tf__a,type,
    design8640656491286871389ocks_a: set_a > multiset_set_a > multiset_set_a ).

thf(sy_c_Design__Basics_Oincidence__system_Odesign__support_001t__Nat__Onat,type,
    design4862117536649126062rt_nat: multiset_set_nat > set_set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Odesign__support_001tf__a,type,
    design5397942185814921632port_a: multiset_set_a > set_set_a ).

thf(sy_c_Design__Basics_Oincidence__system_Oincident_001t__Nat__Onat,type,
    design8502206366797944887nt_nat: multiset_set_nat > nat > set_nat > $o ).

thf(sy_c_Design__Basics_Oincidence__system_Oincident_001tf__a,type,
    design3210447939978979927dent_a: multiset_set_a > a > set_a > $o ).

thf(sy_c_Design__Basics_Oincidence__system_Ointersection__numbers_001t__Nat__Onat,type,
    design9164904592607734462rs_nat: multiset_set_nat > set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Ointersection__numbers_001tf__a,type,
    design3761797438660848528bers_a: multiset_set_a > set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Opoint__indices_001t__Nat__Onat,type,
    design1227534709319296284es_nat: set_nat > multiset_set_nat > nat > set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Opoint__indices_001tf__a,type,
    design328527185268214962ices_a: set_a > multiset_set_a > nat > set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Oreplication__numbers_001t__Nat__Onat,type,
    design3853898657598026467rs_nat: set_nat > multiset_set_nat > set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Oreplication__numbers_001tf__a,type,
    design8835372594653258411bers_a: set_a > multiset_set_a > set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Osys__block__sizes_001t__Nat__Onat,type,
    design8152002643121538447es_nat: multiset_set_nat > set_nat ).

thf(sy_c_Design__Basics_Oincidence__system_Osys__block__sizes_001tf__a,type,
    design1769254222028858111izes_a: multiset_set_a > set_nat ).

thf(sy_c_Design__Basics_Ointersection__number_001t__Nat__Onat,type,
    design7485525362727208274er_nat: set_nat > set_nat > nat ).

thf(sy_c_Design__Basics_Ointersection__number_001t__Set__Oset_It__Nat__Onat_J,type,
    design6421228673553131784et_nat: set_set_nat > set_set_nat > nat ).

thf(sy_c_Design__Basics_Ointersection__number_001t__Set__Oset_Itf__a_J,type,
    design3520961687418077020_set_a: set_set_a > set_set_a > nat ).

thf(sy_c_Design__Basics_Ointersection__number_001tf__a,type,
    design7842873109100088828mber_a: set_a > set_a > nat ).

thf(sy_c_Design__Basics_On__intersect__number_001t__Nat__Onat,type,
    design5554526424970975290er_nat: set_nat > nat > set_nat > nat ).

thf(sy_c_Design__Basics_On__intersect__number_001t__Set__Oset_It__Nat__Onat_J,type,
    design1060797113401760752et_nat: set_set_nat > nat > set_set_nat > nat ).

thf(sy_c_Design__Basics_On__intersect__number_001t__Set__Oset_Itf__a_J,type,
    design3674606213912786548_set_a: set_set_a > nat > set_set_a > nat ).

thf(sy_c_Design__Basics_On__intersect__number_001tf__a,type,
    design735257067508376852mber_a: set_a > nat > set_a > nat ).

thf(sy_c_Design__Basics_Opoint__replication__number_001t__Nat__Onat,type,
    design3571518413069006949er_nat: multiset_set_nat > nat > nat ).

thf(sy_c_Design__Basics_Opoint__replication__number_001t__Set__Oset_It__Nat__Onat_J,type,
    design7496494955846209563et_nat: multiset_set_set_nat > set_nat > nat ).

thf(sy_c_Design__Basics_Opoint__replication__number_001t__Set__Oset_Itf__a_J,type,
    design5008467512594872073_set_a: multiset_set_set_a > set_a > nat ).

thf(sy_c_Design__Basics_Opoint__replication__number_001tf__a,type,
    design6637022207325878697mber_a: multiset_set_a > a > nat ).

thf(sy_c_Design__Basics_Opoints__index_001t__Nat__Onat,type,
    design6574611146354332593ex_nat: multiset_set_nat > set_nat > nat ).

thf(sy_c_Design__Basics_Opoints__index_001tf__a,type,
    design254580327166089565ndex_a: multiset_set_a > set_a > nat ).

thf(sy_c_Design__Basics_Oproper__design_001t__Nat__Onat,type,
    design435815215503836206gn_nat: set_nat > multiset_set_nat > $o ).

thf(sy_c_Design__Basics_Oproper__design_001tf__a,type,
    design7287791228148780576sign_a: set_a > multiset_set_a > $o ).

thf(sy_c_Design__Basics_Osimple__design_001t__Nat__Onat,type,
    design7861764274488435984gn_nat: set_nat > multiset_set_nat > $o ).

thf(sy_c_Design__Basics_Osimple__design_001t__Set__Oset_Itf__a_J,type,
    design1835266114905787166_set_a: set_set_a > multiset_set_set_a > $o ).

thf(sy_c_Design__Basics_Osimple__design_001tf__a,type,
    design3982635895484621246sign_a: set_a > multiset_set_a > $o ).

thf(sy_c_Design__Basics_Osimple__incidence__system_001t__Nat__Onat,type,
    design164292856788568387em_nat: set_nat > multiset_set_nat > $o ).

thf(sy_c_Design__Basics_Osimple__incidence__system_001tf__a,type,
    design1338723777345758283stem_a: set_a > multiset_set_a > $o ).

thf(sy_c_Design__Extras_Oconst__intersect__design_001t__Nat__Onat,type,
    design137120128173859224gn_nat: set_nat > multiset_set_nat > nat > $o ).

thf(sy_c_Design__Extras_Oconst__intersect__design_001tf__a,type,
    design9190424834980853558sign_a: set_a > multiset_set_a > nat > $o ).

thf(sy_c_Design__Extras_Osimple__const__intersect__design_001t__Nat__Onat,type,
    design8545500683235687882gn_nat: set_nat > multiset_set_nat > nat > $o ).

thf(sy_c_Design__Operations_Oincidence__system_Oadd__block_001t__Nat__Onat,type,
    design4725324266511619850ck_nat: multiset_set_nat > set_nat > multiset_set_nat ).

thf(sy_c_Design__Operations_Oincidence__system_Oadd__block_001tf__a,type,
    design4001997691126659652lock_a: multiset_set_a > set_a > multiset_set_a ).

thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001t__Nat__Onat,type,
    design8239173135376323853nt_nat: set_nat > nat > set_nat ).

thf(sy_c_Design__Operations_Oincidence__system_Oadd__point_001tf__a,type,
    design2964366272795260673oint_a: set_a > a > set_a ).

thf(sy_c_Design__Operations_Oincidence__system_Oadd__point__to__blocks_001t__Nat__Onat,type,
    design5698312687278145166ks_nat: multiset_set_nat > nat > set_set_nat > multiset_set_nat ).

thf(sy_c_Design__Operations_Oincidence__system_Oadd__point__to__blocks_001tf__a,type,
    design2935547469388721088ocks_a: multiset_set_a > a > set_set_a > multiset_set_a ).

thf(sy_c_Design__Operations_Oincidence__system_Odel__block_001t__Nat__Onat,type,
    design755385109423264192ck_nat: multiset_set_nat > set_nat > multiset_set_nat ).

thf(sy_c_Design__Operations_Oincidence__system_Odel__block_001tf__a,type,
    design1146539425385464078lock_a: multiset_set_a > set_a > multiset_set_a ).

thf(sy_c_Design__Operations_Oincidence__system_Odel__point_001t__Nat__Onat,type,
    design4269233978287968195nt_nat: set_nat > nat > set_nat ).

thf(sy_c_Design__Operations_Oincidence__system_Odel__point_001tf__a,type,
    design108908007054065099oint_a: set_a > a > set_a ).

thf(sy_c_Design__Operations_Oincidence__system_Odel__point__blocks_001t__Nat__Onat,type,
    design4832208198062110345ks_nat: multiset_set_nat > nat > multiset_set_nat ).

thf(sy_c_Design__Operations_Oincidence__system_Odel__point__blocks_001tf__a,type,
    design6411949732824333445ocks_a: multiset_set_a > a > multiset_set_a ).

thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__block_001t__Nat__Onat,type,
    design3550126062406151447ck_nat: multiset_set_nat > set_nat > multiset_set_nat ).

thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__block_001tf__a,type,
    design4241783006516448631lock_a: multiset_set_a > set_a > multiset_set_a ).

thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__point__blocks_001t__Nat__Onat,type,
    design3278834155446248416ks_nat: multiset_set_nat > nat > multiset_set_nat ).

thf(sy_c_Design__Operations_Oincidence__system_Ostr__del__point__blocks_001tf__a,type,
    design5657747894866638574ocks_a: multiset_set_a > a > multiset_set_a ).

thf(sy_c_Dual__Systems_Odual__blocks_001t__Set__Oset_Itf__a_J,type,
    dual_d359914979145368543_set_a: set_set_a > list_set_set_a > multiset_set_nat ).

thf(sy_c_Dual__Systems_Odual__blocks_001tf__a,type,
    dual_dual_blocks_a: set_a > list_set_a > multiset_set_nat ).

thf(sy_c_Dual__Systems_Oordered__incidence__system_Odual__blocks__ordered_001tf__a,type,
    dual_o8653602421375082564ered_a: list_a > list_set_a > list_set_nat ).

thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
    finite_card_nat: set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_Itf__a_J,type,
    finite_card_set_a: set_set_a > nat ).

thf(sy_c_Finite__Set_Ocard_001tf__a,type,
    finite_card_a: set_a > nat ).

thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
    finite_finite_nat: set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
    finite_finite_set_a: set_set_a > $o ).

thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
    finite_finite_a: set_a > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    minus_8522176038001411705et_nat: multiset_nat > multiset_nat > multiset_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_7237264121398869807et_nat: multiset_set_nat > multiset_set_nat > multiset_set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
    minus_706656509937749387_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
    minus_3765977307040488491iset_a: multiset_a > multiset_a > multiset_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
    minus_minus_set_a: set_a > set_a > set_a ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    plus_p6334493942879108393et_nat: multiset_nat > multiset_nat > multiset_nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
    plus_p8712254050562127327et_nat: multiset_set_nat > multiset_set_nat > multiset_set_nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
    plus_p2331992037799027419_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
    plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    zero_z7348594199698428585et_nat: multiset_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
    zero_z3157962936165190495et_nat: multiset_set_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
    zero_z6127839489552129301et_nat: multiset_set_set_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
    zero_z6396401802697562811_set_a: multiset_set_set_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
    zero_z5079479921072680283_set_a: multiset_set_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
    zero_zero_multiset_a: multiset_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_Incidence__Matrices_Oordered__design_001t__Nat__Onat,type,
    incide8999572217031194378gn_nat: list_nat > list_set_nat > $o ).

thf(sy_c_Incidence__Matrices_Oordered__design_001t__Set__Oset_Itf__a_J,type,
    incide7014649564523408292_set_a: list_set_a > list_set_set_a > $o ).

thf(sy_c_Incidence__Matrices_Oordered__design_001tf__a,type,
    incide2848671379600480836sign_a: list_a > list_set_a > $o ).

thf(sy_c_Incidence__Matrices_Oordered__pairwise__balance_001t__Nat__Onat,type,
    incide3388802471754236788ce_nat: list_nat > list_set_nat > nat > $o ).

thf(sy_c_Incidence__Matrices_Oordered__pairwise__balance_001t__Set__Oset_Itf__a_J,type,
    incide4449361439798955450_set_a: list_set_a > list_set_set_a > nat > $o ).

thf(sy_c_Incidence__Matrices_Oordered__pairwise__balance_001tf__a,type,
    incide6880889959311561818ance_a: list_a > list_set_a > nat > $o ).

thf(sy_c_Incidence__Matrices_Oordered__proper__design_001t__Set__Oset_Itf__a_J,type,
    incide2999377533768400724_set_a: list_set_a > list_set_set_a > $o ).

thf(sy_c_Incidence__Matrices_Oordered__proper__design_001tf__a,type,
    incide3676903341588786676sign_a: list_a > list_set_a > $o ).

thf(sy_c_Incidence__Matrices_Oordered__simple__design_001t__Nat__Onat,type,
    incide8427317466731264060gn_nat: list_nat > list_set_nat > $o ).

thf(sy_c_Incidence__Matrices_Oordered__simple__design_001t__Set__Oset_Itf__a_J,type,
    incide5137607047756421874_set_a: list_set_a > list_set_set_a > $o ).

thf(sy_c_Incidence__Matrices_Oordered__simple__design_001tf__a,type,
    incide371748008924627346sign_a: list_a > list_set_a > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
    sup_sup_set_a: set_a > set_a > set_a ).

thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
    distinct_nat: list_nat > $o ).

thf(sy_c_List_Odistinct_001t__Set__Oset_It__Nat__Onat_J,type,
    distinct_set_nat: list_set_nat > $o ).

thf(sy_c_List_Odistinct_001t__Set__Oset_Itf__a_J,type,
    distinct_set_a: list_set_a > $o ).

thf(sy_c_List_Odistinct_001tf__a,type,
    distinct_a: list_a > $o ).

thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
    set_nat2: list_nat > set_nat ).

thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_set_nat2: list_set_nat > set_set_nat ).

thf(sy_c_List_Olist_Oset_001t__Set__Oset_Itf__a_J,type,
    set_set_a2: list_set_a > set_set_a ).

thf(sy_c_List_Olist_Oset_001tf__a,type,
    set_a2: list_a > set_a ).

thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
    nth_set_nat: list_set_nat > nat > set_nat ).

thf(sy_c_List_Onth_001t__Set__Oset_Itf__a_J,type,
    nth_set_a: list_set_a > nat > set_a ).

thf(sy_c_List_Onth_001tf__a,type,
    nth_a: list_a > nat > a ).

thf(sy_c_Multiset_Oadd__mset_001t__Nat__Onat,type,
    add_mset_nat: nat > multiset_nat > multiset_nat ).

thf(sy_c_Multiset_Oadd__mset_001t__Set__Oset_It__Nat__Onat_J,type,
    add_mset_set_nat: set_nat > multiset_set_nat > multiset_set_nat ).

thf(sy_c_Multiset_Oadd__mset_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    add_mset_set_set_nat: set_set_nat > multiset_set_set_nat > multiset_set_set_nat ).

thf(sy_c_Multiset_Oadd__mset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    add_mset_set_set_a: set_set_a > multiset_set_set_a > multiset_set_set_a ).

thf(sy_c_Multiset_Oadd__mset_001t__Set__Oset_Itf__a_J,type,
    add_mset_set_a: set_a > multiset_set_a > multiset_set_a ).

thf(sy_c_Multiset_Oadd__mset_001tf__a,type,
    add_mset_a: a > multiset_a > multiset_a ).

thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
    mset_nat: list_nat > multiset_nat ).

thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Nat__Onat_J,type,
    mset_set_nat: list_set_nat > multiset_set_nat ).

thf(sy_c_Multiset_Omset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    mset_set_set_a: list_set_set_a > multiset_set_set_a ).

thf(sy_c_Multiset_Omset_001t__Set__Oset_Itf__a_J,type,
    mset_set_a: list_set_a > multiset_set_a ).

thf(sy_c_Multiset_Omset_001tf__a,type,
    mset_a: list_a > multiset_a ).

thf(sy_c_Multiset_Omultiset_Ocount_001t__Set__Oset_It__Nat__Onat_J,type,
    count_set_nat: multiset_set_nat > set_nat > nat ).

thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
    set_mset_nat: multiset_nat > set_nat ).

thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__Nat__Onat_J,type,
    set_mset_set_nat: multiset_set_nat > set_set_nat ).

thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    set_mset_set_set_nat: multiset_set_set_nat > set_set_set_nat ).

thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    set_mset_set_set_a: multiset_set_set_a > set_set_set_a ).

thf(sy_c_Multiset_Oset__mset_001t__Set__Oset_Itf__a_J,type,
    set_mset_set_a: multiset_set_a > set_set_a ).

thf(sy_c_Multiset_Oset__mset_001tf__a,type,
    set_mset_a: multiset_a > set_a ).

thf(sy_c_Multiset_Osubset__mset_001t__Set__Oset_Itf__a_J,type,
    subset_mset_set_a: multiset_set_a > multiset_set_a > $o ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
    size_size_list_set_a: list_set_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
    size_size_list_a: list_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    size_s5917832649809541300et_nat: multiset_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
    size_s7462436076474991978et_nat: multiset_set_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
    size_s6566526139600085008_set_a: multiset_set_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
    size_size_multiset_a: multiset_a > nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    bot_bot_set_set_a: set_set_a ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
    bot_bot_set_a: set_a ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
    ord_less_set_a: set_a > set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    ord_le6602235886369790592et_nat: multiset_nat > multiset_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le4034546139768944438et_nat: multiset_set_nat > multiset_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
    ord_le7905258569527593284_set_a: multiset_set_a > multiset_set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
    ord_less_eq_set_a: set_a > set_a > $o ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
    collect_set_a: ( set_a > $o ) > set_set_a ).

thf(sy_c_Set_OCollect_001tf__a,type,
    collect_a: ( a > $o ) > set_a ).

thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
    insert_nat: nat > set_nat > set_nat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
    insert_set_a: set_a > set_set_a > set_set_a ).

thf(sy_c_Set_Oinsert_001tf__a,type,
    insert_a: a > set_a > set_a ).

thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
    is_singleton_nat: set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Set__Oset_It__Nat__Onat_J,type,
    is_singleton_set_nat: set_set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Set__Oset_Itf__a_J,type,
    is_singleton_set_a: set_set_a > $o ).

thf(sy_c_Set_Ois__singleton_001tf__a,type,
    is_singleton_a: set_a > $o ).

thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(sy_c_Sub__Designs_Osub__design_001tf__a,type,
    sub_sub_design_a: set_a > multiset_set_a > set_a > multiset_set_a > $o ).

thf(sy_c_Sub__Designs_Osub__incidence__system_001t__Nat__Onat,type,
    sub_su5953739893325741765em_nat: set_nat > multiset_set_nat > set_nat > multiset_set_nat > $o ).

thf(sy_c_Sub__Designs_Osub__incidence__system_001tf__a,type,
    sub_su7923802003039619913stem_a: set_a > multiset_set_a > set_a > multiset_set_a > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    member_set_set_nat: set_set_nat > set_set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
    member_set_set_a: set_set_a > set_set_set_a > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v__092_060B_062s,type,
    b_s: list_set_a ).

thf(sy_v__092_060Lambda_062,type,
    lambda: nat ).

thf(sy_v__092_060V_062s,type,
    v_s: list_a ).

thf(sy_v_i1____,type,
    i1: nat ).

thf(sy_v_i2____,type,
    i2: nat ).

thf(sy_v_ps,type,
    ps: set_nat ).

% Relevant facts (1276)
thf(fact_0_inb1,axiom,
    member_set_a @ ( nth_set_a @ b_s @ i1 ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ).

% inb1
thf(fact_1_inter,axiom,
    ( ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ps )
    = ( design7842873109100088828mber_a @ ( nth_set_a @ b_s @ i1 ) @ ( nth_set_a @ b_s @ i2 ) ) ) ).

% inter
thf(fact_2_design__blocks__nempty,axiom,
    ( ( mset_set_a @ b_s )
   != zero_z5079479921072680283_set_a ) ).

% design_blocks_nempty
thf(fact_3_wf__invalid__point,axiom,
    ! [X: a,B: set_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ v_s ) )
     => ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ~ ( member_a @ X @ B ) ) ) ).

% wf_invalid_point
thf(fact_4_dual__sys_Oadd__point__existing__blocks,axiom,
    ! [Bs: set_set_nat,P: nat] :
      ( ! [Bl: set_nat] :
          ( ( member_set_nat @ Bl @ Bs )
         => ( member_nat @ P @ Bl ) )
     => ( ( design5698312687278145166ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P @ Bs )
        = ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.add_point_existing_blocks
thf(fact_5_ordered__pairwise__balance__axioms,axiom,
    incide6880889959311561818ance_a @ v_s @ b_s @ lambda ).

% ordered_pairwise_balance_axioms
thf(fact_6_ordered__design__axioms,axiom,
    incide2848671379600480836sign_a @ v_s @ b_s ).

% ordered_design_axioms
thf(fact_7_ordered__simple__design__axioms,axiom,
    incide371748008924627346sign_a @ v_s @ b_s ).

% ordered_simple_design_axioms
thf(fact_8_ordered__proper__design__axioms,axiom,
    incide3676903341588786676sign_a @ v_s @ b_s ).

% ordered_proper_design_axioms
thf(fact_9_const__intersect__design__axioms,axiom,
    design9190424834980853558sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ lambda ).

% const_intersect_design_axioms
thf(fact_10_block__set__nempty__imp__block__ex,axiom,
    ( ( ( mset_set_a @ b_s )
     != zero_z5079479921072680283_set_a )
   => ? [Bl: set_a] : ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).

% block_set_nempty_imp_block_ex
thf(fact_11_inb2,axiom,
    member_set_a @ ( nth_set_a @ b_s @ i2 ) @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ ( nth_set_a @ b_s @ i1 ) @ zero_z5079479921072680283_set_a ) ) ) ).

% inb2
thf(fact_12_simple__design__axioms,axiom,
    design3982635895484621246sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).

% simple_design_axioms
thf(fact_13_simple__incidence__system__axioms,axiom,
    design1338723777345758283stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).

% simple_incidence_system_axioms
thf(fact_14_pairwise__balance__axioms,axiom,
    block_5355636846524985331ance_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ lambda ).

% pairwise_balance_axioms
thf(fact_15_del__invalid__point,axiom,
    ! [P: a] :
      ( ~ ( member_a @ P @ ( set_a2 @ v_s ) )
     => ( ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P )
        = ( set_a2 @ v_s ) ) ) ).

% del_invalid_point
thf(fact_16_neq,axiom,
    i1 != i2 ).

% neq
thf(fact_17_sym__block__intersections__index,axiom,
    ! [B1: set_a,B2: set_a] :
      ( ( member_set_a @ B1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( member_set_a @ B2 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B1 @ zero_z5079479921072680283_set_a ) ) ) )
       => ( ( design7842873109100088828mber_a @ B1 @ B2 )
          = lambda ) ) ) ).

% sym_block_intersections_index
thf(fact_18_incidence__alt__def,axiom,
    ! [P: a,B: set_a] :
      ( ( member_a @ P @ ( set_a2 @ v_s ) )
     => ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( design3210447939978979927dent_a @ ( mset_set_a @ b_s ) @ P @ B )
          = ( member_a @ P @ B ) ) ) ) ).

% incidence_alt_def
thf(fact_19_obtain__blocks__intersect__num,axiom,
    ! [N: nat] :
      ( ( member_nat @ N @ ( design3761797438660848528bers_a @ ( mset_set_a @ b_s ) ) )
     => ? [B12: set_a,B22: set_a] :
          ( ( member_set_a @ B12 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
          & ( member_set_a @ B22 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B12 @ zero_z5079479921072680283_set_a ) ) ) )
          & ( ( design7842873109100088828mber_a @ B12 @ B22 )
            = N ) ) ) ).

% obtain_blocks_intersect_num
thf(fact_20_intersect__num__in__set,axiom,
    ! [B1: set_a,B2: set_a] :
      ( ( member_set_a @ B1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( member_set_a @ B2 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B1 @ zero_z5079479921072680283_set_a ) ) ) )
       => ( member_nat @ ( design7842873109100088828mber_a @ B1 @ B2 ) @ ( design3761797438660848528bers_a @ ( mset_set_a @ b_s ) ) ) ) ) ).

% intersect_num_in_set
thf(fact_21_design__support__def,axiom,
    ( ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) )
    = ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ).

% design_support_def
thf(fact_22_add__point__existing__blocks,axiom,
    ! [Bs: set_set_a,P: a] :
      ( ! [Bl: set_a] :
          ( ( member_set_a @ Bl @ Bs )
         => ( member_a @ P @ Bl ) )
     => ( ( design2935547469388721088ocks_a @ ( mset_set_a @ b_s ) @ P @ Bs )
        = ( mset_set_a @ b_s ) ) ) ).

% add_point_existing_blocks
thf(fact_23_insert__DiffM,axiom,
    ! [X: a,M: multiset_a] :
      ( ( member_a @ X @ ( set_mset_a @ M ) )
     => ( ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
        = M ) ) ).

% insert_DiffM
thf(fact_24_insert__DiffM,axiom,
    ! [X: nat,M: multiset_nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ M ) )
     => ( ( add_mset_nat @ X @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
        = M ) ) ).

% insert_DiffM
thf(fact_25_insert__DiffM,axiom,
    ! [X: set_a,M: multiset_set_a] :
      ( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
     => ( ( add_mset_set_a @ X @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
        = M ) ) ).

% insert_DiffM
thf(fact_26_insert__DiffM,axiom,
    ! [X: set_nat,M: multiset_set_nat] :
      ( ( member_set_nat @ X @ ( set_mset_set_nat @ M ) )
     => ( ( add_mset_set_nat @ X @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) ) )
        = M ) ) ).

% insert_DiffM
thf(fact_27_diff__union__swap2,axiom,
    ! [Y: a,M: multiset_a,X: a] :
      ( ( member_a @ Y @ ( set_mset_a @ M ) )
     => ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ X @ M ) @ ( add_mset_a @ Y @ zero_zero_multiset_a ) )
        = ( add_mset_a @ X @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ Y @ zero_zero_multiset_a ) ) ) ) ) ).

% diff_union_swap2
thf(fact_28_diff__union__swap2,axiom,
    ! [Y: nat,M: multiset_nat,X: nat] :
      ( ( member_nat @ Y @ ( set_mset_nat @ M ) )
     => ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ X @ M ) @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
        = ( add_mset_nat @ X @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) ) ) ) ) ).

% diff_union_swap2
thf(fact_29_diff__union__swap2,axiom,
    ! [Y: set_a,M: multiset_set_a,X: set_a] :
      ( ( member_set_a @ Y @ ( set_mset_set_a @ M ) )
     => ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ X @ M ) @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) )
        = ( add_mset_set_a @ X @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) ) ) ) ) ).

% diff_union_swap2
thf(fact_30_diff__union__swap2,axiom,
    ! [Y: set_nat,M: multiset_set_nat,X: set_nat] :
      ( ( member_set_nat @ Y @ ( set_mset_set_nat @ M ) )
     => ( ( minus_7237264121398869807et_nat @ ( add_mset_set_nat @ X @ M ) @ ( add_mset_set_nat @ Y @ zero_z3157962936165190495et_nat ) )
        = ( add_mset_set_nat @ X @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ Y @ zero_z3157962936165190495et_nat ) ) ) ) ) ).

% diff_union_swap2
thf(fact_31_diff__single__trivial,axiom,
    ! [X: a,M: multiset_a] :
      ( ~ ( member_a @ X @ ( set_mset_a @ M ) )
     => ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
        = M ) ) ).

% diff_single_trivial
thf(fact_32_diff__single__trivial,axiom,
    ! [X: nat,M: multiset_nat] :
      ( ~ ( member_nat @ X @ ( set_mset_nat @ M ) )
     => ( ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
        = M ) ) ).

% diff_single_trivial
thf(fact_33_diff__single__trivial,axiom,
    ! [X: set_a,M: multiset_set_a] :
      ( ~ ( member_set_a @ X @ ( set_mset_set_a @ M ) )
     => ( ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
        = M ) ) ).

% diff_single_trivial
thf(fact_34_diff__single__trivial,axiom,
    ! [X: set_nat,M: multiset_set_nat] :
      ( ~ ( member_set_nat @ X @ ( set_mset_set_nat @ M ) )
     => ( ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) )
        = M ) ) ).

% diff_single_trivial
thf(fact_35_add__mset__remove__trivial,axiom,
    ! [X: set_a,M: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ X @ M ) @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
      = M ) ).

% add_mset_remove_trivial
thf(fact_36_add__mset__remove__trivial,axiom,
    ! [X: set_nat,M: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( add_mset_set_nat @ X @ M ) @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) )
      = M ) ).

% add_mset_remove_trivial
thf(fact_37_remove1__single__empty__iff,axiom,
    ! [L: set_a,L2: set_a] :
      ( ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ L @ zero_z5079479921072680283_set_a ) @ ( add_mset_set_a @ L2 @ zero_z5079479921072680283_set_a ) )
        = zero_z5079479921072680283_set_a )
      = ( L2 = L ) ) ).

% remove1_single_empty_iff
thf(fact_38_remove1__single__empty__iff,axiom,
    ! [L: set_nat,L2: set_nat] :
      ( ( ( minus_7237264121398869807et_nat @ ( add_mset_set_nat @ L @ zero_z3157962936165190495et_nat ) @ ( add_mset_set_nat @ L2 @ zero_z3157962936165190495et_nat ) )
        = zero_z3157962936165190495et_nat )
      = ( L2 = L ) ) ).

% remove1_single_empty_iff
thf(fact_39_diff__add__mset__swap,axiom,
    ! [B: a,A: multiset_a,M: multiset_a] :
      ( ~ ( member_a @ B @ ( set_mset_a @ A ) )
     => ( ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ M ) @ A )
        = ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ M @ A ) ) ) ) ).

% diff_add_mset_swap
thf(fact_40_diff__add__mset__swap,axiom,
    ! [B: nat,A: multiset_nat,M: multiset_nat] :
      ( ~ ( member_nat @ B @ ( set_mset_nat @ A ) )
     => ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ M ) @ A )
        = ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ M @ A ) ) ) ) ).

% diff_add_mset_swap
thf(fact_41_diff__add__mset__swap,axiom,
    ! [B: set_a,A: multiset_set_a,M: multiset_set_a] :
      ( ~ ( member_set_a @ B @ ( set_mset_set_a @ A ) )
     => ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ B @ M ) @ A )
        = ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ M @ A ) ) ) ) ).

% diff_add_mset_swap
thf(fact_42_diff__add__mset__swap,axiom,
    ! [B: set_nat,A: multiset_set_nat,M: multiset_set_nat] :
      ( ~ ( member_set_nat @ B @ ( set_mset_set_nat @ A ) )
     => ( ( minus_7237264121398869807et_nat @ ( add_mset_set_nat @ B @ M ) @ A )
        = ( add_mset_set_nat @ B @ ( minus_7237264121398869807et_nat @ M @ A ) ) ) ) ).

% diff_add_mset_swap
thf(fact_43_remove__diff__multiset,axiom,
    ! [X13: a,A: multiset_a,B3: multiset_a] :
      ( ~ ( member_a @ X13 @ ( set_mset_a @ A ) )
     => ( ( minus_3765977307040488491iset_a @ A @ ( add_mset_a @ X13 @ B3 ) )
        = ( minus_3765977307040488491iset_a @ A @ B3 ) ) ) ).

% remove_diff_multiset
thf(fact_44_remove__diff__multiset,axiom,
    ! [X13: nat,A: multiset_nat,B3: multiset_nat] :
      ( ~ ( member_nat @ X13 @ ( set_mset_nat @ A ) )
     => ( ( minus_8522176038001411705et_nat @ A @ ( add_mset_nat @ X13 @ B3 ) )
        = ( minus_8522176038001411705et_nat @ A @ B3 ) ) ) ).

% remove_diff_multiset
thf(fact_45_remove__diff__multiset,axiom,
    ! [X13: set_a,A: multiset_set_a,B3: multiset_set_a] :
      ( ~ ( member_set_a @ X13 @ ( set_mset_set_a @ A ) )
     => ( ( minus_706656509937749387_set_a @ A @ ( add_mset_set_a @ X13 @ B3 ) )
        = ( minus_706656509937749387_set_a @ A @ B3 ) ) ) ).

% remove_diff_multiset
thf(fact_46_remove__diff__multiset,axiom,
    ! [X13: set_nat,A: multiset_set_nat,B3: multiset_set_nat] :
      ( ~ ( member_set_nat @ X13 @ ( set_mset_set_nat @ A ) )
     => ( ( minus_7237264121398869807et_nat @ A @ ( add_mset_set_nat @ X13 @ B3 ) )
        = ( minus_7237264121398869807et_nat @ A @ B3 ) ) ) ).

% remove_diff_multiset
thf(fact_47_multi__self__add__other__not__self,axiom,
    ! [M: multiset_set_a,X: set_a] :
      ( M
     != ( add_mset_set_a @ X @ M ) ) ).

% multi_self_add_other_not_self
thf(fact_48_multi__self__add__other__not__self,axiom,
    ! [M: multiset_set_nat,X: set_nat] :
      ( M
     != ( add_mset_set_nat @ X @ M ) ) ).

% multi_self_add_other_not_self
thf(fact_49_add__mset__add__mset__same__iff,axiom,
    ! [A2: set_a,A: multiset_set_a,B3: multiset_set_a] :
      ( ( ( add_mset_set_a @ A2 @ A )
        = ( add_mset_set_a @ A2 @ B3 ) )
      = ( A = B3 ) ) ).

% add_mset_add_mset_same_iff
thf(fact_50_add__mset__add__mset__same__iff,axiom,
    ! [A2: set_nat,A: multiset_set_nat,B3: multiset_set_nat] :
      ( ( ( add_mset_set_nat @ A2 @ A )
        = ( add_mset_set_nat @ A2 @ B3 ) )
      = ( A = B3 ) ) ).

% add_mset_add_mset_same_iff
thf(fact_51_dual__sys_Oblock__set__nempty__imp__block__ex,axiom,
    ( ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
     != zero_z3157962936165190495et_nat )
   => ? [Bl: set_nat] : ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).

% dual_sys.block_set_nempty_imp_block_ex
thf(fact_52_add__mset__eq__singleton__iff,axiom,
    ! [X: set_a,M: multiset_set_a,Y: set_a] :
      ( ( ( add_mset_set_a @ X @ M )
        = ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) )
      = ( ( M = zero_z5079479921072680283_set_a )
        & ( X = Y ) ) ) ).

% add_mset_eq_singleton_iff
thf(fact_53_add__mset__eq__singleton__iff,axiom,
    ! [X: set_nat,M: multiset_set_nat,Y: set_nat] :
      ( ( ( add_mset_set_nat @ X @ M )
        = ( add_mset_set_nat @ Y @ zero_z3157962936165190495et_nat ) )
      = ( ( M = zero_z3157962936165190495et_nat )
        & ( X = Y ) ) ) ).

% add_mset_eq_singleton_iff
thf(fact_54_single__eq__add__mset,axiom,
    ! [A2: set_a,B: set_a,M: multiset_set_a] :
      ( ( ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a )
        = ( add_mset_set_a @ B @ M ) )
      = ( ( B = A2 )
        & ( M = zero_z5079479921072680283_set_a ) ) ) ).

% single_eq_add_mset
thf(fact_55_single__eq__add__mset,axiom,
    ! [A2: set_nat,B: set_nat,M: multiset_set_nat] :
      ( ( ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat )
        = ( add_mset_set_nat @ B @ M ) )
      = ( ( B = A2 )
        & ( M = zero_z3157962936165190495et_nat ) ) ) ).

% single_eq_add_mset
thf(fact_56_add__mset__eq__single,axiom,
    ! [B: set_a,M: multiset_set_a,A2: set_a] :
      ( ( ( add_mset_set_a @ B @ M )
        = ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) )
      = ( ( B = A2 )
        & ( M = zero_z5079479921072680283_set_a ) ) ) ).

% add_mset_eq_single
thf(fact_57_add__mset__eq__single,axiom,
    ! [B: set_nat,M: multiset_set_nat,A2: set_nat] :
      ( ( ( add_mset_set_nat @ B @ M )
        = ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) )
      = ( ( B = A2 )
        & ( M = zero_z3157962936165190495et_nat ) ) ) ).

% add_mset_eq_single
thf(fact_58_single__eq__single,axiom,
    ! [A2: set_a,B: set_a] :
      ( ( ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a )
        = ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) )
      = ( A2 = B ) ) ).

% single_eq_single
thf(fact_59_single__eq__single,axiom,
    ! [A2: set_nat,B: set_nat] :
      ( ( ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat )
        = ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) )
      = ( A2 = B ) ) ).

% single_eq_single
thf(fact_60_set__mset__mset,axiom,
    ! [Xs: list_a] :
      ( ( set_mset_a @ ( mset_a @ Xs ) )
      = ( set_a2 @ Xs ) ) ).

% set_mset_mset
thf(fact_61_set__mset__mset,axiom,
    ! [Xs: list_set_a] :
      ( ( set_mset_set_a @ ( mset_set_a @ Xs ) )
      = ( set_set_a2 @ Xs ) ) ).

% set_mset_mset
thf(fact_62_set__mset__mset,axiom,
    ! [Xs: list_set_nat] :
      ( ( set_mset_set_nat @ ( mset_set_nat @ Xs ) )
      = ( set_set_nat2 @ Xs ) ) ).

% set_mset_mset
thf(fact_63_in__multiset__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
      = ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_64_in__multiset__in__set,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_mset_a @ ( mset_a @ Xs ) ) )
      = ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_65_in__multiset__in__set,axiom,
    ! [X: set_a,Xs: list_set_a] :
      ( ( member_set_a @ X @ ( set_mset_set_a @ ( mset_set_a @ Xs ) ) )
      = ( member_set_a @ X @ ( set_set_a2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_66_in__multiset__in__set,axiom,
    ! [X: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_mset_set_nat @ ( mset_set_nat @ Xs ) ) )
      = ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_67_dual__sys_Odesign__support__def,axiom,
    ( ( design4862117536649126062rt_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
    = ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.design_support_def
thf(fact_68_list__of__mset__exi,axiom,
    ! [M2: multiset_set_a] :
    ? [L3: list_set_a] :
      ( M2
      = ( mset_set_a @ L3 ) ) ).

% list_of_mset_exi
thf(fact_69_list__of__mset__exi,axiom,
    ! [M2: multiset_set_nat] :
    ? [L3: list_set_nat] :
      ( M2
      = ( mset_set_nat @ L3 ) ) ).

% list_of_mset_exi
thf(fact_70_ex__mset,axiom,
    ! [X2: multiset_set_a] :
    ? [Xs2: list_set_a] :
      ( ( mset_set_a @ Xs2 )
      = X2 ) ).

% ex_mset
thf(fact_71_ex__mset,axiom,
    ! [X2: multiset_set_nat] :
    ? [Xs2: list_set_nat] :
      ( ( mset_set_nat @ Xs2 )
      = X2 ) ).

% ex_mset
thf(fact_72_add__mset__commute,axiom,
    ! [X: set_a,Y: set_a,M: multiset_set_a] :
      ( ( add_mset_set_a @ X @ ( add_mset_set_a @ Y @ M ) )
      = ( add_mset_set_a @ Y @ ( add_mset_set_a @ X @ M ) ) ) ).

% add_mset_commute
thf(fact_73_add__mset__commute,axiom,
    ! [X: set_nat,Y: set_nat,M: multiset_set_nat] :
      ( ( add_mset_set_nat @ X @ ( add_mset_set_nat @ Y @ M ) )
      = ( add_mset_set_nat @ Y @ ( add_mset_set_nat @ X @ M ) ) ) ).

% add_mset_commute
thf(fact_74_add__eq__conv__ex,axiom,
    ! [A2: set_a,M: multiset_set_a,B: set_a,N2: multiset_set_a] :
      ( ( ( add_mset_set_a @ A2 @ M )
        = ( add_mset_set_a @ B @ N2 ) )
      = ( ( ( M = N2 )
          & ( A2 = B ) )
        | ? [K: multiset_set_a] :
            ( ( M
              = ( add_mset_set_a @ B @ K ) )
            & ( N2
              = ( add_mset_set_a @ A2 @ K ) ) ) ) ) ).

% add_eq_conv_ex
thf(fact_75_add__eq__conv__ex,axiom,
    ! [A2: set_nat,M: multiset_set_nat,B: set_nat,N2: multiset_set_nat] :
      ( ( ( add_mset_set_nat @ A2 @ M )
        = ( add_mset_set_nat @ B @ N2 ) )
      = ( ( ( M = N2 )
          & ( A2 = B ) )
        | ? [K: multiset_set_nat] :
            ( ( M
              = ( add_mset_set_nat @ B @ K ) )
            & ( N2
              = ( add_mset_set_nat @ A2 @ K ) ) ) ) ) ).

% add_eq_conv_ex
thf(fact_76_mem__Collect__eq,axiom,
    ! [A2: set_a,P2: set_a > $o] :
      ( ( member_set_a @ A2 @ ( collect_set_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_77_mem__Collect__eq,axiom,
    ! [A2: a,P2: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
    ! [A2: set_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ A2 @ ( collect_set_nat @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
    ! [A2: nat,P2: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_80_Collect__mem__eq,axiom,
    ! [A: set_set_a] :
      ( ( collect_set_a
        @ ^ [X3: set_a] : ( member_set_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X3: a] : ( member_a @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
    ! [A: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_84_Multiset_Odiff__right__commute,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a,Q: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ N2 ) @ Q )
      = ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ Q ) @ N2 ) ) ).

% Multiset.diff_right_commute
thf(fact_85_Multiset_Odiff__right__commute,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat,Q: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ M @ N2 ) @ Q )
      = ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ M @ Q ) @ N2 ) ) ).

% Multiset.diff_right_commute
thf(fact_86_mset__eq__setD,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( mset_a @ Xs )
        = ( mset_a @ Ys ) )
     => ( ( set_a2 @ Xs )
        = ( set_a2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_87_mset__eq__setD,axiom,
    ! [Xs: list_set_a,Ys: list_set_a] :
      ( ( ( mset_set_a @ Xs )
        = ( mset_set_a @ Ys ) )
     => ( ( set_set_a2 @ Xs )
        = ( set_set_a2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_88_mset__eq__setD,axiom,
    ! [Xs: list_set_nat,Ys: list_set_nat] :
      ( ( ( mset_set_nat @ Xs )
        = ( mset_set_nat @ Ys ) )
     => ( ( set_set_nat2 @ Xs )
        = ( set_set_nat2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_89_multiset__nonemptyE,axiom,
    ! [A: multiset_a] :
      ( ( A != zero_zero_multiset_a )
     => ~ ! [X4: a] :
            ~ ( member_a @ X4 @ ( set_mset_a @ A ) ) ) ).

% multiset_nonemptyE
thf(fact_90_multiset__nonemptyE,axiom,
    ! [A: multiset_nat] :
      ( ( A != zero_z7348594199698428585et_nat )
     => ~ ! [X4: nat] :
            ~ ( member_nat @ X4 @ ( set_mset_nat @ A ) ) ) ).

% multiset_nonemptyE
thf(fact_91_multiset__nonemptyE,axiom,
    ! [A: multiset_set_a] :
      ( ( A != zero_z5079479921072680283_set_a )
     => ~ ! [X4: set_a] :
            ~ ( member_set_a @ X4 @ ( set_mset_set_a @ A ) ) ) ).

% multiset_nonemptyE
thf(fact_92_multiset__nonemptyE,axiom,
    ! [A: multiset_set_nat] :
      ( ( A != zero_z3157962936165190495et_nat )
     => ~ ! [X4: set_nat] :
            ~ ( member_set_nat @ X4 @ ( set_mset_set_nat @ A ) ) ) ).

% multiset_nonemptyE
thf(fact_93_union__single__eq__member,axiom,
    ! [X: a,M: multiset_a,N2: multiset_a] :
      ( ( ( add_mset_a @ X @ M )
        = N2 )
     => ( member_a @ X @ ( set_mset_a @ N2 ) ) ) ).

% union_single_eq_member
thf(fact_94_union__single__eq__member,axiom,
    ! [X: nat,M: multiset_nat,N2: multiset_nat] :
      ( ( ( add_mset_nat @ X @ M )
        = N2 )
     => ( member_nat @ X @ ( set_mset_nat @ N2 ) ) ) ).

% union_single_eq_member
thf(fact_95_union__single__eq__member,axiom,
    ! [X: set_a,M: multiset_set_a,N2: multiset_set_a] :
      ( ( ( add_mset_set_a @ X @ M )
        = N2 )
     => ( member_set_a @ X @ ( set_mset_set_a @ N2 ) ) ) ).

% union_single_eq_member
thf(fact_96_union__single__eq__member,axiom,
    ! [X: set_nat,M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( ( add_mset_set_nat @ X @ M )
        = N2 )
     => ( member_set_nat @ X @ ( set_mset_set_nat @ N2 ) ) ) ).

% union_single_eq_member
thf(fact_97_insert__noteq__member,axiom,
    ! [B: a,B3: multiset_a,C: a,C2: multiset_a] :
      ( ( ( add_mset_a @ B @ B3 )
        = ( add_mset_a @ C @ C2 ) )
     => ( ( B != C )
       => ( member_a @ C @ ( set_mset_a @ B3 ) ) ) ) ).

% insert_noteq_member
thf(fact_98_insert__noteq__member,axiom,
    ! [B: nat,B3: multiset_nat,C: nat,C2: multiset_nat] :
      ( ( ( add_mset_nat @ B @ B3 )
        = ( add_mset_nat @ C @ C2 ) )
     => ( ( B != C )
       => ( member_nat @ C @ ( set_mset_nat @ B3 ) ) ) ) ).

% insert_noteq_member
thf(fact_99_insert__noteq__member,axiom,
    ! [B: set_a,B3: multiset_set_a,C: set_a,C2: multiset_set_a] :
      ( ( ( add_mset_set_a @ B @ B3 )
        = ( add_mset_set_a @ C @ C2 ) )
     => ( ( B != C )
       => ( member_set_a @ C @ ( set_mset_set_a @ B3 ) ) ) ) ).

% insert_noteq_member
thf(fact_100_insert__noteq__member,axiom,
    ! [B: set_nat,B3: multiset_set_nat,C: set_nat,C2: multiset_set_nat] :
      ( ( ( add_mset_set_nat @ B @ B3 )
        = ( add_mset_set_nat @ C @ C2 ) )
     => ( ( B != C )
       => ( member_set_nat @ C @ ( set_mset_set_nat @ B3 ) ) ) ) ).

% insert_noteq_member
thf(fact_101_multi__member__split,axiom,
    ! [X: a,M: multiset_a] :
      ( ( member_a @ X @ ( set_mset_a @ M ) )
     => ? [A3: multiset_a] :
          ( M
          = ( add_mset_a @ X @ A3 ) ) ) ).

% multi_member_split
thf(fact_102_multi__member__split,axiom,
    ! [X: nat,M: multiset_nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ M ) )
     => ? [A3: multiset_nat] :
          ( M
          = ( add_mset_nat @ X @ A3 ) ) ) ).

% multi_member_split
thf(fact_103_multi__member__split,axiom,
    ! [X: set_a,M: multiset_set_a] :
      ( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
     => ? [A3: multiset_set_a] :
          ( M
          = ( add_mset_set_a @ X @ A3 ) ) ) ).

% multi_member_split
thf(fact_104_multi__member__split,axiom,
    ! [X: set_nat,M: multiset_set_nat] :
      ( ( member_set_nat @ X @ ( set_mset_set_nat @ M ) )
     => ? [A3: multiset_set_nat] :
          ( M
          = ( add_mset_set_nat @ X @ A3 ) ) ) ).

% multi_member_split
thf(fact_105_mset__add,axiom,
    ! [A2: a,A: multiset_a] :
      ( ( member_a @ A2 @ ( set_mset_a @ A ) )
     => ~ ! [B4: multiset_a] :
            ( A
           != ( add_mset_a @ A2 @ B4 ) ) ) ).

% mset_add
thf(fact_106_mset__add,axiom,
    ! [A2: nat,A: multiset_nat] :
      ( ( member_nat @ A2 @ ( set_mset_nat @ A ) )
     => ~ ! [B4: multiset_nat] :
            ( A
           != ( add_mset_nat @ A2 @ B4 ) ) ) ).

% mset_add
thf(fact_107_mset__add,axiom,
    ! [A2: set_a,A: multiset_set_a] :
      ( ( member_set_a @ A2 @ ( set_mset_set_a @ A ) )
     => ~ ! [B4: multiset_set_a] :
            ( A
           != ( add_mset_set_a @ A2 @ B4 ) ) ) ).

% mset_add
thf(fact_108_mset__add,axiom,
    ! [A2: set_nat,A: multiset_set_nat] :
      ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ A ) )
     => ~ ! [B4: multiset_set_nat] :
            ( A
           != ( add_mset_set_nat @ A2 @ B4 ) ) ) ).

% mset_add
thf(fact_109_multi__nonempty__split,axiom,
    ! [M: multiset_set_a] :
      ( ( M != zero_z5079479921072680283_set_a )
     => ? [A3: multiset_set_a,A4: set_a] :
          ( M
          = ( add_mset_set_a @ A4 @ A3 ) ) ) ).

% multi_nonempty_split
thf(fact_110_multi__nonempty__split,axiom,
    ! [M: multiset_set_nat] :
      ( ( M != zero_z3157962936165190495et_nat )
     => ? [A3: multiset_set_nat,A4: set_nat] :
          ( M
          = ( add_mset_set_nat @ A4 @ A3 ) ) ) ).

% multi_nonempty_split
thf(fact_111_empty__not__add__mset,axiom,
    ! [A2: set_a,A: multiset_set_a] :
      ( zero_z5079479921072680283_set_a
     != ( add_mset_set_a @ A2 @ A ) ) ).

% empty_not_add_mset
thf(fact_112_empty__not__add__mset,axiom,
    ! [A2: set_nat,A: multiset_set_nat] :
      ( zero_z3157962936165190495et_nat
     != ( add_mset_set_nat @ A2 @ A ) ) ).

% empty_not_add_mset
thf(fact_113_multiset__induct2,axiom,
    ! [P2: multiset_set_a > multiset_set_a > $o,M: multiset_set_a,N2: multiset_set_a] :
      ( ( P2 @ zero_z5079479921072680283_set_a @ zero_z5079479921072680283_set_a )
     => ( ! [A4: set_a,M3: multiset_set_a,N3: multiset_set_a] :
            ( ( P2 @ M3 @ N3 )
           => ( P2 @ ( add_mset_set_a @ A4 @ M3 ) @ N3 ) )
       => ( ! [A4: set_a,M3: multiset_set_a,N3: multiset_set_a] :
              ( ( P2 @ M3 @ N3 )
             => ( P2 @ M3 @ ( add_mset_set_a @ A4 @ N3 ) ) )
         => ( P2 @ M @ N2 ) ) ) ) ).

% multiset_induct2
thf(fact_114_multiset__induct2,axiom,
    ! [P2: multiset_set_a > multiset_set_nat > $o,M: multiset_set_a,N2: multiset_set_nat] :
      ( ( P2 @ zero_z5079479921072680283_set_a @ zero_z3157962936165190495et_nat )
     => ( ! [A4: set_a,M3: multiset_set_a,N3: multiset_set_nat] :
            ( ( P2 @ M3 @ N3 )
           => ( P2 @ ( add_mset_set_a @ A4 @ M3 ) @ N3 ) )
       => ( ! [A4: set_nat,M3: multiset_set_a,N3: multiset_set_nat] :
              ( ( P2 @ M3 @ N3 )
             => ( P2 @ M3 @ ( add_mset_set_nat @ A4 @ N3 ) ) )
         => ( P2 @ M @ N2 ) ) ) ) ).

% multiset_induct2
thf(fact_115_multiset__induct2,axiom,
    ! [P2: multiset_set_nat > multiset_set_a > $o,M: multiset_set_nat,N2: multiset_set_a] :
      ( ( P2 @ zero_z3157962936165190495et_nat @ zero_z5079479921072680283_set_a )
     => ( ! [A4: set_nat,M3: multiset_set_nat,N3: multiset_set_a] :
            ( ( P2 @ M3 @ N3 )
           => ( P2 @ ( add_mset_set_nat @ A4 @ M3 ) @ N3 ) )
       => ( ! [A4: set_a,M3: multiset_set_nat,N3: multiset_set_a] :
              ( ( P2 @ M3 @ N3 )
             => ( P2 @ M3 @ ( add_mset_set_a @ A4 @ N3 ) ) )
         => ( P2 @ M @ N2 ) ) ) ) ).

% multiset_induct2
thf(fact_116_multiset__induct2,axiom,
    ! [P2: multiset_set_nat > multiset_set_nat > $o,M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( P2 @ zero_z3157962936165190495et_nat @ zero_z3157962936165190495et_nat )
     => ( ! [A4: set_nat,M3: multiset_set_nat,N3: multiset_set_nat] :
            ( ( P2 @ M3 @ N3 )
           => ( P2 @ ( add_mset_set_nat @ A4 @ M3 ) @ N3 ) )
       => ( ! [A4: set_nat,M3: multiset_set_nat,N3: multiset_set_nat] :
              ( ( P2 @ M3 @ N3 )
             => ( P2 @ M3 @ ( add_mset_set_nat @ A4 @ N3 ) ) )
         => ( P2 @ M @ N2 ) ) ) ) ).

% multiset_induct2
thf(fact_117_multiset__induct,axiom,
    ! [P2: multiset_set_a > $o,M: multiset_set_a] :
      ( ( P2 @ zero_z5079479921072680283_set_a )
     => ( ! [X4: set_a,M3: multiset_set_a] :
            ( ( P2 @ M3 )
           => ( P2 @ ( add_mset_set_a @ X4 @ M3 ) ) )
       => ( P2 @ M ) ) ) ).

% multiset_induct
thf(fact_118_multiset__induct,axiom,
    ! [P2: multiset_set_nat > $o,M: multiset_set_nat] :
      ( ( P2 @ zero_z3157962936165190495et_nat )
     => ( ! [X4: set_nat,M3: multiset_set_nat] :
            ( ( P2 @ M3 )
           => ( P2 @ ( add_mset_set_nat @ X4 @ M3 ) ) )
       => ( P2 @ M ) ) ) ).

% multiset_induct
thf(fact_119_multiset__cases,axiom,
    ! [M: multiset_set_a] :
      ( ( M != zero_z5079479921072680283_set_a )
     => ~ ! [X4: set_a,N3: multiset_set_a] :
            ( M
           != ( add_mset_set_a @ X4 @ N3 ) ) ) ).

% multiset_cases
thf(fact_120_multiset__cases,axiom,
    ! [M: multiset_set_nat] :
      ( ( M != zero_z3157962936165190495et_nat )
     => ~ ! [X4: set_nat,N3: multiset_set_nat] :
            ( M
           != ( add_mset_set_nat @ X4 @ N3 ) ) ) ).

% multiset_cases
thf(fact_121_in__diffD,axiom,
    ! [A2: a,M: multiset_a,N2: multiset_a] :
      ( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N2 ) ) )
     => ( member_a @ A2 @ ( set_mset_a @ M ) ) ) ).

% in_diffD
thf(fact_122_in__diffD,axiom,
    ! [A2: nat,M: multiset_nat,N2: multiset_nat] :
      ( ( member_nat @ A2 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ N2 ) ) )
     => ( member_nat @ A2 @ ( set_mset_nat @ M ) ) ) ).

% in_diffD
thf(fact_123_in__diffD,axiom,
    ! [A2: set_a,M: multiset_set_a,N2: multiset_set_a] :
      ( ( member_set_a @ A2 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N2 ) ) )
     => ( member_set_a @ A2 @ ( set_mset_set_a @ M ) ) ) ).

% in_diffD
thf(fact_124_in__diffD,axiom,
    ! [A2: set_nat,M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ M @ N2 ) ) )
     => ( member_set_nat @ A2 @ ( set_mset_set_nat @ M ) ) ) ).

% in_diffD
thf(fact_125_Multiset_Odiff__cancel,axiom,
    ! [A: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ A @ A )
      = zero_z5079479921072680283_set_a ) ).

% Multiset.diff_cancel
thf(fact_126_Multiset_Odiff__cancel,axiom,
    ! [A: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ A @ A )
      = zero_z3157962936165190495et_nat ) ).

% Multiset.diff_cancel
thf(fact_127_diff__empty,axiom,
    ! [M: multiset_set_a] :
      ( ( ( minus_706656509937749387_set_a @ M @ zero_z5079479921072680283_set_a )
        = M )
      & ( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ M )
        = zero_z5079479921072680283_set_a ) ) ).

% diff_empty
thf(fact_128_diff__empty,axiom,
    ! [M: multiset_set_nat] :
      ( ( ( minus_7237264121398869807et_nat @ M @ zero_z3157962936165190495et_nat )
        = M )
      & ( ( minus_7237264121398869807et_nat @ zero_z3157962936165190495et_nat @ M )
        = zero_z3157962936165190495et_nat ) ) ).

% diff_empty
thf(fact_129_add__mset__diff__bothsides,axiom,
    ! [A2: set_a,M: multiset_set_a,A: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ A2 @ M ) @ ( add_mset_set_a @ A2 @ A ) )
      = ( minus_706656509937749387_set_a @ M @ A ) ) ).

% add_mset_diff_bothsides
thf(fact_130_add__mset__diff__bothsides,axiom,
    ! [A2: set_nat,M: multiset_set_nat,A: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( add_mset_set_nat @ A2 @ M ) @ ( add_mset_set_nat @ A2 @ A ) )
      = ( minus_7237264121398869807et_nat @ M @ A ) ) ).

% add_mset_diff_bothsides
thf(fact_131_multi__member__last,axiom,
    ! [X: a] : ( member_a @ X @ ( set_mset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) ) ).

% multi_member_last
thf(fact_132_multi__member__last,axiom,
    ! [X: nat] : ( member_nat @ X @ ( set_mset_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) ) ).

% multi_member_last
thf(fact_133_multi__member__last,axiom,
    ! [X: set_a] : ( member_set_a @ X @ ( set_mset_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ).

% multi_member_last
thf(fact_134_multi__member__last,axiom,
    ! [X: set_nat] : ( member_set_nat @ X @ ( set_mset_set_nat @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) ) ) ).

% multi_member_last
thf(fact_135_union__single__eq__diff,axiom,
    ! [X: set_a,M: multiset_set_a,N2: multiset_set_a] :
      ( ( ( add_mset_set_a @ X @ M )
        = N2 )
     => ( M
        = ( minus_706656509937749387_set_a @ N2 @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) ) ).

% union_single_eq_diff
thf(fact_136_union__single__eq__diff,axiom,
    ! [X: set_nat,M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( ( add_mset_set_nat @ X @ M )
        = N2 )
     => ( M
        = ( minus_7237264121398869807et_nat @ N2 @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) ) ) ) ).

% union_single_eq_diff
thf(fact_137_add__eq__conv__diff,axiom,
    ! [A2: set_a,M: multiset_set_a,B: set_a,N2: multiset_set_a] :
      ( ( ( add_mset_set_a @ A2 @ M )
        = ( add_mset_set_a @ B @ N2 ) )
      = ( ( ( M = N2 )
          & ( A2 = B ) )
        | ( ( M
            = ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ N2 @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ) )
          & ( N2
            = ( add_mset_set_a @ A2 @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).

% add_eq_conv_diff
thf(fact_138_add__eq__conv__diff,axiom,
    ! [A2: set_nat,M: multiset_set_nat,B: set_nat,N2: multiset_set_nat] :
      ( ( ( add_mset_set_nat @ A2 @ M )
        = ( add_mset_set_nat @ B @ N2 ) )
      = ( ( ( M = N2 )
          & ( A2 = B ) )
        | ( ( M
            = ( add_mset_set_nat @ B @ ( minus_7237264121398869807et_nat @ N2 @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ) )
          & ( N2
            = ( add_mset_set_nat @ A2 @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) ) ) ) ) ) ) ).

% add_eq_conv_diff
thf(fact_139_diff__union__swap,axiom,
    ! [A2: set_a,B: set_a,M: multiset_set_a] :
      ( ( A2 != B )
     => ( ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) )
        = ( minus_706656509937749387_set_a @ ( add_mset_set_a @ B @ M ) @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ) ) ).

% diff_union_swap
thf(fact_140_diff__union__swap,axiom,
    ! [A2: set_nat,B: set_nat,M: multiset_set_nat] :
      ( ( A2 != B )
     => ( ( add_mset_set_nat @ B @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) )
        = ( minus_7237264121398869807et_nat @ ( add_mset_set_nat @ B @ M ) @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ) ) ).

% diff_union_swap
thf(fact_141_trivial__add__mset__remove__iff,axiom,
    ! [A2: a,N2: multiset_a,B: a] :
      ( ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) )
        = N2 )
      = ( ( member_a @ A2 @ ( set_mset_a @ N2 ) )
        & ( A2 = B ) ) ) ).

% trivial_add_mset_remove_iff
thf(fact_142_trivial__add__mset__remove__iff,axiom,
    ! [A2: nat,N2: multiset_nat,B: nat] :
      ( ( ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N2 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) )
        = N2 )
      = ( ( member_nat @ A2 @ ( set_mset_nat @ N2 ) )
        & ( A2 = B ) ) ) ).

% trivial_add_mset_remove_iff
thf(fact_143_trivial__add__mset__remove__iff,axiom,
    ! [A2: set_a,N2: multiset_set_a,B: set_a] :
      ( ( ( add_mset_set_a @ A2 @ ( minus_706656509937749387_set_a @ N2 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) )
        = N2 )
      = ( ( member_set_a @ A2 @ ( set_mset_set_a @ N2 ) )
        & ( A2 = B ) ) ) ).

% trivial_add_mset_remove_iff
thf(fact_144_trivial__add__mset__remove__iff,axiom,
    ! [A2: set_nat,N2: multiset_set_nat,B: set_nat] :
      ( ( ( add_mset_set_nat @ A2 @ ( minus_7237264121398869807et_nat @ N2 @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) ) )
        = N2 )
      = ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ N2 ) )
        & ( A2 = B ) ) ) ).

% trivial_add_mset_remove_iff
thf(fact_145_add__mset__remove__trivial__iff,axiom,
    ! [N2: multiset_a,A2: a,B: a] :
      ( ( N2
        = ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) )
      = ( ( member_a @ A2 @ ( set_mset_a @ N2 ) )
        & ( A2 = B ) ) ) ).

% add_mset_remove_trivial_iff
thf(fact_146_add__mset__remove__trivial__iff,axiom,
    ! [N2: multiset_nat,A2: nat,B: nat] :
      ( ( N2
        = ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N2 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) ) )
      = ( ( member_nat @ A2 @ ( set_mset_nat @ N2 ) )
        & ( A2 = B ) ) ) ).

% add_mset_remove_trivial_iff
thf(fact_147_add__mset__remove__trivial__iff,axiom,
    ! [N2: multiset_set_a,A2: set_a,B: set_a] :
      ( ( N2
        = ( add_mset_set_a @ A2 @ ( minus_706656509937749387_set_a @ N2 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) )
      = ( ( member_set_a @ A2 @ ( set_mset_set_a @ N2 ) )
        & ( A2 = B ) ) ) ).

% add_mset_remove_trivial_iff
thf(fact_148_add__mset__remove__trivial__iff,axiom,
    ! [N2: multiset_set_nat,A2: set_nat,B: set_nat] :
      ( ( N2
        = ( add_mset_set_nat @ A2 @ ( minus_7237264121398869807et_nat @ N2 @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) ) ) )
      = ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ N2 ) )
        & ( A2 = B ) ) ) ).

% add_mset_remove_trivial_iff
thf(fact_149_remove__1__mset__id__iff__notin,axiom,
    ! [M: multiset_a,A2: a] :
      ( ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
        = M )
      = ( ~ ( member_a @ A2 @ ( set_mset_a @ M ) ) ) ) ).

% remove_1_mset_id_iff_notin
thf(fact_150_remove__1__mset__id__iff__notin,axiom,
    ! [M: multiset_nat,A2: nat] :
      ( ( ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) )
        = M )
      = ( ~ ( member_nat @ A2 @ ( set_mset_nat @ M ) ) ) ) ).

% remove_1_mset_id_iff_notin
thf(fact_151_remove__1__mset__id__iff__notin,axiom,
    ! [M: multiset_set_a,A2: set_a] :
      ( ( ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) )
        = M )
      = ( ~ ( member_set_a @ A2 @ ( set_mset_set_a @ M ) ) ) ) ).

% remove_1_mset_id_iff_notin
thf(fact_152_remove__1__mset__id__iff__notin,axiom,
    ! [M: multiset_set_nat,A2: set_nat] :
      ( ( ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) )
        = M )
      = ( ~ ( member_set_nat @ A2 @ ( set_mset_set_nat @ M ) ) ) ) ).

% remove_1_mset_id_iff_notin
thf(fact_153_id__remove__1__mset__iff__notin,axiom,
    ! [M: multiset_a,A2: a] :
      ( ( M
        = ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
      = ( ~ ( member_a @ A2 @ ( set_mset_a @ M ) ) ) ) ).

% id_remove_1_mset_iff_notin
thf(fact_154_id__remove__1__mset__iff__notin,axiom,
    ! [M: multiset_nat,A2: nat] :
      ( ( M
        = ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
      = ( ~ ( member_nat @ A2 @ ( set_mset_nat @ M ) ) ) ) ).

% id_remove_1_mset_iff_notin
thf(fact_155_id__remove__1__mset__iff__notin,axiom,
    ! [M: multiset_set_a,A2: set_a] :
      ( ( M
        = ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) )
      = ( ~ ( member_set_a @ A2 @ ( set_mset_set_a @ M ) ) ) ) ).

% id_remove_1_mset_iff_notin
thf(fact_156_id__remove__1__mset__iff__notin,axiom,
    ! [M: multiset_set_nat,A2: set_nat] :
      ( ( M
        = ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) )
      = ( ~ ( member_set_nat @ A2 @ ( set_mset_set_nat @ M ) ) ) ) ).

% id_remove_1_mset_iff_notin
thf(fact_157_add__mset__eq__add__mset__ne,axiom,
    ! [A2: a,B: a,A: multiset_a,B3: multiset_a] :
      ( ( A2 != B )
     => ( ( ( add_mset_a @ A2 @ A )
          = ( add_mset_a @ B @ B3 ) )
        = ( ( member_a @ A2 @ ( set_mset_a @ B3 ) )
          & ( member_a @ B @ ( set_mset_a @ A ) )
          & ( A
            = ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ) ) ).

% add_mset_eq_add_mset_ne
thf(fact_158_add__mset__eq__add__mset__ne,axiom,
    ! [A2: nat,B: nat,A: multiset_nat,B3: multiset_nat] :
      ( ( A2 != B )
     => ( ( ( add_mset_nat @ A2 @ A )
          = ( add_mset_nat @ B @ B3 ) )
        = ( ( member_nat @ A2 @ ( set_mset_nat @ B3 ) )
          & ( member_nat @ B @ ( set_mset_nat @ A ) )
          & ( A
            = ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ B3 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ) ) ).

% add_mset_eq_add_mset_ne
thf(fact_159_add__mset__eq__add__mset__ne,axiom,
    ! [A2: set_a,B: set_a,A: multiset_set_a,B3: multiset_set_a] :
      ( ( A2 != B )
     => ( ( ( add_mset_set_a @ A2 @ A )
          = ( add_mset_set_a @ B @ B3 ) )
        = ( ( member_set_a @ A2 @ ( set_mset_set_a @ B3 ) )
          & ( member_set_a @ B @ ( set_mset_set_a @ A ) )
          & ( A
            = ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ B3 @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).

% add_mset_eq_add_mset_ne
thf(fact_160_add__mset__eq__add__mset__ne,axiom,
    ! [A2: set_nat,B: set_nat,A: multiset_set_nat,B3: multiset_set_nat] :
      ( ( A2 != B )
     => ( ( ( add_mset_set_nat @ A2 @ A )
          = ( add_mset_set_nat @ B @ B3 ) )
        = ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ B3 ) )
          & ( member_set_nat @ B @ ( set_mset_set_nat @ A ) )
          & ( A
            = ( add_mset_set_nat @ B @ ( minus_7237264121398869807et_nat @ B3 @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ) ) ) ) ) ).

% add_mset_eq_add_mset_ne
thf(fact_161_more__than__one__mset__mset__diff,axiom,
    ! [A2: a,M: multiset_a] :
      ( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) )
     => ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
        = ( set_mset_a @ M ) ) ) ).

% more_than_one_mset_mset_diff
thf(fact_162_more__than__one__mset__mset__diff,axiom,
    ! [A2: nat,M: multiset_nat] :
      ( ( member_nat @ A2 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) )
     => ( ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
        = ( set_mset_nat @ M ) ) ) ).

% more_than_one_mset_mset_diff
thf(fact_163_more__than__one__mset__mset__diff,axiom,
    ! [A2: set_a,M: multiset_set_a] :
      ( ( member_set_a @ A2 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ) )
     => ( ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) )
        = ( set_mset_set_a @ M ) ) ) ).

% more_than_one_mset_mset_diff
thf(fact_164_more__than__one__mset__mset__diff,axiom,
    ! [A2: set_nat,M: multiset_set_nat] :
      ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ) )
     => ( ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) )
        = ( set_mset_set_nat @ M ) ) ) ).

% more_than_one_mset_mset_diff
thf(fact_165_multiset__add__sub__el__shuffle,axiom,
    ! [C: a,B3: multiset_a,B: a] :
      ( ( member_a @ C @ ( set_mset_a @ B3 ) )
     => ( ( B != C )
       => ( ( add_mset_a @ B @ ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) )
          = ( minus_3765977307040488491iset_a @ ( add_mset_a @ B @ B3 ) @ ( add_mset_a @ C @ zero_zero_multiset_a ) ) ) ) ) ).

% multiset_add_sub_el_shuffle
thf(fact_166_multiset__add__sub__el__shuffle,axiom,
    ! [C: nat,B3: multiset_nat,B: nat] :
      ( ( member_nat @ C @ ( set_mset_nat @ B3 ) )
     => ( ( B != C )
       => ( ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ B3 @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) )
          = ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ B3 ) @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) ) ) ) ) ).

% multiset_add_sub_el_shuffle
thf(fact_167_multiset__add__sub__el__shuffle,axiom,
    ! [C: set_a,B3: multiset_set_a,B: set_a] :
      ( ( member_set_a @ C @ ( set_mset_set_a @ B3 ) )
     => ( ( B != C )
       => ( ( add_mset_set_a @ B @ ( minus_706656509937749387_set_a @ B3 @ ( add_mset_set_a @ C @ zero_z5079479921072680283_set_a ) ) )
          = ( minus_706656509937749387_set_a @ ( add_mset_set_a @ B @ B3 ) @ ( add_mset_set_a @ C @ zero_z5079479921072680283_set_a ) ) ) ) ) ).

% multiset_add_sub_el_shuffle
thf(fact_168_multiset__add__sub__el__shuffle,axiom,
    ! [C: set_nat,B3: multiset_set_nat,B: set_nat] :
      ( ( member_set_nat @ C @ ( set_mset_set_nat @ B3 ) )
     => ( ( B != C )
       => ( ( add_mset_set_nat @ B @ ( minus_7237264121398869807et_nat @ B3 @ ( add_mset_set_nat @ C @ zero_z3157962936165190495et_nat ) ) )
          = ( minus_7237264121398869807et_nat @ ( add_mset_set_nat @ B @ B3 ) @ ( add_mset_set_nat @ C @ zero_z3157962936165190495et_nat ) ) ) ) ) ).

% multiset_add_sub_el_shuffle
thf(fact_169_add__mset__remove__trivial__eq,axiom,
    ! [N2: multiset_a,A2: a] :
      ( ( N2
        = ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) )
      = ( member_a @ A2 @ ( set_mset_a @ N2 ) ) ) ).

% add_mset_remove_trivial_eq
thf(fact_170_add__mset__remove__trivial__eq,axiom,
    ! [N2: multiset_nat,A2: nat] :
      ( ( N2
        = ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N2 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) )
      = ( member_nat @ A2 @ ( set_mset_nat @ N2 ) ) ) ).

% add_mset_remove_trivial_eq
thf(fact_171_add__mset__remove__trivial__eq,axiom,
    ! [N2: multiset_set_a,A2: set_a] :
      ( ( N2
        = ( add_mset_set_a @ A2 @ ( minus_706656509937749387_set_a @ N2 @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ) )
      = ( member_set_a @ A2 @ ( set_mset_set_a @ N2 ) ) ) ).

% add_mset_remove_trivial_eq
thf(fact_172_add__mset__remove__trivial__eq,axiom,
    ! [N2: multiset_set_nat,A2: set_nat] :
      ( ( N2
        = ( add_mset_set_nat @ A2 @ ( minus_7237264121398869807et_nat @ N2 @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ) )
      = ( member_set_nat @ A2 @ ( set_mset_set_nat @ N2 ) ) ) ).

% add_mset_remove_trivial_eq
thf(fact_173_add__mset__remove__trivial__If,axiom,
    ! [A2: a,N2: multiset_a] :
      ( ( ( member_a @ A2 @ ( set_mset_a @ N2 ) )
       => ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
          = N2 ) )
      & ( ~ ( member_a @ A2 @ ( set_mset_a @ N2 ) )
       => ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ N2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
          = ( add_mset_a @ A2 @ N2 ) ) ) ) ).

% add_mset_remove_trivial_If
thf(fact_174_add__mset__remove__trivial__If,axiom,
    ! [A2: nat,N2: multiset_nat] :
      ( ( ( member_nat @ A2 @ ( set_mset_nat @ N2 ) )
       => ( ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N2 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
          = N2 ) )
      & ( ~ ( member_nat @ A2 @ ( set_mset_nat @ N2 ) )
       => ( ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ N2 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
          = ( add_mset_nat @ A2 @ N2 ) ) ) ) ).

% add_mset_remove_trivial_If
thf(fact_175_add__mset__remove__trivial__If,axiom,
    ! [A2: set_a,N2: multiset_set_a] :
      ( ( ( member_set_a @ A2 @ ( set_mset_set_a @ N2 ) )
       => ( ( add_mset_set_a @ A2 @ ( minus_706656509937749387_set_a @ N2 @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) )
          = N2 ) )
      & ( ~ ( member_set_a @ A2 @ ( set_mset_set_a @ N2 ) )
       => ( ( add_mset_set_a @ A2 @ ( minus_706656509937749387_set_a @ N2 @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) )
          = ( add_mset_set_a @ A2 @ N2 ) ) ) ) ).

% add_mset_remove_trivial_If
thf(fact_176_add__mset__remove__trivial__If,axiom,
    ! [A2: set_nat,N2: multiset_set_nat] :
      ( ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ N2 ) )
       => ( ( add_mset_set_nat @ A2 @ ( minus_7237264121398869807et_nat @ N2 @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) )
          = N2 ) )
      & ( ~ ( member_set_nat @ A2 @ ( set_mset_set_nat @ N2 ) )
       => ( ( add_mset_set_nat @ A2 @ ( minus_7237264121398869807et_nat @ N2 @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) )
          = ( add_mset_set_nat @ A2 @ N2 ) ) ) ) ).

% add_mset_remove_trivial_If
thf(fact_177_add__mset__eq__add__mset,axiom,
    ! [A2: a,M: multiset_a,B: a,M4: multiset_a] :
      ( ( ( add_mset_a @ A2 @ M )
        = ( add_mset_a @ B @ M4 ) )
      = ( ( ( A2 = B )
          & ( M = M4 ) )
        | ( ( A2 != B )
          & ( member_a @ B @ ( set_mset_a @ M ) )
          & ( ( add_mset_a @ A2 @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) )
            = M4 ) ) ) ) ).

% add_mset_eq_add_mset
thf(fact_178_add__mset__eq__add__mset,axiom,
    ! [A2: nat,M: multiset_nat,B: nat,M4: multiset_nat] :
      ( ( ( add_mset_nat @ A2 @ M )
        = ( add_mset_nat @ B @ M4 ) )
      = ( ( ( A2 = B )
          & ( M = M4 ) )
        | ( ( A2 != B )
          & ( member_nat @ B @ ( set_mset_nat @ M ) )
          & ( ( add_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) )
            = M4 ) ) ) ) ).

% add_mset_eq_add_mset
thf(fact_179_add__mset__eq__add__mset,axiom,
    ! [A2: set_a,M: multiset_set_a,B: set_a,M4: multiset_set_a] :
      ( ( ( add_mset_set_a @ A2 @ M )
        = ( add_mset_set_a @ B @ M4 ) )
      = ( ( ( A2 = B )
          & ( M = M4 ) )
        | ( ( A2 != B )
          & ( member_set_a @ B @ ( set_mset_set_a @ M ) )
          & ( ( add_mset_set_a @ A2 @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) )
            = M4 ) ) ) ) ).

% add_mset_eq_add_mset
thf(fact_180_add__mset__eq__add__mset,axiom,
    ! [A2: set_nat,M: multiset_set_nat,B: set_nat,M4: multiset_set_nat] :
      ( ( ( add_mset_set_nat @ A2 @ M )
        = ( add_mset_set_nat @ B @ M4 ) )
      = ( ( ( A2 = B )
          & ( M = M4 ) )
        | ( ( A2 != B )
          & ( member_set_nat @ B @ ( set_mset_set_nat @ M ) )
          & ( ( add_mset_set_nat @ A2 @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) ) )
            = M4 ) ) ) ) ).

% add_mset_eq_add_mset
thf(fact_181_in__remove1__mset__neq,axiom,
    ! [A2: a,B: a,C2: multiset_a] :
      ( ( A2 != B )
     => ( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ C2 @ ( add_mset_a @ B @ zero_zero_multiset_a ) ) ) )
        = ( member_a @ A2 @ ( set_mset_a @ C2 ) ) ) ) ).

% in_remove1_mset_neq
thf(fact_182_in__remove1__mset__neq,axiom,
    ! [A2: nat,B: nat,C2: multiset_nat] :
      ( ( A2 != B )
     => ( ( member_nat @ A2 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ C2 @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) ) ) )
        = ( member_nat @ A2 @ ( set_mset_nat @ C2 ) ) ) ) ).

% in_remove1_mset_neq
thf(fact_183_in__remove1__mset__neq,axiom,
    ! [A2: set_a,B: set_a,C2: multiset_set_a] :
      ( ( A2 != B )
     => ( ( member_set_a @ A2 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ C2 @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) )
        = ( member_set_a @ A2 @ ( set_mset_set_a @ C2 ) ) ) ) ).

% in_remove1_mset_neq
thf(fact_184_in__remove1__mset__neq,axiom,
    ! [A2: set_nat,B: set_nat,C2: multiset_set_nat] :
      ( ( A2 != B )
     => ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ C2 @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) ) ) )
        = ( member_set_nat @ A2 @ ( set_mset_set_nat @ C2 ) ) ) ) ).

% in_remove1_mset_neq
thf(fact_185_multi__drop__mem__not__eq,axiom,
    ! [C: a,B3: multiset_a] :
      ( ( member_a @ C @ ( set_mset_a @ B3 ) )
     => ( ( minus_3765977307040488491iset_a @ B3 @ ( add_mset_a @ C @ zero_zero_multiset_a ) )
       != B3 ) ) ).

% multi_drop_mem_not_eq
thf(fact_186_multi__drop__mem__not__eq,axiom,
    ! [C: nat,B3: multiset_nat] :
      ( ( member_nat @ C @ ( set_mset_nat @ B3 ) )
     => ( ( minus_8522176038001411705et_nat @ B3 @ ( add_mset_nat @ C @ zero_z7348594199698428585et_nat ) )
       != B3 ) ) ).

% multi_drop_mem_not_eq
thf(fact_187_multi__drop__mem__not__eq,axiom,
    ! [C: set_a,B3: multiset_set_a] :
      ( ( member_set_a @ C @ ( set_mset_set_a @ B3 ) )
     => ( ( minus_706656509937749387_set_a @ B3 @ ( add_mset_set_a @ C @ zero_z5079479921072680283_set_a ) )
       != B3 ) ) ).

% multi_drop_mem_not_eq
thf(fact_188_multi__drop__mem__not__eq,axiom,
    ! [C: set_nat,B3: multiset_set_nat] :
      ( ( member_set_nat @ C @ ( set_mset_set_nat @ B3 ) )
     => ( ( minus_7237264121398869807et_nat @ B3 @ ( add_mset_set_nat @ C @ zero_z3157962936165190495et_nat ) )
       != B3 ) ) ).

% multi_drop_mem_not_eq
thf(fact_189_diff__single__eq__union,axiom,
    ! [X: a,M: multiset_a,N2: multiset_a] :
      ( ( member_a @ X @ ( set_mset_a @ M ) )
     => ( ( ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
          = N2 )
        = ( M
          = ( add_mset_a @ X @ N2 ) ) ) ) ).

% diff_single_eq_union
thf(fact_190_diff__single__eq__union,axiom,
    ! [X: nat,M: multiset_nat,N2: multiset_nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ M ) )
     => ( ( ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
          = N2 )
        = ( M
          = ( add_mset_nat @ X @ N2 ) ) ) ) ).

% diff_single_eq_union
thf(fact_191_diff__single__eq__union,axiom,
    ! [X: set_a,M: multiset_set_a,N2: multiset_set_a] :
      ( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
     => ( ( ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
          = N2 )
        = ( M
          = ( add_mset_set_a @ X @ N2 ) ) ) ) ).

% diff_single_eq_union
thf(fact_192_diff__single__eq__union,axiom,
    ! [X: set_nat,M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( member_set_nat @ X @ ( set_mset_set_nat @ M ) )
     => ( ( ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) )
          = N2 )
        = ( M
          = ( add_mset_set_nat @ X @ N2 ) ) ) ) ).

% diff_single_eq_union
thf(fact_193_dual__sys_Odelete__point__p__not__in__bl__blocks,axiom,
    ! [P: nat] :
      ( ! [Bl: set_nat] :
          ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
         => ~ ( member_nat @ P @ Bl ) )
     => ( ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P )
        = ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.delete_point_p_not_in_bl_blocks
thf(fact_194_del__invalid__point__blocks,axiom,
    ! [P: a] :
      ( ~ ( member_a @ P @ ( set_a2 @ v_s ) )
     => ( ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P )
        = ( mset_set_a @ b_s ) ) ) ).

% del_invalid_point_blocks
thf(fact_195_dual__sys_Odelete__point__strong__block__in,axiom,
    ! [P: nat,Bl2: set_nat] :
      ( ~ ( member_nat @ P @ Bl2 )
     => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
       => ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) ) ) ) ).

% dual_sys.delete_point_strong_block_in
thf(fact_196_dual__sys_Odelete__point__strong__block__in__iff,axiom,
    ! [Bl2: set_nat,P: nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) )
        = ( ~ ( member_nat @ P @ Bl2 ) ) ) ) ).

% dual_sys.delete_point_strong_block_in_iff
thf(fact_197_dual__sys_Odelete__point__strong__block__in__orig,axiom,
    ! [Bl2: set_nat,P: nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) )
     => ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).

% dual_sys.delete_point_strong_block_in_orig
thf(fact_198_dual__sys_Odelete__point__strong__block__not__in,axiom,
    ! [P: nat,Bl2: set_nat] :
      ( ( member_nat @ P @ Bl2 )
     => ~ ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) ) ) ).

% dual_sys.delete_point_strong_block_not_in
thf(fact_199_add__delete__point__inv,axiom,
    ! [P: a] :
      ( ~ ( member_a @ P @ ( set_a2 @ v_s ) )
     => ( ( design108908007054065099oint_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P ) @ P )
        = ( set_a2 @ v_s ) ) ) ).

% add_delete_point_inv
thf(fact_200_const__intersect__design_Oconst__intersect,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a,M5: nat,B1: set_a,B2: set_a] :
      ( ( design9190424834980853558sign_a @ Point_set @ Block_collection @ M5 )
     => ( ( member_set_a @ B1 @ ( set_mset_set_a @ Block_collection ) )
       => ( ( member_set_a @ B2 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ Block_collection @ ( add_mset_set_a @ B1 @ zero_z5079479921072680283_set_a ) ) ) )
         => ( ( design7842873109100088828mber_a @ B1 @ B2 )
            = M5 ) ) ) ) ).

% const_intersect_design.const_intersect
thf(fact_201_const__intersect__design_Oconst__intersect,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat,M5: nat,B1: set_nat,B2: set_nat] :
      ( ( design137120128173859224gn_nat @ Point_set @ Block_collection @ M5 )
     => ( ( member_set_nat @ B1 @ ( set_mset_set_nat @ Block_collection ) )
       => ( ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ Block_collection @ ( add_mset_set_nat @ B1 @ zero_z3157962936165190495et_nat ) ) ) )
         => ( ( design7485525362727208274er_nat @ B1 @ B2 )
            = M5 ) ) ) ) ).

% const_intersect_design.const_intersect
thf(fact_202_delete__invalid__pt__strong__eq,axiom,
    ! [P: a] :
      ( ~ ( member_a @ P @ ( set_a2 @ v_s ) )
     => ( ( mset_set_a @ b_s )
        = ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) ) ).

% delete_invalid_pt_strong_eq
thf(fact_203_del__block__def,axiom,
    ! [B: set_a] :
      ( ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B )
      = ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) ).

% del_block_def
thf(fact_204_ordered__simple__design__def,axiom,
    ( incide5137607047756421874_set_a
    = ( ^ [V_s: list_set_a,B_s: list_set_set_a] :
          ( ( incide7014649564523408292_set_a @ V_s @ B_s )
          & ( design1835266114905787166_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) ) ) ) ) ).

% ordered_simple_design_def
thf(fact_205_ordered__simple__design__def,axiom,
    ( incide8427317466731264060gn_nat
    = ( ^ [V_s: list_nat,B_s: list_set_nat] :
          ( ( incide8999572217031194378gn_nat @ V_s @ B_s )
          & ( design7861764274488435984gn_nat @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) ) ) ) ) ).

% ordered_simple_design_def
thf(fact_206_ordered__simple__design__def,axiom,
    ( incide371748008924627346sign_a
    = ( ^ [V_s: list_a,B_s: list_set_a] :
          ( ( incide2848671379600480836sign_a @ V_s @ B_s )
          & ( design3982635895484621246sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) ) ) ) ) ).

% ordered_simple_design_def
thf(fact_207_delete__invalid__block__eq,axiom,
    ! [B: set_a] :
      ( ~ ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B )
        = ( mset_set_a @ b_s ) ) ) ).

% delete_invalid_block_eq
thf(fact_208_delete__point__strong__block__not__in,axiom,
    ! [P: a,Bl2: set_a] :
      ( ( member_a @ P @ Bl2 )
     => ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) ) ) ).

% delete_point_strong_block_not_in
thf(fact_209_delete__point__strong__block__in__orig,axiom,
    ! [Bl2: set_a,P: a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) )
     => ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).

% delete_point_strong_block_in_orig
thf(fact_210_delete__point__strong__block__in__iff,axiom,
    ! [Bl2: set_a,P: a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) )
        = ( ~ ( member_a @ P @ Bl2 ) ) ) ) ).

% delete_point_strong_block_in_iff
thf(fact_211_delete__point__strong__block__in,axiom,
    ! [P: a,Bl2: set_a] :
      ( ~ ( member_a @ P @ Bl2 )
     => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) ) ) ) ).

% delete_point_strong_block_in
thf(fact_212_delete__point__p__not__in__bl__blocks,axiom,
    ! [P: a] :
      ( ! [Bl: set_a] :
          ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
         => ~ ( member_a @ P @ Bl ) )
     => ( ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P )
        = ( mset_set_a @ b_s ) ) ) ).

% delete_point_p_not_in_bl_blocks
thf(fact_213_add__existing__point,axiom,
    ! [P: a] :
      ( ( member_a @ P @ ( set_a2 @ v_s ) )
     => ( ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P )
        = ( set_a2 @ v_s ) ) ) ).

% add_existing_point
thf(fact_214_ordered__proper__design_Oaxioms_I1_J,axiom,
    ! [V_s2: list_a,B_s2: list_set_a] :
      ( ( incide3676903341588786676sign_a @ V_s2 @ B_s2 )
     => ( incide2848671379600480836sign_a @ V_s2 @ B_s2 ) ) ).

% ordered_proper_design.axioms(1)
thf(fact_215_ordered__simple__design_Oaxioms_I1_J,axiom,
    ! [V_s2: list_a,B_s2: list_set_a] :
      ( ( incide371748008924627346sign_a @ V_s2 @ B_s2 )
     => ( incide2848671379600480836sign_a @ V_s2 @ B_s2 ) ) ).

% ordered_simple_design.axioms(1)
thf(fact_216_ordered__pairwise__balance_Oaxioms_I2_J,axiom,
    ! [V_s2: list_set_a,B_s2: list_set_set_a,Lambda: nat] :
      ( ( incide4449361439798955450_set_a @ V_s2 @ B_s2 @ Lambda )
     => ( block_6207159848980890963_set_a @ ( set_set_a2 @ V_s2 ) @ ( mset_set_set_a @ B_s2 ) @ Lambda ) ) ).

% ordered_pairwise_balance.axioms(2)
thf(fact_217_ordered__pairwise__balance_Oaxioms_I2_J,axiom,
    ! [V_s2: list_nat,B_s2: list_set_nat,Lambda: nat] :
      ( ( incide3388802471754236788ce_nat @ V_s2 @ B_s2 @ Lambda )
     => ( block_1456364645985477531ce_nat @ ( set_nat2 @ V_s2 ) @ ( mset_set_nat @ B_s2 ) @ Lambda ) ) ).

% ordered_pairwise_balance.axioms(2)
thf(fact_218_ordered__pairwise__balance_Oaxioms_I2_J,axiom,
    ! [V_s2: list_a,B_s2: list_set_a,Lambda: nat] :
      ( ( incide6880889959311561818ance_a @ V_s2 @ B_s2 @ Lambda )
     => ( block_5355636846524985331ance_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) @ Lambda ) ) ).

% ordered_pairwise_balance.axioms(2)
thf(fact_219_ordered__simple__design_Oaxioms_I2_J,axiom,
    ! [V_s2: list_set_a,B_s2: list_set_set_a] :
      ( ( incide5137607047756421874_set_a @ V_s2 @ B_s2 )
     => ( design1835266114905787166_set_a @ ( set_set_a2 @ V_s2 ) @ ( mset_set_set_a @ B_s2 ) ) ) ).

% ordered_simple_design.axioms(2)
thf(fact_220_ordered__simple__design_Oaxioms_I2_J,axiom,
    ! [V_s2: list_nat,B_s2: list_set_nat] :
      ( ( incide8427317466731264060gn_nat @ V_s2 @ B_s2 )
     => ( design7861764274488435984gn_nat @ ( set_nat2 @ V_s2 ) @ ( mset_set_nat @ B_s2 ) ) ) ).

% ordered_simple_design.axioms(2)
thf(fact_221_ordered__simple__design_Oaxioms_I2_J,axiom,
    ! [V_s2: list_a,B_s2: list_set_a] :
      ( ( incide371748008924627346sign_a @ V_s2 @ B_s2 )
     => ( design3982635895484621246sign_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) ) ) ).

% ordered_simple_design.axioms(2)
thf(fact_222_ordered__simple__design_Ointro,axiom,
    ! [V_s2: list_set_a,B_s2: list_set_set_a] :
      ( ( incide7014649564523408292_set_a @ V_s2 @ B_s2 )
     => ( ( design1835266114905787166_set_a @ ( set_set_a2 @ V_s2 ) @ ( mset_set_set_a @ B_s2 ) )
       => ( incide5137607047756421874_set_a @ V_s2 @ B_s2 ) ) ) ).

% ordered_simple_design.intro
thf(fact_223_ordered__simple__design_Ointro,axiom,
    ! [V_s2: list_nat,B_s2: list_set_nat] :
      ( ( incide8999572217031194378gn_nat @ V_s2 @ B_s2 )
     => ( ( design7861764274488435984gn_nat @ ( set_nat2 @ V_s2 ) @ ( mset_set_nat @ B_s2 ) )
       => ( incide8427317466731264060gn_nat @ V_s2 @ B_s2 ) ) ) ).

% ordered_simple_design.intro
thf(fact_224_ordered__simple__design_Ointro,axiom,
    ! [V_s2: list_a,B_s2: list_set_a] :
      ( ( incide2848671379600480836sign_a @ V_s2 @ B_s2 )
     => ( ( design3982635895484621246sign_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) )
       => ( incide371748008924627346sign_a @ V_s2 @ B_s2 ) ) ) ).

% ordered_simple_design.intro
thf(fact_225_dual__sys_Oobtain__blocks__intersect__num,axiom,
    ! [N: nat] :
      ( ( member_nat @ N @ ( design9164904592607734462rs_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ? [B12: set_nat,B22: set_nat] :
          ( ( member_set_nat @ B12 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
          & ( member_set_nat @ B22 @ ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( add_mset_set_nat @ B12 @ zero_z3157962936165190495et_nat ) ) ) )
          & ( ( design7485525362727208274er_nat @ B12 @ B22 )
            = N ) ) ) ).

% dual_sys.obtain_blocks_intersect_num
thf(fact_226_dual__sys_Ointersect__num__in__set,axiom,
    ! [B1: set_nat,B2: set_nat] :
      ( ( member_set_nat @ B1 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( member_set_nat @ B2 @ ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( add_mset_set_nat @ B1 @ zero_z3157962936165190495et_nat ) ) ) )
       => ( member_nat @ ( design7485525362727208274er_nat @ B1 @ B2 ) @ ( design9164904592607734462rs_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ).

% dual_sys.intersect_num_in_set
thf(fact_227_strong__del__point__sub__des,axiom,
    ! [P: a] : ( sub_sub_design_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).

% strong_del_point_sub_des
thf(fact_228_add__point__sub__des,axiom,
    ! [P: a] : ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P ) @ ( mset_set_a @ b_s ) ) ).

% add_point_sub_des
thf(fact_229_delete__block__sub__des,axiom,
    ! [B: set_a] : ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).

% delete_block_sub_des
thf(fact_230_strong__del__point__sub__sys,axiom,
    ! [P: a] : ( sub_su7923802003039619913stem_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).

% strong_del_point_sub_sys
thf(fact_231_add__point__sub__sys,axiom,
    ! [P: a] : ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P ) @ ( mset_set_a @ b_s ) ) ).

% add_point_sub_sys
thf(fact_232_delete__block__sub__sys,axiom,
    ! [B: set_a] : ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B ) @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).

% delete_block_sub_sys
thf(fact_233_dual__sys_Odel__block__def,axiom,
    ! [B: set_nat] :
      ( ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B )
      = ( minus_7237264121398869807et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) ) ) ).

% dual_sys.del_block_def
thf(fact_234_dual__sys_Odelete__invalid__block__eq,axiom,
    ! [B: set_nat] :
      ( ~ ( member_set_nat @ B @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B )
        = ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.delete_invalid_block_eq
thf(fact_235_dual__sys_Odel__add__block__inv,axiom,
    ! [Bl2: set_nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( design4725324266511619850ck_nat @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) @ Bl2 )
        = ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.del_add_block_inv
thf(fact_236_dual__sys_Odel__invalid__add__block__eq,axiom,
    ! [Bl2: set_nat] :
      ( ~ ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( design4725324266511619850ck_nat @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) @ Bl2 )
        = ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) ) ) ).

% dual_sys.del_invalid_add_block_eq
thf(fact_237_dual__sys_Oadd__block__alt,axiom,
    ! [B: set_nat] :
      ( ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B )
      = ( add_mset_set_nat @ B @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.add_block_alt
thf(fact_238_dual__sys_Odelete__point__blocks__sub,axiom,
    ! [B: set_nat,P: nat] :
      ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) )
     => ~ ! [Bl: set_nat] :
            ~ ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
              & ( ord_less_eq_set_nat @ B @ Bl ) ) ) ).

% dual_sys.delete_point_blocks_sub
thf(fact_239_ordered__pbdI,axiom,
    ! [B_s2: list_set_a,V_s2: list_a] :
      ( ( ( mset_set_a @ b_s )
        = ( mset_set_a @ B_s2 ) )
     => ( ( ( set_a2 @ v_s )
          = ( set_a2 @ V_s2 ) )
       => ( ( distinct_a @ V_s2 )
         => ( incide6880889959311561818ance_a @ V_s2 @ B_s2 @ lambda ) ) ) ) ).

% ordered_pbdI
thf(fact_240_dual__sys_Ofinite__design__support,axiom,
    finite1152437895449049373et_nat @ ( design4862117536649126062rt_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.finite_design_support
thf(fact_241_block__set__nempty__imp__points,axiom,
    ( ( ( mset_set_a @ b_s )
     != zero_z5079479921072680283_set_a )
   => ( ( set_a2 @ v_s )
     != bot_bot_set_a ) ) ).

% block_set_nempty_imp_points
thf(fact_242_add__point__to__blocks__finite,axiom,
    ! [P: a,Bs: set_set_a] : ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P ) @ ( design2935547469388721088ocks_a @ ( mset_set_a @ b_s ) @ P @ Bs ) ) ).

% add_point_to_blocks_finite
thf(fact_243_delete__point__finite,axiom,
    ! [P: a] : ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P ) @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P ) ) ).

% delete_point_finite
thf(fact_244_distinct,axiom,
    distinct_a @ v_s ).

% distinct
thf(fact_245_design__points__nempty,axiom,
    ( ( set_a2 @ v_s )
   != bot_bot_set_a ) ).

% design_points_nempty
thf(fact_246_blocks__nempty__alt,axiom,
    ! [X5: set_a] :
      ( ( member_set_a @ X5 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( X5 != bot_bot_set_a ) ) ).

% blocks_nempty_alt
thf(fact_247_blocks__nempty,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( Bl2 != bot_bot_set_a ) ) ).

% blocks_nempty
thf(fact_248_finite__incidence__system__axioms,axiom,
    design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).

% finite_incidence_system_axioms
thf(fact_249_delete__block__fin__incidence__sys,axiom,
    ! [B: set_a] : ( design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B ) ) ).

% delete_block_fin_incidence_sys
thf(fact_250_add__point__finite,axiom,
    ! [P: a] : ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P ) @ ( mset_set_a @ b_s ) ) ).

% add_point_finite
thf(fact_251_finite__set__mset,axiom,
    ! [M: multiset_a] : ( finite_finite_a @ ( set_mset_a @ M ) ) ).

% finite_set_mset
thf(fact_252_finite__set__mset,axiom,
    ! [M: multiset_nat] : ( finite_finite_nat @ ( set_mset_nat @ M ) ) ).

% finite_set_mset
thf(fact_253_finite__set__mset,axiom,
    ! [M: multiset_set_a] : ( finite_finite_set_a @ ( set_mset_set_a @ M ) ) ).

% finite_set_mset
thf(fact_254_finite__set__mset,axiom,
    ! [M: multiset_set_nat] : ( finite1152437895449049373et_nat @ ( set_mset_set_nat @ M ) ) ).

% finite_set_mset
thf(fact_255_dual__sys_Oadd__block__index__not__in,axiom,
    ! [Ps: set_nat,B: set_nat] :
      ( ~ ( ord_less_eq_set_nat @ Ps @ B )
     => ( ( design6574611146354332593ex_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) @ Ps )
        = ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps ) ) ) ).

% dual_sys.add_block_index_not_in
thf(fact_256_strong__del__point__finite,axiom,
    ! [P: a] : ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) ).

% strong_del_point_finite
thf(fact_257_set__mset__empty,axiom,
    ( ( set_mset_set_a @ zero_z5079479921072680283_set_a )
    = bot_bot_set_set_a ) ).

% set_mset_empty
thf(fact_258_set__mset__empty,axiom,
    ( ( set_mset_set_nat @ zero_z3157962936165190495et_nat )
    = bot_bot_set_set_nat ) ).

% set_mset_empty
thf(fact_259_set__mset__empty,axiom,
    ( ( set_mset_a @ zero_zero_multiset_a )
    = bot_bot_set_a ) ).

% set_mset_empty
thf(fact_260_set__mset__empty,axiom,
    ( ( set_mset_nat @ zero_z7348594199698428585et_nat )
    = bot_bot_set_nat ) ).

% set_mset_empty
thf(fact_261_set__mset__eq__empty__iff,axiom,
    ! [M: multiset_set_a] :
      ( ( ( set_mset_set_a @ M )
        = bot_bot_set_set_a )
      = ( M = zero_z5079479921072680283_set_a ) ) ).

% set_mset_eq_empty_iff
thf(fact_262_set__mset__eq__empty__iff,axiom,
    ! [M: multiset_set_nat] :
      ( ( ( set_mset_set_nat @ M )
        = bot_bot_set_set_nat )
      = ( M = zero_z3157962936165190495et_nat ) ) ).

% set_mset_eq_empty_iff
thf(fact_263_set__mset__eq__empty__iff,axiom,
    ! [M: multiset_a] :
      ( ( ( set_mset_a @ M )
        = bot_bot_set_a )
      = ( M = zero_zero_multiset_a ) ) ).

% set_mset_eq_empty_iff
thf(fact_264_set__mset__eq__empty__iff,axiom,
    ! [M: multiset_nat] :
      ( ( ( set_mset_nat @ M )
        = bot_bot_set_nat )
      = ( M = zero_z7348594199698428585et_nat ) ) ).

% set_mset_eq_empty_iff
thf(fact_265_mset__eq__imp__distinct__iff,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( mset_a @ Xs )
        = ( mset_a @ Ys ) )
     => ( ( distinct_a @ Xs )
        = ( distinct_a @ Ys ) ) ) ).

% mset_eq_imp_distinct_iff
thf(fact_266_mset__eq__imp__distinct__iff,axiom,
    ! [Xs: list_set_a,Ys: list_set_a] :
      ( ( ( mset_set_a @ Xs )
        = ( mset_set_a @ Ys ) )
     => ( ( distinct_set_a @ Xs )
        = ( distinct_set_a @ Ys ) ) ) ).

% mset_eq_imp_distinct_iff
thf(fact_267_mset__eq__imp__distinct__iff,axiom,
    ! [Xs: list_set_nat,Ys: list_set_nat] :
      ( ( ( mset_set_nat @ Xs )
        = ( mset_set_nat @ Ys ) )
     => ( ( distinct_set_nat @ Xs )
        = ( distinct_set_nat @ Ys ) ) ) ).

% mset_eq_imp_distinct_iff
thf(fact_268_same__mset__distinct__iff,axiom,
    ! [M: list_a,M4: list_a] :
      ( ( ( mset_a @ M )
        = ( mset_a @ M4 ) )
     => ( ( distinct_a @ M )
        = ( distinct_a @ M4 ) ) ) ).

% same_mset_distinct_iff
thf(fact_269_same__mset__distinct__iff,axiom,
    ! [M: list_set_a,M4: list_set_a] :
      ( ( ( mset_set_a @ M )
        = ( mset_set_a @ M4 ) )
     => ( ( distinct_set_a @ M )
        = ( distinct_set_a @ M4 ) ) ) ).

% same_mset_distinct_iff
thf(fact_270_same__mset__distinct__iff,axiom,
    ! [M: list_set_nat,M4: list_set_nat] :
      ( ( ( mset_set_nat @ M )
        = ( mset_set_nat @ M4 ) )
     => ( ( distinct_set_nat @ M )
        = ( distinct_set_nat @ M4 ) ) ) ).

% same_mset_distinct_iff
thf(fact_271_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( distinct_a @ X )
     => ( ( distinct_a @ Y )
       => ( ( ( set_a2 @ X )
            = ( set_a2 @ Y ) )
          = ( ( mset_a @ X )
            = ( mset_a @ Y ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_272_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_set_a,Y: list_set_a] :
      ( ( distinct_set_a @ X )
     => ( ( distinct_set_a @ Y )
       => ( ( ( set_set_a2 @ X )
            = ( set_set_a2 @ Y ) )
          = ( ( mset_set_a @ X )
            = ( mset_set_a @ Y ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_273_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_set_nat,Y: list_set_nat] :
      ( ( distinct_set_nat @ X )
     => ( ( distinct_set_nat @ Y )
       => ( ( ( set_set_nat2 @ X )
            = ( set_set_nat2 @ Y ) )
          = ( ( mset_set_nat @ X )
            = ( mset_set_nat @ Y ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_274_multiset__induct__max,axiom,
    ! [P2: multiset_nat > $o,M: multiset_nat] :
      ( ( P2 @ zero_z7348594199698428585et_nat )
     => ( ! [X4: nat,M3: multiset_nat] :
            ( ( P2 @ M3 )
           => ( ! [Xa: nat] :
                  ( ( member_nat @ Xa @ ( set_mset_nat @ M3 ) )
                 => ( ord_less_eq_nat @ Xa @ X4 ) )
             => ( P2 @ ( add_mset_nat @ X4 @ M3 ) ) ) )
       => ( P2 @ M ) ) ) ).

% multiset_induct_max
thf(fact_275_multiset__induct__min,axiom,
    ! [P2: multiset_nat > $o,M: multiset_nat] :
      ( ( P2 @ zero_z7348594199698428585et_nat )
     => ( ! [X4: nat,M3: multiset_nat] :
            ( ( P2 @ M3 )
           => ( ! [Xa: nat] :
                  ( ( member_nat @ Xa @ ( set_mset_nat @ M3 ) )
                 => ( ord_less_eq_nat @ X4 @ Xa ) )
             => ( P2 @ ( add_mset_nat @ X4 @ M3 ) ) ) )
       => ( P2 @ M ) ) ) ).

% multiset_induct_min
thf(fact_276_pairwise__balance_Oordered__pbdI,axiom,
    ! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Index: nat,B_s2: list_set_set_a,V_s2: list_set_a] :
      ( ( block_6207159848980890963_set_a @ Point_set @ Block_collection @ Index )
     => ( ( Block_collection
          = ( mset_set_set_a @ B_s2 ) )
       => ( ( Point_set
            = ( set_set_a2 @ V_s2 ) )
         => ( ( distinct_set_a @ V_s2 )
           => ( incide4449361439798955450_set_a @ V_s2 @ B_s2 @ Index ) ) ) ) ) ).

% pairwise_balance.ordered_pbdI
thf(fact_277_pairwise__balance_Oordered__pbdI,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat,Index: nat,B_s2: list_set_nat,V_s2: list_nat] :
      ( ( block_1456364645985477531ce_nat @ Point_set @ Block_collection @ Index )
     => ( ( Block_collection
          = ( mset_set_nat @ B_s2 ) )
       => ( ( Point_set
            = ( set_nat2 @ V_s2 ) )
         => ( ( distinct_nat @ V_s2 )
           => ( incide3388802471754236788ce_nat @ V_s2 @ B_s2 @ Index ) ) ) ) ) ).

% pairwise_balance.ordered_pbdI
thf(fact_278_pairwise__balance_Oordered__pbdI,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a,Index: nat,B_s2: list_set_a,V_s2: list_a] :
      ( ( block_5355636846524985331ance_a @ Point_set @ Block_collection @ Index )
     => ( ( Block_collection
          = ( mset_set_a @ B_s2 ) )
       => ( ( Point_set
            = ( set_a2 @ V_s2 ) )
         => ( ( distinct_a @ V_s2 )
           => ( incide6880889959311561818ance_a @ V_s2 @ B_s2 @ Index ) ) ) ) ) ).

% pairwise_balance.ordered_pbdI
thf(fact_279_mset__le__single__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_le4034546139768944438et_nat @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) @ ( add_mset_set_nat @ Y @ zero_z3157962936165190495et_nat ) )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% mset_le_single_iff
thf(fact_280_mset__le__single__iff,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_le7905258569527593284_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) )
      = ( ord_less_eq_set_a @ X @ Y ) ) ).

% mset_le_single_iff
thf(fact_281_mset__le__single__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_le6602235886369790592et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% mset_le_single_iff
thf(fact_282_dual__sys_Oadd__block__def,axiom,
    ! [B: set_nat] :
      ( ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B )
      = ( plus_p8712254050562127327et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) ) ) ).

% dual_sys.add_block_def
thf(fact_283_complement__finite,axiom,
    design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).

% complement_finite
thf(fact_284_del__invalid__add__block__eq,axiom,
    ! [Bl2: set_a] :
      ( ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ Bl2 )
        = ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ) ).

% del_invalid_add_block_eq
thf(fact_285_del__add__block__inv,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ Bl2 )
        = ( mset_set_a @ b_s ) ) ) ).

% del_add_block_inv
thf(fact_286_dual__sys_Odel__block__b_I2_J,axiom,
    ! [Bl2: set_nat] :
      ( ~ ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( size_s7462436076474991978et_nat @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) )
        = ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).

% dual_sys.del_block_b(2)
thf(fact_287_wf__list,axiom,
    ! [B: set_a] :
      ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ord_less_eq_set_a @ B @ ( set_a2 @ v_s ) ) ) ).

% wf_list
thf(fact_288_block__complement__inv,axiom,
    ! [Bl2: set_a,Bl22: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 )
          = Bl22 )
       => ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl22 )
          = Bl2 ) ) ) ).

% block_complement_inv
thf(fact_289_block__list__distinct,axiom,
    distinct_set_a @ b_s ).

% block_list_distinct
thf(fact_290_finite__sets,axiom,
    finite_finite_a @ ( set_a2 @ v_s ) ).

% finite_sets
thf(fact_291_finite__blocks,axiom,
    ! [B: set_a] :
      ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( finite_finite_a @ B ) ) ).

% finite_blocks
thf(fact_292_block__complement__def,axiom,
    ! [B: set_a] :
      ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ B )
      = ( minus_minus_set_a @ ( set_a2 @ v_s ) @ B ) ) ).

% block_complement_def
thf(fact_293_finite__design__support,axiom,
    finite_finite_set_a @ ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) ) ).

% finite_design_support
thf(fact_294_block__comp__elem__alt__left,axiom,
    ! [X: a,Bl2: set_a,Ps: set_a] :
      ( ( member_a @ X @ Bl2 )
     => ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) )
       => ~ ( member_a @ X @ Ps ) ) ) ).

% block_comp_elem_alt_left
thf(fact_295_block__comp__elem__alt__right,axiom,
    ! [Ps: set_a,Bl2: set_a] :
      ( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ Ps )
           => ~ ( member_a @ X4 @ Bl2 ) )
       => ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) ) ) ) ).

% block_comp_elem_alt_right
thf(fact_296_block__complement__elem__iff,axiom,
    ! [Ps: set_a,Bl2: set_a] :
      ( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
     => ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) )
        = ( ! [X3: a] :
              ( ( member_a @ X3 @ Ps )
             => ~ ( member_a @ X3 @ Bl2 ) ) ) ) ) ).

% block_complement_elem_iff
thf(fact_297_block__complement__subset__points,axiom,
    ! [Ps: set_a,Bl2: set_a] :
      ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) )
     => ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) ) ) ).

% block_complement_subset_points
thf(fact_298_add__block__alt,axiom,
    ! [B: set_a] :
      ( ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B )
      = ( add_mset_set_a @ B @ ( mset_set_a @ b_s ) ) ) ).

% add_block_alt
thf(fact_299_delete__point__blocks__sub,axiom,
    ! [B: set_a,P: a] :
      ( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P ) ) )
     => ~ ! [Bl: set_a] :
            ~ ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
              & ( ord_less_eq_set_a @ B @ Bl ) ) ) ).

% delete_point_blocks_sub
thf(fact_300_dual__sys_Ofinite__blocks,axiom,
    ! [B: set_nat] :
      ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( finite_finite_nat @ B ) ) ).

% dual_sys.finite_blocks
thf(fact_301_dual__sys__b__non__zero,axiom,
    ( ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
   != zero_zero_nat ) ).

% dual_sys_b_non_zero
thf(fact_302_dual__sys_Odel__point__block__count,axiom,
    ! [P: nat] :
      ( ( size_s7462436076474991978et_nat @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) )
      = ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.del_point_block_count
thf(fact_303_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_304_add__0,axiom,
    ! [A2: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A2 )
      = A2 ) ).

% add_0
thf(fact_305_add__0,axiom,
    ! [A2: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_306_add__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_307_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_308_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_309_add__cancel__right__right,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( A2
        = ( plus_p2331992037799027419_set_a @ A2 @ B ) )
      = ( B = zero_z5079479921072680283_set_a ) ) ).

% add_cancel_right_right
thf(fact_310_add__cancel__right__right,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( A2
        = ( plus_p8712254050562127327et_nat @ A2 @ B ) )
      = ( B = zero_z3157962936165190495et_nat ) ) ).

% add_cancel_right_right
thf(fact_311_add__cancel__right__right,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ A2 @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_312_add__cancel__right__left,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( A2
        = ( plus_p2331992037799027419_set_a @ B @ A2 ) )
      = ( B = zero_z5079479921072680283_set_a ) ) ).

% add_cancel_right_left
thf(fact_313_add__cancel__right__left,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( A2
        = ( plus_p8712254050562127327et_nat @ B @ A2 ) )
      = ( B = zero_z3157962936165190495et_nat ) ) ).

% add_cancel_right_left
thf(fact_314_add__cancel__right__left,axiom,
    ! [A2: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ B @ A2 ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_315_add__cancel__left__right,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( ( plus_p2331992037799027419_set_a @ A2 @ B )
        = A2 )
      = ( B = zero_z5079479921072680283_set_a ) ) ).

% add_cancel_left_right
thf(fact_316_add__cancel__left__right,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( ( plus_p8712254050562127327et_nat @ A2 @ B )
        = A2 )
      = ( B = zero_z3157962936165190495et_nat ) ) ).

% add_cancel_left_right
thf(fact_317_add__cancel__left__right,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( plus_plus_nat @ A2 @ B )
        = A2 )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_318_add__cancel__left__left,axiom,
    ! [B: multiset_set_a,A2: multiset_set_a] :
      ( ( ( plus_p2331992037799027419_set_a @ B @ A2 )
        = A2 )
      = ( B = zero_z5079479921072680283_set_a ) ) ).

% add_cancel_left_left
thf(fact_319_add__cancel__left__left,axiom,
    ! [B: multiset_set_nat,A2: multiset_set_nat] :
      ( ( ( plus_p8712254050562127327et_nat @ B @ A2 )
        = A2 )
      = ( B = zero_z3157962936165190495et_nat ) ) ).

% add_cancel_left_left
thf(fact_320_add__cancel__left__left,axiom,
    ! [B: nat,A2: nat] :
      ( ( ( plus_plus_nat @ B @ A2 )
        = A2 )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_321_add_Oright__neutral,axiom,
    ! [A2: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ A2 @ zero_z5079479921072680283_set_a )
      = A2 ) ).

% add.right_neutral
thf(fact_322_add_Oright__neutral,axiom,
    ! [A2: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ A2 @ zero_z3157962936165190495et_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_323_add_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_324_add__le__cancel__right,axiom,
    ! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ C ) )
      = ( ord_le4034546139768944438et_nat @ A2 @ B ) ) ).

% add_le_cancel_right
thf(fact_325_add__le__cancel__right,axiom,
    ! [A2: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
      = ( ord_le7905258569527593284_set_a @ A2 @ B ) ) ).

% add_le_cancel_right
thf(fact_326_add__le__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A2 @ B ) ) ).

% add_le_cancel_right
thf(fact_327_add__le__cancel__left,axiom,
    ! [C: multiset_set_nat,A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ ( plus_p8712254050562127327et_nat @ C @ B ) )
      = ( ord_le4034546139768944438et_nat @ A2 @ B ) ) ).

% add_le_cancel_left
thf(fact_328_add__le__cancel__left,axiom,
    ! [C: multiset_set_a,A2: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ C @ A2 ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
      = ( ord_le7905258569527593284_set_a @ A2 @ B ) ) ).

% add_le_cancel_left
thf(fact_329_add__le__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A2 @ B ) ) ).

% add_le_cancel_left
thf(fact_330_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ A2 @ A2 )
      = zero_z5079479921072680283_set_a ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_331_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ A2 @ A2 )
      = zero_z3157962936165190495et_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_332_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_333_diff__zero,axiom,
    ! [A2: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ A2 @ zero_z5079479921072680283_set_a )
      = A2 ) ).

% diff_zero
thf(fact_334_diff__zero,axiom,
    ! [A2: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ A2 @ zero_z3157962936165190495et_nat )
      = A2 ) ).

% diff_zero
thf(fact_335_diff__zero,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% diff_zero
thf(fact_336_zero__diff,axiom,
    ! [A2: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ A2 )
      = zero_z5079479921072680283_set_a ) ).

% zero_diff
thf(fact_337_zero__diff,axiom,
    ! [A2: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ zero_z3157962936165190495et_nat @ A2 )
      = zero_z3157962936165190495et_nat ) ).

% zero_diff
thf(fact_338_zero__diff,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_339_add__diff__cancel__right_H,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B ) @ B )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_340_add__diff__cancel__right_H,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) @ B )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_341_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_342_add__diff__cancel__right,axiom,
    ! [A2: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
      = ( minus_706656509937749387_set_a @ A2 @ B ) ) ).

% add_diff_cancel_right
thf(fact_343_add__diff__cancel__right,axiom,
    ! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ C ) )
      = ( minus_7237264121398869807et_nat @ A2 @ B ) ) ).

% add_diff_cancel_right
thf(fact_344_add__diff__cancel__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A2 @ B ) ) ).

% add_diff_cancel_right
thf(fact_345_add__diff__cancel__left_H,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B ) @ A2 )
      = B ) ).

% add_diff_cancel_left'
thf(fact_346_add__diff__cancel__left_H,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) @ A2 )
      = B ) ).

% add_diff_cancel_left'
thf(fact_347_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B ) @ A2 )
      = B ) ).

% add_diff_cancel_left'
thf(fact_348_add__diff__cancel__left,axiom,
    ! [C: multiset_set_a,A2: multiset_set_a,B: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ A2 ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
      = ( minus_706656509937749387_set_a @ A2 @ B ) ) ).

% add_diff_cancel_left
thf(fact_349_add__diff__cancel__left,axiom,
    ! [C: multiset_set_nat,A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ ( plus_p8712254050562127327et_nat @ C @ B ) )
      = ( minus_7237264121398869807et_nat @ A2 @ B ) ) ).

% add_diff_cancel_left
thf(fact_350_add__diff__cancel__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A2 @ B ) ) ).

% add_diff_cancel_left
thf(fact_351_size__eq__0__iff__empty,axiom,
    ! [M: multiset_set_a] :
      ( ( ( size_s6566526139600085008_set_a @ M )
        = zero_zero_nat )
      = ( M = zero_z5079479921072680283_set_a ) ) ).

% size_eq_0_iff_empty
thf(fact_352_size__eq__0__iff__empty,axiom,
    ! [M: multiset_set_nat] :
      ( ( ( size_s7462436076474991978et_nat @ M )
        = zero_zero_nat )
      = ( M = zero_z3157962936165190495et_nat ) ) ).

% size_eq_0_iff_empty
thf(fact_353_size__empty,axiom,
    ( ( size_s6566526139600085008_set_a @ zero_z5079479921072680283_set_a )
    = zero_zero_nat ) ).

% size_empty
thf(fact_354_size__empty,axiom,
    ( ( size_s7462436076474991978et_nat @ zero_z3157962936165190495et_nat )
    = zero_zero_nat ) ).

% size_empty
thf(fact_355_complement__blocks__wf,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) )
     => ( ord_less_eq_set_a @ Bl2 @ ( set_a2 @ v_s ) ) ) ).

% complement_blocks_wf
thf(fact_356_union__eq__empty,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a] :
      ( ( ( plus_p2331992037799027419_set_a @ M @ N2 )
        = zero_z5079479921072680283_set_a )
      = ( ( M = zero_z5079479921072680283_set_a )
        & ( N2 = zero_z5079479921072680283_set_a ) ) ) ).

% union_eq_empty
thf(fact_357_union__eq__empty,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( ( plus_p8712254050562127327et_nat @ M @ N2 )
        = zero_z3157962936165190495et_nat )
      = ( ( M = zero_z3157962936165190495et_nat )
        & ( N2 = zero_z3157962936165190495et_nat ) ) ) ).

% union_eq_empty
thf(fact_358_empty__eq__union,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a] :
      ( ( zero_z5079479921072680283_set_a
        = ( plus_p2331992037799027419_set_a @ M @ N2 ) )
      = ( ( M = zero_z5079479921072680283_set_a )
        & ( N2 = zero_z5079479921072680283_set_a ) ) ) ).

% empty_eq_union
thf(fact_359_empty__eq__union,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( zero_z3157962936165190495et_nat
        = ( plus_p8712254050562127327et_nat @ M @ N2 ) )
      = ( ( M = zero_z3157962936165190495et_nat )
        & ( N2 = zero_z3157962936165190495et_nat ) ) ) ).

% empty_eq_union
thf(fact_360_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [X: multiset_set_a,Y: multiset_set_a] :
      ( ( zero_z5079479921072680283_set_a
        = ( plus_p2331992037799027419_set_a @ X @ Y ) )
      = ( ( X = zero_z5079479921072680283_set_a )
        & ( Y = zero_z5079479921072680283_set_a ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_361_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [X: multiset_set_nat,Y: multiset_set_nat] :
      ( ( zero_z3157962936165190495et_nat
        = ( plus_p8712254050562127327et_nat @ X @ Y ) )
      = ( ( X = zero_z3157962936165190495et_nat )
        & ( Y = zero_z3157962936165190495et_nat ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_362_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [X: multiset_set_a,Y: multiset_set_a] :
      ( ( ( plus_p2331992037799027419_set_a @ X @ Y )
        = zero_z5079479921072680283_set_a )
      = ( ( X = zero_z5079479921072680283_set_a )
        & ( Y = zero_z5079479921072680283_set_a ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_363_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [X: multiset_set_nat,Y: multiset_set_nat] :
      ( ( ( plus_p8712254050562127327et_nat @ X @ Y )
        = zero_z3157962936165190495et_nat )
      = ( ( X = zero_z3157962936165190495et_nat )
        & ( Y = zero_z3157962936165190495et_nat ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_364_union__mset__add__mset__right,axiom,
    ! [A: multiset_set_nat,A2: set_nat,B3: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ A @ ( add_mset_set_nat @ A2 @ B3 ) )
      = ( add_mset_set_nat @ A2 @ ( plus_p8712254050562127327et_nat @ A @ B3 ) ) ) ).

% union_mset_add_mset_right
thf(fact_365_union__mset__add__mset__right,axiom,
    ! [A: multiset_set_a,A2: set_a,B3: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ A @ ( add_mset_set_a @ A2 @ B3 ) )
      = ( add_mset_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ A @ B3 ) ) ) ).

% union_mset_add_mset_right
thf(fact_366_union__mset__add__mset__left,axiom,
    ! [A2: set_nat,A: multiset_set_nat,B3: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ ( add_mset_set_nat @ A2 @ A ) @ B3 )
      = ( add_mset_set_nat @ A2 @ ( plus_p8712254050562127327et_nat @ A @ B3 ) ) ) ).

% union_mset_add_mset_left
thf(fact_367_union__mset__add__mset__left,axiom,
    ! [A2: set_a,A: multiset_set_a,B3: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ A2 @ A ) @ B3 )
      = ( add_mset_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ A @ B3 ) ) ) ).

% union_mset_add_mset_left
thf(fact_368_obtain__comp__block__orig,axiom,
    ! [Bl1: set_a] :
      ( ( member_set_a @ Bl1 @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) )
     => ~ ! [Bl23: set_a] :
            ( ( member_set_a @ Bl23 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
           => ( Bl1
             != ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl23 ) ) ) ) ).

% obtain_comp_block_orig
thf(fact_369_diff__diff__add__mset,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a,P2: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ N2 ) @ P2 )
      = ( minus_706656509937749387_set_a @ M @ ( plus_p2331992037799027419_set_a @ N2 @ P2 ) ) ) ).

% diff_diff_add_mset
thf(fact_370_diff__diff__add__mset,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat,P2: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ M @ N2 ) @ P2 )
      = ( minus_7237264121398869807et_nat @ M @ ( plus_p8712254050562127327et_nat @ N2 @ P2 ) ) ) ).

% diff_diff_add_mset
thf(fact_371_add__del__block__inv,axiom,
    ! [Bl2: set_a] :
      ( ( ord_less_eq_set_a @ Bl2 @ ( set_a2 @ v_s ) )
     => ( ( design1146539425385464078lock_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ Bl2 )
        = ( mset_set_a @ b_s ) ) ) ).

% add_del_block_inv
thf(fact_372_add__block__fin__cond,axiom,
    ! [B: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( set_a2 @ v_s ) )
     => ( design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) ) ) ).

% add_block_fin_cond
thf(fact_373_add__le__same__cancel1,axiom,
    ! [B: multiset_set_a,A2: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ B @ A2 ) @ B )
      = ( ord_le7905258569527593284_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ).

% add_le_same_cancel1
thf(fact_374_add__le__same__cancel1,axiom,
    ! [B: multiset_set_nat,A2: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ B @ A2 ) @ B )
      = ( ord_le4034546139768944438et_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ).

% add_le_same_cancel1
thf(fact_375_add__le__same__cancel1,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_376_add__le__same__cancel2,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B ) @ B )
      = ( ord_le7905258569527593284_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ).

% add_le_same_cancel2
thf(fact_377_add__le__same__cancel2,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) @ B )
      = ( ord_le4034546139768944438et_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ).

% add_le_same_cancel2
thf(fact_378_add__le__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
      = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_379_le__add__same__cancel1,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ A2 @ B ) )
      = ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).

% le_add_same_cancel1
thf(fact_380_le__add__same__cancel1,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ A2 @ B ) )
      = ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_381_le__add__same__cancel1,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_382_le__add__same__cancel2,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ B @ A2 ) )
      = ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).

% le_add_same_cancel2
thf(fact_383_le__add__same__cancel2,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ B @ A2 ) )
      = ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_384_le__add__same__cancel2,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B @ A2 ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_385_diff__add__zero,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ A2 @ B ) )
      = zero_z5079479921072680283_set_a ) ).

% diff_add_zero
thf(fact_386_diff__add__zero,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ A2 @ B ) )
      = zero_z3157962936165190495et_nat ) ).

% diff_add_zero
thf(fact_387_diff__add__zero,axiom,
    ! [A2: nat,B: nat] :
      ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_388_finite__sysI,axiom,
    ( ( finite_finite_a @ ( set_a2 @ v_s ) )
   => ( design9187838744727572296stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ).

% finite_sysI
thf(fact_389_psin,axiom,
    ( ps
    = ( insert_nat @ i1 @ ( insert_nat @ i2 @ bot_bot_set_nat ) ) ) ).

% psin
thf(fact_390_diff__size__le__size__Diff,axiom,
    ! [M: multiset_set_a,M4: multiset_set_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M ) @ ( size_s6566526139600085008_set_a @ M4 ) ) @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ M4 ) ) ) ).

% diff_size_le_size_Diff
thf(fact_391_diff__size__le__size__Diff,axiom,
    ! [M: multiset_set_nat,M4: multiset_set_nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ M ) @ ( size_s7462436076474991978et_nat @ M4 ) ) @ ( size_s7462436076474991978et_nat @ ( minus_7237264121398869807et_nat @ M @ M4 ) ) ) ).

% diff_size_le_size_Diff
thf(fact_392_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: multiset_set_nat,J: multiset_set_nat,K2: multiset_set_nat,L4: multiset_set_nat] :
      ( ( ( ord_le4034546139768944438et_nat @ I @ J )
        & ( K2 = L4 ) )
     => ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ I @ K2 ) @ ( plus_p8712254050562127327et_nat @ J @ L4 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_393_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: multiset_set_a,J: multiset_set_a,K2: multiset_set_a,L4: multiset_set_a] :
      ( ( ( ord_le7905258569527593284_set_a @ I @ J )
        & ( K2 = L4 ) )
     => ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I @ K2 ) @ ( plus_p2331992037799027419_set_a @ J @ L4 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_394_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L4: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K2 = L4 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L4 ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_395_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: multiset_set_nat,J: multiset_set_nat,K2: multiset_set_nat,L4: multiset_set_nat] :
      ( ( ( I = J )
        & ( ord_le4034546139768944438et_nat @ K2 @ L4 ) )
     => ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ I @ K2 ) @ ( plus_p8712254050562127327et_nat @ J @ L4 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_396_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: multiset_set_a,J: multiset_set_a,K2: multiset_set_a,L4: multiset_set_a] :
      ( ( ( I = J )
        & ( ord_le7905258569527593284_set_a @ K2 @ L4 ) )
     => ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I @ K2 ) @ ( plus_p2331992037799027419_set_a @ J @ L4 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_397_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L4: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K2 @ L4 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L4 ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_398_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: multiset_set_nat,J: multiset_set_nat,K2: multiset_set_nat,L4: multiset_set_nat] :
      ( ( ( ord_le4034546139768944438et_nat @ I @ J )
        & ( ord_le4034546139768944438et_nat @ K2 @ L4 ) )
     => ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ I @ K2 ) @ ( plus_p8712254050562127327et_nat @ J @ L4 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_399_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: multiset_set_a,J: multiset_set_a,K2: multiset_set_a,L4: multiset_set_a] :
      ( ( ( ord_le7905258569527593284_set_a @ I @ J )
        & ( ord_le7905258569527593284_set_a @ K2 @ L4 ) )
     => ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ I @ K2 ) @ ( plus_p2331992037799027419_set_a @ J @ L4 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_400_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L4: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K2 @ L4 ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L4 ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_401_add__mono,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat,D: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ A2 @ B )
     => ( ( ord_le4034546139768944438et_nat @ C @ D )
       => ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_402_add__mono,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a,D: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ A2 @ B )
     => ( ( ord_le7905258569527593284_set_a @ C @ D )
       => ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ ( plus_p2331992037799027419_set_a @ B @ D ) ) ) ) ).

% add_mono
thf(fact_403_add__mono,axiom,
    ! [A2: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_404_add__left__mono,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ A2 @ B )
     => ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ ( plus_p8712254050562127327et_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_405_add__left__mono,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ A2 @ B )
     => ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ C @ A2 ) @ ( plus_p2331992037799027419_set_a @ C @ B ) ) ) ).

% add_left_mono
thf(fact_406_add__left__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_407_less__eqE,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A2 @ C3 ) ) ) ).

% less_eqE
thf(fact_408_add__right__mono,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ A2 @ B )
     => ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_409_add__right__mono,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ A2 @ B )
     => ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).

% add_right_mono
thf(fact_410_add__right__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_411_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B5: nat] :
        ? [C4: nat] :
          ( B5
          = ( plus_plus_nat @ A5 @ C4 ) ) ) ) ).

% le_iff_add
thf(fact_412_add__le__imp__le__left,axiom,
    ! [C: multiset_set_nat,A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ ( plus_p8712254050562127327et_nat @ C @ B ) )
     => ( ord_le4034546139768944438et_nat @ A2 @ B ) ) ).

% add_le_imp_le_left
thf(fact_413_add__le__imp__le__left,axiom,
    ! [C: multiset_set_a,A2: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ C @ A2 ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
     => ( ord_le7905258569527593284_set_a @ A2 @ B ) ) ).

% add_le_imp_le_left
thf(fact_414_add__le__imp__le__left,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A2 @ B ) ) ).

% add_le_imp_le_left
thf(fact_415_add__le__imp__le__right,axiom,
    ! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ C ) )
     => ( ord_le4034546139768944438et_nat @ A2 @ B ) ) ).

% add_le_imp_le_right
thf(fact_416_add__le__imp__le__right,axiom,
    ! [A2: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
     => ( ord_le7905258569527593284_set_a @ A2 @ B ) ) ).

% add_le_imp_le_right
thf(fact_417_add__le__imp__le__right,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A2 @ B ) ) ).

% add_le_imp_le_right
thf(fact_418_less__eq__multiset__plus__left,axiom,
    ! [N2: multiset_set_nat,M: multiset_set_nat] : ( ord_le4034546139768944438et_nat @ N2 @ ( plus_p8712254050562127327et_nat @ M @ N2 ) ) ).

% less_eq_multiset_plus_left
thf(fact_419_less__eq__multiset__plus__left,axiom,
    ! [N2: multiset_set_a,M: multiset_set_a] : ( ord_le7905258569527593284_set_a @ N2 @ ( plus_p2331992037799027419_set_a @ M @ N2 ) ) ).

% less_eq_multiset_plus_left
thf(fact_420_less__eq__multiset__plus__right,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat] : ( ord_le4034546139768944438et_nat @ M @ ( plus_p8712254050562127327et_nat @ M @ N2 ) ) ).

% less_eq_multiset_plus_right
thf(fact_421_less__eq__multiset__plus__right,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a] : ( ord_le7905258569527593284_set_a @ M @ ( plus_p2331992037799027419_set_a @ M @ N2 ) ) ).

% less_eq_multiset_plus_right
thf(fact_422_add__implies__diff,axiom,
    ! [C: multiset_set_a,B: multiset_set_a,A2: multiset_set_a] :
      ( ( ( plus_p2331992037799027419_set_a @ C @ B )
        = A2 )
     => ( C
        = ( minus_706656509937749387_set_a @ A2 @ B ) ) ) ).

% add_implies_diff
thf(fact_423_add__implies__diff,axiom,
    ! [C: multiset_set_nat,B: multiset_set_nat,A2: multiset_set_nat] :
      ( ( ( plus_p8712254050562127327et_nat @ C @ B )
        = A2 )
     => ( C
        = ( minus_7237264121398869807et_nat @ A2 @ B ) ) ) ).

% add_implies_diff
thf(fact_424_add__implies__diff,axiom,
    ! [C: nat,B: nat,A2: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A2 )
     => ( C
        = ( minus_minus_nat @ A2 @ B ) ) ) ).

% add_implies_diff
thf(fact_425_diff__diff__eq,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A2 @ B ) @ C )
      = ( minus_706656509937749387_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_426_diff__diff__eq,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ A2 @ B ) @ C )
      = ( minus_7237264121398869807et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_427_diff__diff__eq,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C )
      = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_428_union__assoc,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat,K3: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ ( plus_p8712254050562127327et_nat @ M @ N2 ) @ K3 )
      = ( plus_p8712254050562127327et_nat @ M @ ( plus_p8712254050562127327et_nat @ N2 @ K3 ) ) ) ).

% union_assoc
thf(fact_429_union__assoc,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a,K3: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ ( plus_p2331992037799027419_set_a @ M @ N2 ) @ K3 )
      = ( plus_p2331992037799027419_set_a @ M @ ( plus_p2331992037799027419_set_a @ N2 @ K3 ) ) ) ).

% union_assoc
thf(fact_430_union__lcomm,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat,K3: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ M @ ( plus_p8712254050562127327et_nat @ N2 @ K3 ) )
      = ( plus_p8712254050562127327et_nat @ N2 @ ( plus_p8712254050562127327et_nat @ M @ K3 ) ) ) ).

% union_lcomm
thf(fact_431_union__lcomm,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a,K3: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ M @ ( plus_p2331992037799027419_set_a @ N2 @ K3 ) )
      = ( plus_p2331992037799027419_set_a @ N2 @ ( plus_p2331992037799027419_set_a @ M @ K3 ) ) ) ).

% union_lcomm
thf(fact_432_union__commute,axiom,
    ( plus_p8712254050562127327et_nat
    = ( ^ [M6: multiset_set_nat,N4: multiset_set_nat] : ( plus_p8712254050562127327et_nat @ N4 @ M6 ) ) ) ).

% union_commute
thf(fact_433_union__commute,axiom,
    ( plus_p2331992037799027419_set_a
    = ( ^ [M6: multiset_set_a,N4: multiset_set_a] : ( plus_p2331992037799027419_set_a @ N4 @ M6 ) ) ) ).

% union_commute
thf(fact_434_union__left__cancel,axiom,
    ! [K3: multiset_set_nat,M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( ( plus_p8712254050562127327et_nat @ K3 @ M )
        = ( plus_p8712254050562127327et_nat @ K3 @ N2 ) )
      = ( M = N2 ) ) ).

% union_left_cancel
thf(fact_435_union__left__cancel,axiom,
    ! [K3: multiset_set_a,M: multiset_set_a,N2: multiset_set_a] :
      ( ( ( plus_p2331992037799027419_set_a @ K3 @ M )
        = ( plus_p2331992037799027419_set_a @ K3 @ N2 ) )
      = ( M = N2 ) ) ).

% union_left_cancel
thf(fact_436_union__right__cancel,axiom,
    ! [M: multiset_set_nat,K3: multiset_set_nat,N2: multiset_set_nat] :
      ( ( ( plus_p8712254050562127327et_nat @ M @ K3 )
        = ( plus_p8712254050562127327et_nat @ N2 @ K3 ) )
      = ( M = N2 ) ) ).

% union_right_cancel
thf(fact_437_union__right__cancel,axiom,
    ! [M: multiset_set_a,K3: multiset_set_a,N2: multiset_set_a] :
      ( ( ( plus_p2331992037799027419_set_a @ M @ K3 )
        = ( plus_p2331992037799027419_set_a @ N2 @ K3 ) )
      = ( M = N2 ) ) ).

% union_right_cancel
thf(fact_438_multi__union__self__other__eq,axiom,
    ! [A: multiset_set_nat,X2: multiset_set_nat,Y2: multiset_set_nat] :
      ( ( ( plus_p8712254050562127327et_nat @ A @ X2 )
        = ( plus_p8712254050562127327et_nat @ A @ Y2 ) )
     => ( X2 = Y2 ) ) ).

% multi_union_self_other_eq
thf(fact_439_multi__union__self__other__eq,axiom,
    ! [A: multiset_set_a,X2: multiset_set_a,Y2: multiset_set_a] :
      ( ( ( plus_p2331992037799027419_set_a @ A @ X2 )
        = ( plus_p2331992037799027419_set_a @ A @ Y2 ) )
     => ( X2 = Y2 ) ) ).

% multi_union_self_other_eq
thf(fact_440_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_441_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_442_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_443_add_Ocomm__neutral,axiom,
    ! [A2: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ A2 @ zero_z5079479921072680283_set_a )
      = A2 ) ).

% add.comm_neutral
thf(fact_444_add_Ocomm__neutral,axiom,
    ! [A2: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ A2 @ zero_z3157962936165190495et_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_445_add_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_446_pairwise__balance_Ob__gt__index,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat,Index: nat] :
      ( ( block_1456364645985477531ce_nat @ Point_set @ Block_collection @ Index )
     => ( ord_less_eq_nat @ Index @ ( size_s7462436076474991978et_nat @ Block_collection ) ) ) ).

% pairwise_balance.b_gt_index
thf(fact_447_pairwise__balance_Ob__gt__index,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a,Index: nat] :
      ( ( block_5355636846524985331ance_a @ Point_set @ Block_collection @ Index )
     => ( ord_less_eq_nat @ Index @ ( size_s6566526139600085008_set_a @ Block_collection ) ) ) ).

% pairwise_balance.b_gt_index
thf(fact_448_less__eq__multiset__empty__right,axiom,
    ! [M: multiset_set_a] :
      ( ( M != zero_z5079479921072680283_set_a )
     => ~ ( ord_le7905258569527593284_set_a @ M @ zero_z5079479921072680283_set_a ) ) ).

% less_eq_multiset_empty_right
thf(fact_449_less__eq__multiset__empty__right,axiom,
    ! [M: multiset_set_nat] :
      ( ( M != zero_z3157962936165190495et_nat )
     => ~ ( ord_le4034546139768944438et_nat @ M @ zero_z3157962936165190495et_nat ) ) ).

% less_eq_multiset_empty_right
thf(fact_450_less__eq__multiset__empty__left,axiom,
    ! [M: multiset_set_a] : ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ M ) ).

% less_eq_multiset_empty_left
thf(fact_451_less__eq__multiset__empty__left,axiom,
    ! [M: multiset_set_nat] : ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ M ) ).

% less_eq_multiset_empty_left
thf(fact_452_union__iff,axiom,
    ! [A2: a,A: multiset_a,B3: multiset_a] :
      ( ( member_a @ A2 @ ( set_mset_a @ ( plus_plus_multiset_a @ A @ B3 ) ) )
      = ( ( member_a @ A2 @ ( set_mset_a @ A ) )
        | ( member_a @ A2 @ ( set_mset_a @ B3 ) ) ) ) ).

% union_iff
thf(fact_453_union__iff,axiom,
    ! [A2: nat,A: multiset_nat,B3: multiset_nat] :
      ( ( member_nat @ A2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ B3 ) ) )
      = ( ( member_nat @ A2 @ ( set_mset_nat @ A ) )
        | ( member_nat @ A2 @ ( set_mset_nat @ B3 ) ) ) ) ).

% union_iff
thf(fact_454_union__iff,axiom,
    ! [A2: set_nat,A: multiset_set_nat,B3: multiset_set_nat] :
      ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ ( plus_p8712254050562127327et_nat @ A @ B3 ) ) )
      = ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ A ) )
        | ( member_set_nat @ A2 @ ( set_mset_set_nat @ B3 ) ) ) ) ).

% union_iff
thf(fact_455_union__iff,axiom,
    ! [A2: set_a,A: multiset_set_a,B3: multiset_set_a] :
      ( ( member_set_a @ A2 @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ B3 ) ) )
      = ( ( member_set_a @ A2 @ ( set_mset_set_a @ A ) )
        | ( member_set_a @ A2 @ ( set_mset_set_a @ B3 ) ) ) ) ).

% union_iff
thf(fact_456_empty__neutral_I1_J,axiom,
    ! [X: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ X )
      = X ) ).

% empty_neutral(1)
thf(fact_457_empty__neutral_I1_J,axiom,
    ! [X: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat @ X )
      = X ) ).

% empty_neutral(1)
thf(fact_458_empty__neutral_I2_J,axiom,
    ! [X: multiset_set_a] :
      ( ( plus_p2331992037799027419_set_a @ X @ zero_z5079479921072680283_set_a )
      = X ) ).

% empty_neutral(2)
thf(fact_459_empty__neutral_I2_J,axiom,
    ! [X: multiset_set_nat] :
      ( ( plus_p8712254050562127327et_nat @ X @ zero_z3157962936165190495et_nat )
      = X ) ).

% empty_neutral(2)
thf(fact_460_diff__union__cancelR,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ M @ N2 ) @ N2 )
      = M ) ).

% diff_union_cancelR
thf(fact_461_diff__union__cancelR,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_union_cancelR
thf(fact_462_diff__union__cancelL,axiom,
    ! [N2: multiset_set_a,M: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ N2 @ M ) @ N2 )
      = M ) ).

% diff_union_cancelL
thf(fact_463_diff__union__cancelL,axiom,
    ! [N2: multiset_set_nat,M: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_union_cancelL
thf(fact_464_Multiset_Odiff__add,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a,Q: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ M @ ( plus_p2331992037799027419_set_a @ N2 @ Q ) )
      = ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ N2 ) @ Q ) ) ).

% Multiset.diff_add
thf(fact_465_Multiset_Odiff__add,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat,Q: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ M @ ( plus_p8712254050562127327et_nat @ N2 @ Q ) )
      = ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ M @ N2 ) @ Q ) ) ).

% Multiset.diff_add
thf(fact_466_add__decreasing,axiom,
    ! [A2: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ A2 @ zero_z5079479921072680283_set_a )
     => ( ( ord_le7905258569527593284_set_a @ C @ B )
       => ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_467_add__decreasing,axiom,
    ! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ A2 @ zero_z3157962936165190495et_nat )
     => ( ( ord_le4034546139768944438et_nat @ C @ B )
       => ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_468_add__decreasing,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_469_add__increasing,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A2 )
     => ( ( ord_le7905258569527593284_set_a @ B @ C )
       => ( ord_le7905258569527593284_set_a @ B @ ( plus_p2331992037799027419_set_a @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_470_add__increasing,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ A2 )
     => ( ( ord_le4034546139768944438et_nat @ B @ C )
       => ( ord_le4034546139768944438et_nat @ B @ ( plus_p8712254050562127327et_nat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_471_add__increasing,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing
thf(fact_472_add__decreasing2,axiom,
    ! [C: multiset_set_a,A2: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ C @ zero_z5079479921072680283_set_a )
     => ( ( ord_le7905258569527593284_set_a @ A2 @ B )
       => ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_473_add__decreasing2,axiom,
    ! [C: multiset_set_nat,A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ C @ zero_z3157962936165190495et_nat )
     => ( ( ord_le4034546139768944438et_nat @ A2 @ B )
       => ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_474_add__decreasing2,axiom,
    ! [C: nat,A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_475_add__increasing2,axiom,
    ! [C: multiset_set_a,B: multiset_set_a,A2: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ C )
     => ( ( ord_le7905258569527593284_set_a @ B @ A2 )
       => ( ord_le7905258569527593284_set_a @ B @ ( plus_p2331992037799027419_set_a @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_476_add__increasing2,axiom,
    ! [C: multiset_set_nat,B: multiset_set_nat,A2: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ C )
     => ( ( ord_le4034546139768944438et_nat @ B @ A2 )
       => ( ord_le4034546139768944438et_nat @ B @ ( plus_p8712254050562127327et_nat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_477_add__increasing2,axiom,
    ! [C: nat,B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% add_increasing2
thf(fact_478_add__nonneg__nonneg,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A2 )
     => ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B )
       => ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_479_add__nonneg__nonneg,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ A2 )
     => ( ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ B )
       => ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_480_add__nonneg__nonneg,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_481_add__nonpos__nonpos,axiom,
    ! [A2: multiset_set_a,B: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ A2 @ zero_z5079479921072680283_set_a )
     => ( ( ord_le7905258569527593284_set_a @ B @ zero_z5079479921072680283_set_a )
       => ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).

% add_nonpos_nonpos
thf(fact_482_add__nonpos__nonpos,axiom,
    ! [A2: multiset_set_nat,B: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ A2 @ zero_z3157962936165190495et_nat )
     => ( ( ord_le4034546139768944438et_nat @ B @ zero_z3157962936165190495et_nat )
       => ( ord_le4034546139768944438et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) @ zero_z3157962936165190495et_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_483_add__nonpos__nonpos,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_484_add__nonneg__eq__0__iff,axiom,
    ! [X: multiset_set_a,Y: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ X )
     => ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ Y )
       => ( ( ( plus_p2331992037799027419_set_a @ X @ Y )
            = zero_z5079479921072680283_set_a )
          = ( ( X = zero_z5079479921072680283_set_a )
            & ( Y = zero_z5079479921072680283_set_a ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_485_add__nonneg__eq__0__iff,axiom,
    ! [X: multiset_set_nat,Y: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ X )
     => ( ( ord_le4034546139768944438et_nat @ zero_z3157962936165190495et_nat @ Y )
       => ( ( ( plus_p8712254050562127327et_nat @ X @ Y )
            = zero_z3157962936165190495et_nat )
          = ( ( X = zero_z3157962936165190495et_nat )
            & ( Y = zero_z3157962936165190495et_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_486_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_487_add__nonpos__eq__0__iff,axiom,
    ! [X: multiset_set_a,Y: multiset_set_a] :
      ( ( ord_le7905258569527593284_set_a @ X @ zero_z5079479921072680283_set_a )
     => ( ( ord_le7905258569527593284_set_a @ Y @ zero_z5079479921072680283_set_a )
       => ( ( ( plus_p2331992037799027419_set_a @ X @ Y )
            = zero_z5079479921072680283_set_a )
          = ( ( X = zero_z5079479921072680283_set_a )
            & ( Y = zero_z5079479921072680283_set_a ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_488_add__nonpos__eq__0__iff,axiom,
    ! [X: multiset_set_nat,Y: multiset_set_nat] :
      ( ( ord_le4034546139768944438et_nat @ X @ zero_z3157962936165190495et_nat )
     => ( ( ord_le4034546139768944438et_nat @ Y @ zero_z3157962936165190495et_nat )
       => ( ( ( plus_p8712254050562127327et_nat @ X @ Y )
            = zero_z3157962936165190495et_nat )
          = ( ( X = zero_z3157962936165190495et_nat )
            & ( Y = zero_z3157962936165190495et_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_489_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_490_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ A2 @ B )
       => ( ( ( minus_minus_nat @ B @ A2 )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_491_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B @ A2 ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_492_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_493_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_494_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_495_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_496_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_497_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_498_le__add__diff,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A2 ) ) ) ).

% le_add_diff
thf(fact_499_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A2 ) @ A2 )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_500_ordered__simple__design_Oblock__list__distinct,axiom,
    ! [V_s2: list_a,B_s2: list_set_a] :
      ( ( incide371748008924627346sign_a @ V_s2 @ B_s2 )
     => ( distinct_set_a @ B_s2 ) ) ).

% ordered_simple_design.block_list_distinct
thf(fact_501_size__Diff1__le,axiom,
    ! [M: multiset_set_a,X: set_a] : ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) ) @ ( size_s6566526139600085008_set_a @ M ) ) ).

% size_Diff1_le
thf(fact_502_size__Diff1__le,axiom,
    ! [M: multiset_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( size_s7462436076474991978et_nat @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) ) ) @ ( size_s7462436076474991978et_nat @ M ) ) ).

% size_Diff1_le
thf(fact_503_ordered__proper__design_Odual__sys__b__non__zero,axiom,
    ! [V_s2: list_set_a,B_s2: list_set_set_a] :
      ( ( incide2999377533768400724_set_a @ V_s2 @ B_s2 )
     => ( ( size_s7462436076474991978et_nat @ ( dual_d359914979145368543_set_a @ ( set_set_a2 @ V_s2 ) @ B_s2 ) )
       != zero_zero_nat ) ) ).

% ordered_proper_design.dual_sys_b_non_zero
thf(fact_504_ordered__proper__design_Odual__sys__b__non__zero,axiom,
    ! [V_s2: list_a,B_s2: list_set_a] :
      ( ( incide3676903341588786676sign_a @ V_s2 @ B_s2 )
     => ( ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ V_s2 ) @ B_s2 ) )
       != zero_zero_nat ) ) ).

% ordered_proper_design.dual_sys_b_non_zero
thf(fact_505_add__mset__add__single,axiom,
    ( add_mset_set_a
    = ( ^ [A5: set_a,A6: multiset_set_a] : ( plus_p2331992037799027419_set_a @ A6 @ ( add_mset_set_a @ A5 @ zero_z5079479921072680283_set_a ) ) ) ) ).

% add_mset_add_single
thf(fact_506_add__mset__add__single,axiom,
    ( add_mset_set_nat
    = ( ^ [A5: set_nat,A6: multiset_set_nat] : ( plus_p8712254050562127327et_nat @ A6 @ ( add_mset_set_nat @ A5 @ zero_z3157962936165190495et_nat ) ) ) ) ).

% add_mset_add_single
thf(fact_507_union__is__single,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a,A2: set_a] :
      ( ( ( plus_p2331992037799027419_set_a @ M @ N2 )
        = ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) )
      = ( ( ( M
            = ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) )
          & ( N2 = zero_z5079479921072680283_set_a ) )
        | ( ( M = zero_z5079479921072680283_set_a )
          & ( N2
            = ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ) ) ) ).

% union_is_single
thf(fact_508_union__is__single,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat,A2: set_nat] :
      ( ( ( plus_p8712254050562127327et_nat @ M @ N2 )
        = ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) )
      = ( ( ( M
            = ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) )
          & ( N2 = zero_z3157962936165190495et_nat ) )
        | ( ( M = zero_z3157962936165190495et_nat )
          & ( N2
            = ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ) ) ) ).

% union_is_single
thf(fact_509_single__is__union,axiom,
    ! [A2: set_a,M: multiset_set_a,N2: multiset_set_a] :
      ( ( ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a )
        = ( plus_p2331992037799027419_set_a @ M @ N2 ) )
      = ( ( ( ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a )
            = M )
          & ( N2 = zero_z5079479921072680283_set_a ) )
        | ( ( M = zero_z5079479921072680283_set_a )
          & ( ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a )
            = N2 ) ) ) ) ).

% single_is_union
thf(fact_510_single__is__union,axiom,
    ! [A2: set_nat,M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat )
        = ( plus_p8712254050562127327et_nat @ M @ N2 ) )
      = ( ( ( ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat )
            = M )
          & ( N2 = zero_z3157962936165190495et_nat ) )
        | ( ( M = zero_z3157962936165190495et_nat )
          & ( ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat )
            = N2 ) ) ) ) ).

% single_is_union
thf(fact_511_zero__reorient,axiom,
    ! [X: multiset_set_a] :
      ( ( zero_z5079479921072680283_set_a = X )
      = ( X = zero_z5079479921072680283_set_a ) ) ).

% zero_reorient
thf(fact_512_zero__reorient,axiom,
    ! [X: multiset_set_nat] :
      ( ( zero_z3157962936165190495et_nat = X )
      = ( X = zero_z3157962936165190495et_nat ) ) ).

% zero_reorient
thf(fact_513_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_514_multi__member__this,axiom,
    ! [X: a,XS: multiset_a] : ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ X @ zero_zero_multiset_a ) @ XS ) ) ) ).

% multi_member_this
thf(fact_515_multi__member__this,axiom,
    ! [X: nat,XS: multiset_nat] : ( member_nat @ X @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ).

% multi_member_this
thf(fact_516_multi__member__this,axiom,
    ! [X: set_a,XS: multiset_set_a] : ( member_set_a @ X @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) @ XS ) ) ) ).

% multi_member_this
thf(fact_517_multi__member__this,axiom,
    ! [X: set_nat,XS: multiset_set_nat] : ( member_set_nat @ X @ ( set_mset_set_nat @ ( plus_p8712254050562127327et_nat @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) @ XS ) ) ) ).

% multi_member_this
thf(fact_518_multi__member__skip,axiom,
    ! [X: a,XS: multiset_a,Y: a] :
      ( ( member_a @ X @ ( set_mset_a @ XS ) )
     => ( member_a @ X @ ( set_mset_a @ ( plus_plus_multiset_a @ ( add_mset_a @ Y @ zero_zero_multiset_a ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_519_multi__member__skip,axiom,
    ! [X: nat,XS: multiset_nat,Y: nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ XS ) )
     => ( member_nat @ X @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ Y @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_520_multi__member__skip,axiom,
    ! [X: set_a,XS: multiset_set_a,Y: set_a] :
      ( ( member_set_a @ X @ ( set_mset_set_a @ XS ) )
     => ( member_set_a @ X @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( add_mset_set_a @ Y @ zero_z5079479921072680283_set_a ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_521_multi__member__skip,axiom,
    ! [X: set_nat,XS: multiset_set_nat,Y: set_nat] :
      ( ( member_set_nat @ X @ ( set_mset_set_nat @ XS ) )
     => ( member_set_nat @ X @ ( set_mset_set_nat @ ( plus_p8712254050562127327et_nat @ ( add_mset_set_nat @ Y @ zero_z3157962936165190495et_nat ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_522_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
      ( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A2 @ C ) @ B )
      = ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ A2 @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_523_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
      ( ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ A2 @ C ) @ B )
      = ( minus_7237264121398869807et_nat @ ( minus_7237264121398869807et_nat @ A2 @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_524_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_525_remove1__mset__add__mset__If,axiom,
    ! [L2: set_a,L: set_a,C2: multiset_set_a] :
      ( ( ( L2 = L )
       => ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ L @ C2 ) @ ( add_mset_set_a @ L2 @ zero_z5079479921072680283_set_a ) )
          = C2 ) )
      & ( ( L2 != L )
       => ( ( minus_706656509937749387_set_a @ ( add_mset_set_a @ L @ C2 ) @ ( add_mset_set_a @ L2 @ zero_z5079479921072680283_set_a ) )
          = ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ C2 @ ( add_mset_set_a @ L2 @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ L @ zero_z5079479921072680283_set_a ) ) ) ) ) ).

% remove1_mset_add_mset_If
thf(fact_526_remove1__mset__add__mset__If,axiom,
    ! [L2: set_nat,L: set_nat,C2: multiset_set_nat] :
      ( ( ( L2 = L )
       => ( ( minus_7237264121398869807et_nat @ ( add_mset_set_nat @ L @ C2 ) @ ( add_mset_set_nat @ L2 @ zero_z3157962936165190495et_nat ) )
          = C2 ) )
      & ( ( L2 != L )
       => ( ( minus_7237264121398869807et_nat @ ( add_mset_set_nat @ L @ C2 ) @ ( add_mset_set_nat @ L2 @ zero_z3157962936165190495et_nat ) )
          = ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ C2 @ ( add_mset_set_nat @ L2 @ zero_z3157962936165190495et_nat ) ) @ ( add_mset_set_nat @ L @ zero_z3157962936165190495et_nat ) ) ) ) ) ).

% remove1_mset_add_mset_If
thf(fact_527_diff__union__single__conv,axiom,
    ! [A2: a,J2: multiset_a,I2: multiset_a] :
      ( ( member_a @ A2 @ ( set_mset_a @ J2 ) )
     => ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ I2 @ J2 ) @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) )
        = ( plus_plus_multiset_a @ I2 @ ( minus_3765977307040488491iset_a @ J2 @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) ) ) ).

% diff_union_single_conv
thf(fact_528_diff__union__single__conv,axiom,
    ! [A2: nat,J2: multiset_nat,I2: multiset_nat] :
      ( ( member_nat @ A2 @ ( set_mset_nat @ J2 ) )
     => ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ I2 @ J2 ) @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) )
        = ( plus_p6334493942879108393et_nat @ I2 @ ( minus_8522176038001411705et_nat @ J2 @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) ) ) ).

% diff_union_single_conv
thf(fact_529_diff__union__single__conv,axiom,
    ! [A2: set_a,J2: multiset_set_a,I2: multiset_set_a] :
      ( ( member_set_a @ A2 @ ( set_mset_set_a @ J2 ) )
     => ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ I2 @ J2 ) @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) )
        = ( plus_p2331992037799027419_set_a @ I2 @ ( minus_706656509937749387_set_a @ J2 @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ) ) ) ).

% diff_union_single_conv
thf(fact_530_diff__union__single__conv,axiom,
    ! [A2: set_nat,J2: multiset_set_nat,I2: multiset_set_nat] :
      ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ J2 ) )
     => ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ I2 @ J2 ) @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) )
        = ( plus_p8712254050562127327et_nat @ I2 @ ( minus_7237264121398869807et_nat @ J2 @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ) ) ) ).

% diff_union_single_conv
thf(fact_531_remove1__mset__eqE,axiom,
    ! [X1: multiset_a,L2: a,M: multiset_a] :
      ( ( ( minus_3765977307040488491iset_a @ X1 @ ( add_mset_a @ L2 @ zero_zero_multiset_a ) )
        = M )
     => ( ( ( member_a @ L2 @ ( set_mset_a @ X1 ) )
         => ( X1
           != ( plus_plus_multiset_a @ M @ ( add_mset_a @ L2 @ zero_zero_multiset_a ) ) ) )
       => ~ ( ~ ( member_a @ L2 @ ( set_mset_a @ X1 ) )
           => ( X1 != M ) ) ) ) ).

% remove1_mset_eqE
thf(fact_532_remove1__mset__eqE,axiom,
    ! [X1: multiset_nat,L2: nat,M: multiset_nat] :
      ( ( ( minus_8522176038001411705et_nat @ X1 @ ( add_mset_nat @ L2 @ zero_z7348594199698428585et_nat ) )
        = M )
     => ( ( ( member_nat @ L2 @ ( set_mset_nat @ X1 ) )
         => ( X1
           != ( plus_p6334493942879108393et_nat @ M @ ( add_mset_nat @ L2 @ zero_z7348594199698428585et_nat ) ) ) )
       => ~ ( ~ ( member_nat @ L2 @ ( set_mset_nat @ X1 ) )
           => ( X1 != M ) ) ) ) ).

% remove1_mset_eqE
thf(fact_533_remove1__mset__eqE,axiom,
    ! [X1: multiset_set_a,L2: set_a,M: multiset_set_a] :
      ( ( ( minus_706656509937749387_set_a @ X1 @ ( add_mset_set_a @ L2 @ zero_z5079479921072680283_set_a ) )
        = M )
     => ( ( ( member_set_a @ L2 @ ( set_mset_set_a @ X1 ) )
         => ( X1
           != ( plus_p2331992037799027419_set_a @ M @ ( add_mset_set_a @ L2 @ zero_z5079479921072680283_set_a ) ) ) )
       => ~ ( ~ ( member_set_a @ L2 @ ( set_mset_set_a @ X1 ) )
           => ( X1 != M ) ) ) ) ).

% remove1_mset_eqE
thf(fact_534_remove1__mset__eqE,axiom,
    ! [X1: multiset_set_nat,L2: set_nat,M: multiset_set_nat] :
      ( ( ( minus_7237264121398869807et_nat @ X1 @ ( add_mset_set_nat @ L2 @ zero_z3157962936165190495et_nat ) )
        = M )
     => ( ( ( member_set_nat @ L2 @ ( set_mset_set_nat @ X1 ) )
         => ( X1
           != ( plus_p8712254050562127327et_nat @ M @ ( add_mset_set_nat @ L2 @ zero_z3157962936165190495et_nat ) ) ) )
       => ~ ( ~ ( member_set_nat @ L2 @ ( set_mset_set_nat @ X1 ) )
           => ( X1 != M ) ) ) ) ).

% remove1_mset_eqE
thf(fact_535_insert__DiffM2,axiom,
    ! [X: a,M: multiset_a] :
      ( ( member_a @ X @ ( set_mset_a @ M ) )
     => ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) @ ( add_mset_a @ X @ zero_zero_multiset_a ) )
        = M ) ) ).

% insert_DiffM2
thf(fact_536_insert__DiffM2,axiom,
    ! [X: nat,M: multiset_nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ M ) )
     => ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) )
        = M ) ) ).

% insert_DiffM2
thf(fact_537_insert__DiffM2,axiom,
    ! [X: set_a,M: multiset_set_a] :
      ( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
     => ( ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) )
        = M ) ) ).

% insert_DiffM2
thf(fact_538_insert__DiffM2,axiom,
    ! [X: set_nat,M: multiset_set_nat] :
      ( ( member_set_nat @ X @ ( set_mset_set_nat @ M ) )
     => ( ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) ) @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) )
        = M ) ) ).

% insert_DiffM2
thf(fact_539_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_540_strong__del__block__fin,axiom,
    ! [B: set_a] : ( design9187838744727572296stem_a @ ( minus_minus_set_a @ ( set_a2 @ v_s ) @ B ) @ ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ B ) ) ).

% strong_del_block_fin
thf(fact_541_replication__numbers__non__empty,axiom,
    ( ( ( set_a2 @ v_s )
     != bot_bot_set_a )
   => ( ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
     != bot_bot_set_nat ) ) ).

% replication_numbers_non_empty
thf(fact_542_replication__numbers__finite,axiom,
    finite_finite_nat @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).

% replication_numbers_finite
thf(fact_543_dual__sys_Oblock__sizes__non__empty__set,axiom,
    ( ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
     != zero_z3157962936165190495et_nat )
   => ( ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
     != bot_bot_set_nat ) ) ).

% dual_sys.block_sizes_non_empty_set
thf(fact_544_dual__sys_Odel__block__b_I1_J,axiom,
    ! [Bl2: set_nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( size_s7462436076474991978et_nat @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) )
        = ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ one_one_nat ) ) ) ).

% dual_sys.del_block_b(1)
thf(fact_545_dual__sys_Ofinite__block__sizes,axiom,
    finite_finite_nat @ ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.finite_block_sizes
thf(fact_546_delete__point__blocks__wf,axiom,
    ! [B: set_a,P: a] :
      ( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P ) ) )
     => ( ord_less_eq_set_a @ B @ ( minus_minus_set_a @ ( set_a2 @ v_s ) @ ( insert_a @ P @ bot_bot_set_a ) ) ) ) ).

% delete_point_blocks_wf
thf(fact_547_le__add__diff__inverse2,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B ) @ B )
        = A2 ) ) ).

% le_add_diff_inverse2
thf(fact_548_le__add__diff__inverse,axiom,
    ! [B: nat,A2: nat] :
      ( ( ord_less_eq_nat @ B @ A2 )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A2 @ B ) )
        = A2 ) ) ).

% le_add_diff_inverse
thf(fact_549_b__non__zero,axiom,
    ( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
   != zero_zero_nat ) ).

% b_non_zero
thf(fact_550_index__not__zero,axiom,
    ord_less_eq_nat @ one_one_nat @ lambda ).

% index_not_zero
thf(fact_551_del__point__block__count,axiom,
    ! [P: a] :
      ( ( size_s6566526139600085008_set_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P ) )
      = ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).

% del_point_block_count
thf(fact_552__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062i1_Ai2_O_A_092_060lbrakk_062ps_A_061_A_123i1_M_Ai2_125_059_Ai1_A_092_060noteq_062_Ai2_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [I1: nat,I22: nat] :
        ( ( ps
          = ( insert_nat @ I1 @ ( insert_nat @ I22 @ bot_bot_set_nat ) ) )
       => ( I1 = I22 ) ) ).

% \<open>\<And>thesis. (\<And>i1 i2. \<lbrakk>ps = {i1, i2}; i1 \<noteq> i2\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_553_del__block__b_I2_J,axiom,
    ! [Bl2: set_a] :
      ( ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
        = ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ).

% del_block_b(2)
thf(fact_554_b__gt__index,axiom,
    ord_less_eq_nat @ lambda @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).

% b_gt_index
thf(fact_555_add__point__def,axiom,
    ! [P: a] :
      ( ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P )
      = ( insert_a @ P @ ( set_a2 @ v_s ) ) ) ).

% add_point_def
thf(fact_556_add__block__def,axiom,
    ! [B: set_a] :
      ( ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B )
      = ( plus_p2331992037799027419_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) ) ) ).

% add_block_def
thf(fact_557_del__block__b_I1_J,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
        = ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ one_one_nat ) ) ) ).

% del_block_b(1)
thf(fact_558_del__point__def,axiom,
    ! [P: a] :
      ( ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P )
      = ( minus_minus_set_a @ ( set_a2 @ v_s ) @ ( insert_a @ P @ bot_bot_set_a ) ) ) ).

% del_point_def
thf(fact_559_size__union,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( size_s7462436076474991978et_nat @ ( plus_p8712254050562127327et_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( size_s7462436076474991978et_nat @ M ) @ ( size_s7462436076474991978et_nat @ N2 ) ) ) ).

% size_union
thf(fact_560_size__union,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a] :
      ( ( size_s6566526139600085008_set_a @ ( plus_p2331992037799027419_set_a @ M @ N2 ) )
      = ( plus_plus_nat @ ( size_s6566526139600085008_set_a @ M ) @ ( size_s6566526139600085008_set_a @ N2 ) ) ) ).

% size_union
thf(fact_561_str__del__block__del__point,axiom,
    ! [X: a] :
      ( ~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ ( insert_a @ X @ bot_bot_set_a ) )
        = ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ X ) ) ) ).

% str_del_block_del_point
thf(fact_562_dual__sys_Oadd__block__index__in,axiom,
    ! [Ps: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ Ps @ B )
     => ( ( design6574611146354332593ex_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) @ Ps )
        = ( plus_plus_nat @ ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps ) @ one_one_nat ) ) ) ).

% dual_sys.add_block_index_in
thf(fact_563_remove__invalid__point__block,axiom,
    ! [P: a,Bl2: set_a] :
      ( ~ ( member_a @ P @ ( set_a2 @ v_s ) )
     => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( minus_minus_set_a @ Bl2 @ ( insert_a @ P @ bot_bot_set_a ) )
          = Bl2 ) ) ) ).

% remove_invalid_point_block
thf(fact_564_set__mset__add__mset__insert,axiom,
    ! [A2: nat,A: multiset_nat] :
      ( ( set_mset_nat @ ( add_mset_nat @ A2 @ A ) )
      = ( insert_nat @ A2 @ ( set_mset_nat @ A ) ) ) ).

% set_mset_add_mset_insert
thf(fact_565_set__mset__add__mset__insert,axiom,
    ! [A2: a,A: multiset_a] :
      ( ( set_mset_a @ ( add_mset_a @ A2 @ A ) )
      = ( insert_a @ A2 @ ( set_mset_a @ A ) ) ) ).

% set_mset_add_mset_insert
thf(fact_566_set__mset__add__mset__insert,axiom,
    ! [A2: set_a,A: multiset_set_a] :
      ( ( set_mset_set_a @ ( add_mset_set_a @ A2 @ A ) )
      = ( insert_set_a @ A2 @ ( set_mset_set_a @ A ) ) ) ).

% set_mset_add_mset_insert
thf(fact_567_set__mset__add__mset__insert,axiom,
    ! [A2: set_nat,A: multiset_set_nat] :
      ( ( set_mset_set_nat @ ( add_mset_set_nat @ A2 @ A ) )
      = ( insert_set_nat @ A2 @ ( set_mset_set_nat @ A ) ) ) ).

% set_mset_add_mset_insert
thf(fact_568_complement__same__b,axiom,
    ( ( size_s6566526139600085008_set_a @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) )
    = ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).

% complement_same_b
thf(fact_569_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_570_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_571_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_572_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_573_size__single,axiom,
    ! [B: set_a] :
      ( ( size_s6566526139600085008_set_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) )
      = one_one_nat ) ).

% size_single
thf(fact_574_size__single,axiom,
    ! [B: set_nat] :
      ( ( size_s7462436076474991978et_nat @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) )
      = one_one_nat ) ).

% size_single
thf(fact_575_size__1__singleton__mset,axiom,
    ! [M: multiset_set_a] :
      ( ( ( size_s6566526139600085008_set_a @ M )
        = one_one_nat )
     => ? [A4: set_a] :
          ( M
          = ( add_mset_set_a @ A4 @ zero_z5079479921072680283_set_a ) ) ) ).

% size_1_singleton_mset
thf(fact_576_size__1__singleton__mset,axiom,
    ! [M: multiset_set_nat] :
      ( ( ( size_s7462436076474991978et_nat @ M )
        = one_one_nat )
     => ? [A4: set_nat] :
          ( M
          = ( add_mset_set_nat @ A4 @ zero_z3157962936165190495et_nat ) ) ) ).

% size_1_singleton_mset
thf(fact_577_set__mset__single,axiom,
    ! [B: set_a] :
      ( ( set_mset_set_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) )
      = ( insert_set_a @ B @ bot_bot_set_set_a ) ) ).

% set_mset_single
thf(fact_578_set__mset__single,axiom,
    ! [B: set_nat] :
      ( ( set_mset_set_nat @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) )
      = ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) ).

% set_mset_single
thf(fact_579_set__mset__single,axiom,
    ! [B: a] :
      ( ( set_mset_a @ ( add_mset_a @ B @ zero_zero_multiset_a ) )
      = ( insert_a @ B @ bot_bot_set_a ) ) ).

% set_mset_single
thf(fact_580_set__mset__single,axiom,
    ! [B: nat] :
      ( ( set_mset_nat @ ( add_mset_nat @ B @ zero_z7348594199698428585et_nat ) )
      = ( insert_nat @ B @ bot_bot_set_nat ) ) ).

% set_mset_single
thf(fact_581_at__most__one__mset__mset__diff,axiom,
    ! [A2: a,M: multiset_a] :
      ( ~ ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) ) )
     => ( ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ A2 @ zero_zero_multiset_a ) ) )
        = ( minus_minus_set_a @ ( set_mset_a @ M ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).

% at_most_one_mset_mset_diff
thf(fact_582_at__most__one__mset__mset__diff,axiom,
    ! [A2: nat,M: multiset_nat] :
      ( ~ ( member_nat @ A2 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ) )
     => ( ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) )
        = ( minus_minus_set_nat @ ( set_mset_nat @ M ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% at_most_one_mset_mset_diff
thf(fact_583_at__most__one__mset__mset__diff,axiom,
    ! [A2: set_a,M: multiset_set_a] :
      ( ~ ( member_set_a @ A2 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ) )
     => ( ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) )
        = ( minus_5736297505244876581_set_a @ ( set_mset_set_a @ M ) @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ).

% at_most_one_mset_mset_diff
thf(fact_584_at__most__one__mset__mset__diff,axiom,
    ! [A2: set_nat,M: multiset_set_nat] :
      ( ~ ( member_set_nat @ A2 @ ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ) )
     => ( ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ A2 @ zero_z3157962936165190495et_nat ) ) )
        = ( minus_2163939370556025621et_nat @ ( set_mset_set_nat @ M ) @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ) ) ).

% at_most_one_mset_mset_diff
thf(fact_585_size__Diff__singleton,axiom,
    ! [X: a,M: multiset_a] :
      ( ( member_a @ X @ ( set_mset_a @ M ) )
     => ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
        = ( minus_minus_nat @ ( size_size_multiset_a @ M ) @ one_one_nat ) ) ) ).

% size_Diff_singleton
thf(fact_586_size__Diff__singleton,axiom,
    ! [X: nat,M: multiset_nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ M ) )
     => ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
        = ( minus_minus_nat @ ( size_s5917832649809541300et_nat @ M ) @ one_one_nat ) ) ) ).

% size_Diff_singleton
thf(fact_587_size__Diff__singleton,axiom,
    ! [X: set_a,M: multiset_set_a] :
      ( ( member_set_a @ X @ ( set_mset_set_a @ M ) )
     => ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
        = ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M ) @ one_one_nat ) ) ) ).

% size_Diff_singleton
thf(fact_588_size__Diff__singleton,axiom,
    ! [X: set_nat,M: multiset_set_nat] :
      ( ( member_set_nat @ X @ ( set_mset_set_nat @ M ) )
     => ( ( size_s7462436076474991978et_nat @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) ) )
        = ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ M ) @ one_one_nat ) ) ) ).

% size_Diff_singleton
thf(fact_589_size__Diff__singleton__if,axiom,
    ! [X: a,A: multiset_a] :
      ( ( ( member_a @ X @ ( set_mset_a @ A ) )
       => ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
          = ( minus_minus_nat @ ( size_size_multiset_a @ A ) @ one_one_nat ) ) )
      & ( ~ ( member_a @ X @ ( set_mset_a @ A ) )
       => ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ A @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
          = ( size_size_multiset_a @ A ) ) ) ) ).

% size_Diff_singleton_if
thf(fact_590_size__Diff__singleton__if,axiom,
    ! [X: nat,A: multiset_nat] :
      ( ( ( member_nat @ X @ ( set_mset_nat @ A ) )
       => ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ A @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
          = ( minus_minus_nat @ ( size_s5917832649809541300et_nat @ A ) @ one_one_nat ) ) )
      & ( ~ ( member_nat @ X @ ( set_mset_nat @ A ) )
       => ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ A @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
          = ( size_s5917832649809541300et_nat @ A ) ) ) ) ).

% size_Diff_singleton_if
thf(fact_591_size__Diff__singleton__if,axiom,
    ! [X: set_a,A: multiset_set_a] :
      ( ( ( member_set_a @ X @ ( set_mset_set_a @ A ) )
       => ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ A @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
          = ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ A ) @ one_one_nat ) ) )
      & ( ~ ( member_set_a @ X @ ( set_mset_set_a @ A ) )
       => ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ A @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
          = ( size_s6566526139600085008_set_a @ A ) ) ) ) ).

% size_Diff_singleton_if
thf(fact_592_size__Diff__singleton__if,axiom,
    ! [X: set_nat,A: multiset_set_nat] :
      ( ( ( member_set_nat @ X @ ( set_mset_set_nat @ A ) )
       => ( ( size_s7462436076474991978et_nat @ ( minus_7237264121398869807et_nat @ A @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) ) )
          = ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ A ) @ one_one_nat ) ) )
      & ( ~ ( member_set_nat @ X @ ( set_mset_set_nat @ A ) )
       => ( ( size_s7462436076474991978et_nat @ ( minus_7237264121398869807et_nat @ A @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) ) )
          = ( size_s7462436076474991978et_nat @ A ) ) ) ) ).

% size_Diff_singleton_if
thf(fact_593_size__remove1__mset__If,axiom,
    ! [M: multiset_a,X: a] :
      ( ( size_size_multiset_a @ ( minus_3765977307040488491iset_a @ M @ ( add_mset_a @ X @ zero_zero_multiset_a ) ) )
      = ( minus_minus_nat @ ( size_size_multiset_a @ M ) @ ( if_nat @ ( member_a @ X @ ( set_mset_a @ M ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).

% size_remove1_mset_If
thf(fact_594_size__remove1__mset__If,axiom,
    ! [M: multiset_nat,X: nat] :
      ( ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M @ ( add_mset_nat @ X @ zero_z7348594199698428585et_nat ) ) )
      = ( minus_minus_nat @ ( size_s5917832649809541300et_nat @ M ) @ ( if_nat @ ( member_nat @ X @ ( set_mset_nat @ M ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).

% size_remove1_mset_If
thf(fact_595_size__remove1__mset__If,axiom,
    ! [M: multiset_set_a,X: set_a] :
      ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ ( add_mset_set_a @ X @ zero_z5079479921072680283_set_a ) ) )
      = ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M ) @ ( if_nat @ ( member_set_a @ X @ ( set_mset_set_a @ M ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).

% size_remove1_mset_If
thf(fact_596_size__remove1__mset__If,axiom,
    ! [M: multiset_set_nat,X: set_nat] :
      ( ( size_s7462436076474991978et_nat @ ( minus_7237264121398869807et_nat @ M @ ( add_mset_set_nat @ X @ zero_z3157962936165190495et_nat ) ) )
      = ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ M ) @ ( if_nat @ ( member_set_nat @ X @ ( set_mset_set_nat @ M ) ) @ one_one_nat @ zero_zero_nat ) ) ) ).

% size_remove1_mset_If
thf(fact_597_add__le__imp__le__diff,axiom,
    ! [I: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_598_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K2: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_599_replication__number__single,axiom,
    is_singleton_nat @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).

% replication_number_single
thf(fact_600_n__inter__num__zero,axiom,
    ! [B1: set_a,B2: set_a] :
      ( ( member_set_a @ B1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( member_set_a @ B2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( design735257067508376852mber_a @ B1 @ zero_zero_nat @ B2 )
          = one_one_nat ) ) ) ).

% n_inter_num_zero
thf(fact_601_dual__sys_Ostr__del__block__del__point,axiom,
    ! [X: nat] :
      ( ~ ( member_set_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( design3550126062406151447ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) ) ) ).

% dual_sys.str_del_block_del_point
thf(fact_602_points__index__empty,axiom,
    ! [Ps: set_nat] :
      ( ( design6574611146354332593ex_nat @ zero_z3157962936165190495et_nat @ Ps )
      = zero_zero_nat ) ).

% points_index_empty
thf(fact_603_points__index__empty,axiom,
    ! [Ps: set_a] :
      ( ( design254580327166089565ndex_a @ zero_z5079479921072680283_set_a @ Ps )
      = zero_zero_nat ) ).

% points_index_empty
thf(fact_604_add__block__sub__des,axiom,
    ! [B: set_a] :
      ( ( finite_finite_a @ B )
     => ( ( B != bot_bot_set_a )
       => ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( sup_sup_set_a @ ( set_a2 @ v_s ) @ B ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) ) ) ) ).

% add_block_sub_des
thf(fact_605_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% diff_is_0_eq
thf(fact_606_add__block__index__not__in,axiom,
    ! [Ps: set_a,B: set_a] :
      ( ~ ( ord_less_eq_set_a @ Ps @ B )
     => ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) @ Ps )
        = ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) ) ) ).

% add_block_index_not_in
thf(fact_607_add__block__index__in,axiom,
    ! [Ps: set_a,B: set_a] :
      ( ( ord_less_eq_set_a @ Ps @ B )
     => ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) @ Ps )
        = ( plus_plus_nat @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) @ one_one_nat ) ) ) ).

% add_block_index_in
thf(fact_608_points__index__ps__nin,axiom,
    ! [Ps: set_a] :
      ( ~ ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
     => ( ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps )
        = zero_zero_nat ) ) ).

% points_index_ps_nin
thf(fact_609_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).

% bot_nat_0.extremum
thf(fact_610_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_611_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_612_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_613_delete__point__index__eq,axiom,
    ! [Ps: set_a,P: a] :
      ( ( ord_less_eq_set_a @ Ps @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P ) )
     => ( ( design254580327166089565ndex_a @ ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ P ) @ Ps )
        = ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) ) ) ).

% delete_point_index_eq
thf(fact_614_add__block__sub__sys,axiom,
    ! [B: set_a] : ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( sup_sup_set_a @ ( set_a2 @ v_s ) @ B ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) ) ).

% add_block_sub_sys
thf(fact_615_add__block__fin,axiom,
    ! [B: set_a] :
      ( ( finite_finite_a @ B )
     => ( design9187838744727572296stem_a @ ( sup_sup_set_a @ ( set_a2 @ v_s ) @ B ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) ) ) ).

% add_block_fin
thf(fact_616_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_617_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_618_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_619_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).

% Nat.add_diff_assoc
thf(fact_620_set__mset__union,axiom,
    ! [M: multiset_set_nat,N2: multiset_set_nat] :
      ( ( set_mset_set_nat @ ( plus_p8712254050562127327et_nat @ M @ N2 ) )
      = ( sup_sup_set_set_nat @ ( set_mset_set_nat @ M ) @ ( set_mset_set_nat @ N2 ) ) ) ).

% set_mset_union
thf(fact_621_set__mset__union,axiom,
    ! [M: multiset_set_a,N2: multiset_set_a] :
      ( ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ M @ N2 ) )
      = ( sup_sup_set_set_a @ ( set_mset_set_a @ M ) @ ( set_mset_set_a @ N2 ) ) ) ).

% set_mset_union
thf(fact_622_set__mset__union,axiom,
    ! [M: multiset_a,N2: multiset_a] :
      ( ( set_mset_a @ ( plus_plus_multiset_a @ M @ N2 ) )
      = ( sup_sup_set_a @ ( set_mset_a @ M ) @ ( set_mset_a @ N2 ) ) ) ).

% set_mset_union
thf(fact_623_set__mset__union,axiom,
    ! [M: multiset_nat,N2: multiset_nat] :
      ( ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ M @ N2 ) )
      = ( sup_sup_set_nat @ ( set_mset_nat @ M ) @ ( set_mset_nat @ N2 ) ) ) ).

% set_mset_union
thf(fact_624_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K2: nat,B: nat] :
      ( ( P2 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X4: nat] :
            ( ( P2 @ X4 )
            & ! [Y4: nat] :
                ( ( P2 @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_625_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M7: nat,N5: nat] :
        ? [K4: nat] :
          ( N5
          = ( plus_plus_nat @ M7 @ K4 ) ) ) ) ).

% nat_le_iff_add
thf(fact_626_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_627_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_628_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_629_add__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_le_mono1
thf(fact_630_add__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L4: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K2 @ L4 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L4 ) ) ) ) ).

% add_le_mono
thf(fact_631_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_632_le__Suc__ex,axiom,
    ! [K2: nat,L4: nat] :
      ( ( ord_less_eq_nat @ K2 @ L4 )
     => ? [N6: nat] :
          ( L4
          = ( plus_plus_nat @ K2 @ N6 ) ) ) ).

% le_Suc_ex
thf(fact_633_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_634_le__trans,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K2 )
       => ( ord_less_eq_nat @ I @ K2 ) ) ) ).

% le_trans
thf(fact_635_add__leD2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ( ord_less_eq_nat @ K2 @ N ) ) ).

% add_leD2
thf(fact_636_add__leD1,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% add_leD1
thf(fact_637_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_638_le__add2,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).

% le_add2
thf(fact_639_le__add1,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).

% le_add1
thf(fact_640_add__leE,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ~ ( ( ord_less_eq_nat @ M2 @ N )
         => ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).

% add_leE
thf(fact_641_n__inter__num__one,axiom,
    ! [B1: set_set_nat,B2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B1 )
     => ( ( finite1152437895449049373et_nat @ B2 )
       => ( ( design1060797113401760752et_nat @ B1 @ one_one_nat @ B2 )
          = ( design6421228673553131784et_nat @ B1 @ B2 ) ) ) ) ).

% n_inter_num_one
thf(fact_642_n__inter__num__one,axiom,
    ! [B1: set_set_a,B2: set_set_a] :
      ( ( finite_finite_set_a @ B1 )
     => ( ( finite_finite_set_a @ B2 )
       => ( ( design3674606213912786548_set_a @ B1 @ one_one_nat @ B2 )
          = ( design3520961687418077020_set_a @ B1 @ B2 ) ) ) ) ).

% n_inter_num_one
thf(fact_643_n__inter__num__one,axiom,
    ! [B1: set_a,B2: set_a] :
      ( ( finite_finite_a @ B1 )
     => ( ( finite_finite_a @ B2 )
       => ( ( design735257067508376852mber_a @ B1 @ one_one_nat @ B2 )
          = ( design7842873109100088828mber_a @ B1 @ B2 ) ) ) ) ).

% n_inter_num_one
thf(fact_644_n__inter__num__one,axiom,
    ! [B1: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ B1 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( design5554526424970975290er_nat @ B1 @ one_one_nat @ B2 )
          = ( design7485525362727208274er_nat @ B1 @ B2 ) ) ) ) ).

% n_inter_num_one
thf(fact_645_intersect__num__commute,axiom,
    ( design7842873109100088828mber_a
    = ( ^ [B13: set_a,B23: set_a] : ( design7842873109100088828mber_a @ B23 @ B13 ) ) ) ).

% intersect_num_commute
thf(fact_646_intersect__num__commute,axiom,
    ( design7485525362727208274er_nat
    = ( ^ [B13: set_nat,B23: set_nat] : ( design7485525362727208274er_nat @ B23 @ B13 ) ) ) ).

% intersect_num_commute
thf(fact_647_incidence__system_Ointersection__numbers_Ocong,axiom,
    design3761797438660848528bers_a = design3761797438660848528bers_a ).

% incidence_system.intersection_numbers.cong
thf(fact_648_incidence__system_Ointersection__numbers_Ocong,axiom,
    design9164904592607734462rs_nat = design9164904592607734462rs_nat ).

% incidence_system.intersection_numbers.cong
thf(fact_649_incidence__system_Odesign__support_Ocong,axiom,
    design5397942185814921632port_a = design5397942185814921632port_a ).

% incidence_system.design_support.cong
thf(fact_650_incidence__system_Odesign__support_Ocong,axiom,
    design4862117536649126062rt_nat = design4862117536649126062rt_nat ).

% incidence_system.design_support.cong
thf(fact_651_incidence__system_Oincident_Ocong,axiom,
    design3210447939978979927dent_a = design3210447939978979927dent_a ).

% incidence_system.incident.cong
thf(fact_652_incidence__system_Oincident_Ocong,axiom,
    design8502206366797944887nt_nat = design8502206366797944887nt_nat ).

% incidence_system.incident.cong
thf(fact_653_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_654_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
      = ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_655_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
     => ( A2 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_656_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_657_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K2 )
        = ( J
          = ( plus_plus_nat @ K2 @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_658_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_659_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_660_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_661_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L4: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L4 @ N ) @ ( minus_minus_nat @ L4 @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_662_le__diff__iff_H,axiom,
    ! [A2: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A2 @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A2 ) ) ) ) ).

% le_diff_iff'
thf(fact_663_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).

% le_diff_conv
thf(fact_664_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_665_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L4: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L4 ) @ ( minus_minus_nat @ N @ L4 ) ) ) ).

% diff_le_mono
thf(fact_666_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_667_le__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_668_eq__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ( minus_minus_nat @ M2 @ K2 )
            = ( minus_minus_nat @ N @ K2 ) )
          = ( M2 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_669_finite__incidence__system_Ofinite__sets,axiom,
    ! [Point_set: set_set_nat,Block_collection: multiset_set_set_nat] :
      ( ( design4015805878629997756et_nat @ Point_set @ Block_collection )
     => ( finite1152437895449049373et_nat @ Point_set ) ) ).

% finite_incidence_system.finite_sets
thf(fact_670_finite__incidence__system_Ofinite__sets,axiom,
    ! [Point_set: set_set_a,Block_collection: multiset_set_set_a] :
      ( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
     => ( finite_finite_set_a @ Point_set ) ) ).

% finite_incidence_system.finite_sets
thf(fact_671_finite__incidence__system_Ofinite__sets,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a] :
      ( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
     => ( finite_finite_a @ Point_set ) ) ).

% finite_incidence_system.finite_sets
thf(fact_672_finite__incidence__system_Ofinite__sets,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat] :
      ( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
     => ( finite_finite_nat @ Point_set ) ) ).

% finite_incidence_system.finite_sets
thf(fact_673_point__index__distrib,axiom,
    ! [B14: multiset_set_nat,B24: multiset_set_nat,Ps: set_nat] :
      ( ( design6574611146354332593ex_nat @ ( plus_p8712254050562127327et_nat @ B14 @ B24 ) @ Ps )
      = ( plus_plus_nat @ ( design6574611146354332593ex_nat @ B14 @ Ps ) @ ( design6574611146354332593ex_nat @ B24 @ Ps ) ) ) ).

% point_index_distrib
thf(fact_674_point__index__distrib,axiom,
    ! [B14: multiset_set_a,B24: multiset_set_a,Ps: set_a] :
      ( ( design254580327166089565ndex_a @ ( plus_p2331992037799027419_set_a @ B14 @ B24 ) @ Ps )
      = ( plus_plus_nat @ ( design254580327166089565ndex_a @ B14 @ Ps ) @ ( design254580327166089565ndex_a @ B24 @ Ps ) ) ) ).

% point_index_distrib
thf(fact_675_finite__incidence__system_Ocomplement__finite,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a] :
      ( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
     => ( design9187838744727572296stem_a @ Point_set @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) ) ).

% finite_incidence_system.complement_finite
thf(fact_676_finite__incidence__system_Ocomplement__finite,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat] :
      ( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
     => ( design5426232790142929158em_nat @ Point_set @ ( design5569578106646884273ks_nat @ Point_set @ Block_collection ) ) ) ).

% finite_incidence_system.complement_finite
thf(fact_677_simple__design_Oaxioms_I2_J,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a] :
      ( ( design3982635895484621246sign_a @ Point_set @ Block_collection )
     => ( design1338723777345758283stem_a @ Point_set @ Block_collection ) ) ).

% simple_design.axioms(2)
thf(fact_678_simple__design_Oaxioms_I2_J,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat] :
      ( ( design7861764274488435984gn_nat @ Point_set @ Block_collection )
     => ( design164292856788568387em_nat @ Point_set @ Block_collection ) ) ).

% simple_design.axioms(2)
thf(fact_679_intersection__num__non__neg,axiom,
    ! [B1: set_a,B2: set_a] : ( ord_less_eq_nat @ zero_zero_nat @ ( design7842873109100088828mber_a @ B1 @ B2 ) ) ).

% intersection_num_non_neg
thf(fact_680_intersection__num__non__neg,axiom,
    ! [B1: set_nat,B2: set_nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( design7485525362727208274er_nat @ B1 @ B2 ) ) ).

% intersection_num_non_neg
thf(fact_681_finite__incidence__system_Ofinite__blocks,axiom,
    ! [Point_set: set_set_nat,Block_collection: multiset_set_set_nat,B: set_set_nat] :
      ( ( design4015805878629997756et_nat @ Point_set @ Block_collection )
     => ( ( member_set_set_nat @ B @ ( set_mset_set_set_nat @ Block_collection ) )
       => ( finite1152437895449049373et_nat @ B ) ) ) ).

% finite_incidence_system.finite_blocks
thf(fact_682_finite__incidence__system_Ofinite__blocks,axiom,
    ! [Point_set: set_set_a,Block_collection: multiset_set_set_a,B: set_set_a] :
      ( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
     => ( ( member_set_set_a @ B @ ( set_mset_set_set_a @ Block_collection ) )
       => ( finite_finite_set_a @ B ) ) ) ).

% finite_incidence_system.finite_blocks
thf(fact_683_finite__incidence__system_Ofinite__blocks,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
      ( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
     => ( ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
       => ( finite_finite_a @ B ) ) ) ).

% finite_incidence_system.finite_blocks
thf(fact_684_finite__incidence__system_Ofinite__blocks,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat] :
      ( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
     => ( ( member_set_nat @ B @ ( set_mset_set_nat @ Block_collection ) )
       => ( finite_finite_nat @ B ) ) ) ).

% finite_incidence_system.finite_blocks
thf(fact_685_point__index__diff,axiom,
    ! [B24: multiset_set_nat] :
      ( design6574611146354332593ex_nat
      = ( ^ [B15: multiset_set_nat,Ps2: set_nat] : ( minus_minus_nat @ ( design6574611146354332593ex_nat @ ( plus_p8712254050562127327et_nat @ B15 @ B24 ) @ Ps2 ) @ ( design6574611146354332593ex_nat @ B24 @ Ps2 ) ) ) ) ).

% point_index_diff
thf(fact_686_point__index__diff,axiom,
    ! [B24: multiset_set_a] :
      ( design254580327166089565ndex_a
      = ( ^ [B15: multiset_set_a,Ps2: set_a] : ( minus_minus_nat @ ( design254580327166089565ndex_a @ ( plus_p2331992037799027419_set_a @ B15 @ B24 ) @ Ps2 ) @ ( design254580327166089565ndex_a @ B24 @ Ps2 ) ) ) ) ).

% point_index_diff
thf(fact_687_finite__incidence__system_Oreplication__numbers__finite,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a] :
      ( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
     => ( finite_finite_nat @ ( design8835372594653258411bers_a @ Point_set @ Block_collection ) ) ) ).

% finite_incidence_system.replication_numbers_finite
thf(fact_688_finite__incidence__system_Oreplication__numbers__finite,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat] :
      ( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
     => ( finite_finite_nat @ ( design3853898657598026467rs_nat @ Point_set @ Block_collection ) ) ) ).

% finite_incidence_system.replication_numbers_finite
thf(fact_689_finite__incidence__system_Ofinite__design__support,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a] :
      ( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
     => ( finite_finite_set_a @ ( design5397942185814921632port_a @ Block_collection ) ) ) ).

% finite_incidence_system.finite_design_support
thf(fact_690_finite__incidence__system_Ofinite__design__support,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat] :
      ( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
     => ( finite1152437895449049373et_nat @ ( design4862117536649126062rt_nat @ Block_collection ) ) ) ).

% finite_incidence_system.finite_design_support
thf(fact_691_points__index__0__right__imp,axiom,
    ! [B3: multiset_set_nat,Ps: set_nat] :
      ( ! [B6: set_nat] :
          ( ( member_set_nat @ B6 @ ( set_mset_set_nat @ B3 ) )
         => ~ ( ord_less_eq_set_nat @ Ps @ B6 ) )
     => ( ( design6574611146354332593ex_nat @ B3 @ Ps )
        = zero_zero_nat ) ) ).

% points_index_0_right_imp
thf(fact_692_points__index__0__right__imp,axiom,
    ! [B3: multiset_set_a,Ps: set_a] :
      ( ! [B6: set_a] :
          ( ( member_set_a @ B6 @ ( set_mset_set_a @ B3 ) )
         => ~ ( ord_less_eq_set_a @ Ps @ B6 ) )
     => ( ( design254580327166089565ndex_a @ B3 @ Ps )
        = zero_zero_nat ) ) ).

% points_index_0_right_imp
thf(fact_693_points__index__0__left__imp,axiom,
    ! [B3: multiset_set_nat,Ps: set_nat,B: set_nat] :
      ( ( ( design6574611146354332593ex_nat @ B3 @ Ps )
        = zero_zero_nat )
     => ( ( member_set_nat @ B @ ( set_mset_set_nat @ B3 ) )
       => ~ ( ord_less_eq_set_nat @ Ps @ B ) ) ) ).

% points_index_0_left_imp
thf(fact_694_points__index__0__left__imp,axiom,
    ! [B3: multiset_set_a,Ps: set_a,B: set_a] :
      ( ( ( design254580327166089565ndex_a @ B3 @ Ps )
        = zero_zero_nat )
     => ( ( member_set_a @ B @ ( set_mset_set_a @ B3 ) )
       => ~ ( ord_less_eq_set_a @ Ps @ B ) ) ) ).

% points_index_0_left_imp
thf(fact_695_points__index__0__iff,axiom,
    ! [B3: multiset_set_nat,Ps: set_nat] :
      ( ( ( design6574611146354332593ex_nat @ B3 @ Ps )
        = zero_zero_nat )
      = ( ! [B5: set_nat] :
            ( ( member_set_nat @ B5 @ ( set_mset_set_nat @ B3 ) )
           => ~ ( ord_less_eq_set_nat @ Ps @ B5 ) ) ) ) ).

% points_index_0_iff
thf(fact_696_points__index__0__iff,axiom,
    ! [B3: multiset_set_a,Ps: set_a] :
      ( ( ( design254580327166089565ndex_a @ B3 @ Ps )
        = zero_zero_nat )
      = ( ! [B5: set_a] :
            ( ( member_set_a @ B5 @ ( set_mset_set_a @ B3 ) )
           => ~ ( ord_less_eq_set_a @ Ps @ B5 ) ) ) ) ).

% points_index_0_iff
thf(fact_697_points__index__one__unique__block,axiom,
    ! [B3: multiset_set_nat,Ps: set_nat] :
      ( ( ( design6574611146354332593ex_nat @ B3 @ Ps )
        = one_one_nat )
     => ? [X4: set_nat] :
          ( ( member_set_nat @ X4 @ ( set_mset_set_nat @ B3 ) )
          & ( ord_less_eq_set_nat @ Ps @ X4 )
          & ! [Y4: set_nat] :
              ( ( ( member_set_nat @ Y4 @ ( set_mset_set_nat @ B3 ) )
                & ( ord_less_eq_set_nat @ Ps @ Y4 ) )
             => ( Y4 = X4 ) ) ) ) ).

% points_index_one_unique_block
thf(fact_698_points__index__one__unique__block,axiom,
    ! [B3: multiset_set_a,Ps: set_a] :
      ( ( ( design254580327166089565ndex_a @ B3 @ Ps )
        = one_one_nat )
     => ? [X4: set_a] :
          ( ( member_set_a @ X4 @ ( set_mset_set_a @ B3 ) )
          & ( ord_less_eq_set_a @ Ps @ X4 )
          & ! [Y4: set_a] :
              ( ( ( member_set_a @ Y4 @ ( set_mset_set_a @ B3 ) )
                & ( ord_less_eq_set_a @ Ps @ Y4 ) )
             => ( Y4 = X4 ) ) ) ) ).

% points_index_one_unique_block
thf(fact_699_points__index__one__unique,axiom,
    ! [B3: multiset_set_nat,Ps: set_nat,Bl2: set_nat,Bl3: set_nat] :
      ( ( ( design6574611146354332593ex_nat @ B3 @ Ps )
        = one_one_nat )
     => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ B3 ) )
       => ( ( ord_less_eq_set_nat @ Ps @ Bl2 )
         => ( ( member_set_nat @ Bl3 @ ( set_mset_set_nat @ B3 ) )
           => ( ( ord_less_eq_set_nat @ Ps @ Bl3 )
             => ( Bl2 = Bl3 ) ) ) ) ) ) ).

% points_index_one_unique
thf(fact_700_points__index__one__unique,axiom,
    ! [B3: multiset_set_a,Ps: set_a,Bl2: set_a,Bl3: set_a] :
      ( ( ( design254580327166089565ndex_a @ B3 @ Ps )
        = one_one_nat )
     => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ B3 ) )
       => ( ( ord_less_eq_set_a @ Ps @ Bl2 )
         => ( ( member_set_a @ Bl3 @ ( set_mset_set_a @ B3 ) )
           => ( ( ord_less_eq_set_a @ Ps @ Bl3 )
             => ( Bl2 = Bl3 ) ) ) ) ) ) ).

% points_index_one_unique
thf(fact_701_points__index__singleton__zero,axiom,
    ! [Ps: set_nat,B: set_nat] :
      ( ~ ( ord_less_eq_set_nat @ Ps @ B )
     => ( ( design6574611146354332593ex_nat @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) @ Ps )
        = zero_zero_nat ) ) ).

% points_index_singleton_zero
thf(fact_702_points__index__singleton__zero,axiom,
    ! [Ps: set_a,B: set_a] :
      ( ~ ( ord_less_eq_set_a @ Ps @ B )
     => ( ( design254580327166089565ndex_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) @ Ps )
        = zero_zero_nat ) ) ).

% points_index_singleton_zero
thf(fact_703_points__index__singleton,axiom,
    ! [B: set_nat,Ps: set_nat] :
      ( ( ( design6574611146354332593ex_nat @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) @ Ps )
        = one_one_nat )
      = ( ord_less_eq_set_nat @ Ps @ B ) ) ).

% points_index_singleton
thf(fact_704_points__index__singleton,axiom,
    ! [B: set_a,Ps: set_a] :
      ( ( ( design254580327166089565ndex_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) @ Ps )
        = one_one_nat )
      = ( ord_less_eq_set_a @ Ps @ B ) ) ).

% points_index_singleton
thf(fact_705_points__index__one__not__unique__block,axiom,
    ! [B3: multiset_set_nat,Ps: set_nat,Bl2: set_nat,Bl3: set_nat] :
      ( ( ( design6574611146354332593ex_nat @ B3 @ Ps )
        = one_one_nat )
     => ( ( ord_less_eq_set_nat @ Ps @ Bl2 )
       => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ B3 ) )
         => ( ( member_set_nat @ Bl3 @ ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ B3 @ ( add_mset_set_nat @ Bl2 @ zero_z3157962936165190495et_nat ) ) ) )
           => ~ ( ord_less_eq_set_nat @ Ps @ Bl3 ) ) ) ) ) ).

% points_index_one_not_unique_block
thf(fact_706_points__index__one__not__unique__block,axiom,
    ! [B3: multiset_set_a,Ps: set_a,Bl2: set_a,Bl3: set_a] :
      ( ( ( design254580327166089565ndex_a @ B3 @ Ps )
        = one_one_nat )
     => ( ( ord_less_eq_set_a @ Ps @ Bl2 )
       => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ B3 ) )
         => ( ( member_set_a @ Bl3 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ B3 @ ( add_mset_set_a @ Bl2 @ zero_z5079479921072680283_set_a ) ) ) )
           => ~ ( ord_less_eq_set_a @ Ps @ Bl3 ) ) ) ) ) ).

% points_index_one_not_unique_block
thf(fact_707_is__singletonI,axiom,
    ! [X: a] : ( is_singleton_a @ ( insert_a @ X @ bot_bot_set_a ) ) ).

% is_singletonI
thf(fact_708_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_709_finite__Diff__insert,axiom,
    ! [A: set_set_nat,A2: set_nat,B3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ B3 ) ) )
      = ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_710_finite__Diff__insert,axiom,
    ! [A: set_set_a,A2: set_a,B3: set_set_a] :
      ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ B3 ) ) )
      = ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_711_finite__Diff__insert,axiom,
    ! [A: set_a,A2: a,B3: set_a] :
      ( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B3 ) ) )
      = ( finite_finite_a @ ( minus_minus_set_a @ A @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_712_finite__Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B3: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B3 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B3 ) ) ) ).

% finite_Diff_insert
thf(fact_713_insert__Diff__single,axiom,
    ! [A2: a,A: set_a] :
      ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
      = ( insert_a @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_714_insert__Diff__single,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_715_Diff__eq__empty__iff,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A @ B3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_716_Diff__eq__empty__iff,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ( minus_minus_set_a @ A @ B3 )
        = bot_bot_set_a )
      = ( ord_less_eq_set_a @ A @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_717_singleton__insert__inj__eq_H,axiom,
    ! [A2: nat,A: set_nat,B: nat] :
      ( ( ( insert_nat @ A2 @ A )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_718_singleton__insert__inj__eq_H,axiom,
    ! [A2: a,A: set_a,B: a] :
      ( ( ( insert_a @ A2 @ A )
        = ( insert_a @ B @ bot_bot_set_a ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_719_singleton__insert__inj__eq,axiom,
    ! [B: nat,A2: nat,A: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_720_singleton__insert__inj__eq,axiom,
    ! [B: a,A2: a,A: set_a] :
      ( ( ( insert_a @ B @ bot_bot_set_a )
        = ( insert_a @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_a @ A @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_721_empty__iff,axiom,
    ! [C: set_a] :
      ~ ( member_set_a @ C @ bot_bot_set_set_a ) ).

% empty_iff
thf(fact_722_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_723_empty__iff,axiom,
    ! [C: a] :
      ~ ( member_a @ C @ bot_bot_set_a ) ).

% empty_iff
thf(fact_724_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_725_all__not__in__conv,axiom,
    ! [A: set_set_a] :
      ( ( ! [X3: set_a] :
            ~ ( member_set_a @ X3 @ A ) )
      = ( A = bot_bot_set_set_a ) ) ).

% all_not_in_conv
thf(fact_726_all__not__in__conv,axiom,
    ! [A: set_set_nat] :
      ( ( ! [X3: set_nat] :
            ~ ( member_set_nat @ X3 @ A ) )
      = ( A = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_727_all__not__in__conv,axiom,
    ! [A: set_a] :
      ( ( ! [X3: a] :
            ~ ( member_a @ X3 @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_728_all__not__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ! [X3: nat] :
            ~ ( member_nat @ X3 @ A ) )
      = ( A = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_729_Collect__empty__eq,axiom,
    ! [P2: a > $o] :
      ( ( ( collect_a @ P2 )
        = bot_bot_set_a )
      = ( ! [X3: a] :
            ~ ( P2 @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_730_Collect__empty__eq,axiom,
    ! [P2: nat > $o] :
      ( ( ( collect_nat @ P2 )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ~ ( P2 @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_731_empty__Collect__eq,axiom,
    ! [P2: a > $o] :
      ( ( bot_bot_set_a
        = ( collect_a @ P2 ) )
      = ( ! [X3: a] :
            ~ ( P2 @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_732_empty__Collect__eq,axiom,
    ! [P2: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P2 ) )
      = ( ! [X3: nat] :
            ~ ( P2 @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_733_subsetI,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ! [X4: set_a] :
          ( ( member_set_a @ X4 @ A )
         => ( member_set_a @ X4 @ B3 ) )
     => ( ord_le3724670747650509150_set_a @ A @ B3 ) ) ).

% subsetI
thf(fact_734_subsetI,axiom,
    ! [A: set_set_nat,B3: set_set_nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ A )
         => ( member_set_nat @ X4 @ B3 ) )
     => ( ord_le6893508408891458716et_nat @ A @ B3 ) ) ).

% subsetI
thf(fact_735_subsetI,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A )
         => ( member_nat @ X4 @ B3 ) )
     => ( ord_less_eq_set_nat @ A @ B3 ) ) ).

% subsetI
thf(fact_736_subsetI,axiom,
    ! [A: set_a,B3: set_a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A )
         => ( member_a @ X4 @ B3 ) )
     => ( ord_less_eq_set_a @ A @ B3 ) ) ).

% subsetI
thf(fact_737_subset__antisym,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A )
       => ( A = B3 ) ) ) ).

% subset_antisym
thf(fact_738_subset__antisym,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ A )
       => ( A = B3 ) ) ) ).

% subset_antisym
thf(fact_739_subset__empty,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_740_subset__empty,axiom,
    ! [A: set_a] :
      ( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
      = ( A = bot_bot_set_a ) ) ).

% subset_empty
thf(fact_741_empty__subsetI,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% empty_subsetI
thf(fact_742_empty__subsetI,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).

% empty_subsetI
thf(fact_743_singletonI,axiom,
    ! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).

% singletonI
thf(fact_744_singletonI,axiom,
    ! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_745_singletonI,axiom,
    ! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_746_singletonI,axiom,
    ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_747_insert__subset,axiom,
    ! [X: set_a,A: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A ) @ B3 )
      = ( ( member_set_a @ X @ B3 )
        & ( ord_le3724670747650509150_set_a @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_748_insert__subset,axiom,
    ! [X: set_nat,A: set_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X @ A ) @ B3 )
      = ( ( member_set_nat @ X @ B3 )
        & ( ord_le6893508408891458716et_nat @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_749_insert__subset,axiom,
    ! [X: nat,A: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B3 )
      = ( ( member_nat @ X @ B3 )
        & ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_750_insert__subset,axiom,
    ! [X: a,A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a @ X @ A ) @ B3 )
      = ( ( member_a @ X @ B3 )
        & ( ord_less_eq_set_a @ A @ B3 ) ) ) ).

% insert_subset
thf(fact_751_finite__insert,axiom,
    ! [A2: set_nat,A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ ( insert_set_nat @ A2 @ A ) )
      = ( finite1152437895449049373et_nat @ A ) ) ).

% finite_insert
thf(fact_752_finite__insert,axiom,
    ! [A2: a,A: set_a] :
      ( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
      = ( finite_finite_a @ A ) ) ).

% finite_insert
thf(fact_753_finite__insert,axiom,
    ! [A2: set_a,A: set_set_a] :
      ( ( finite_finite_set_a @ ( insert_set_a @ A2 @ A ) )
      = ( finite_finite_set_a @ A ) ) ).

% finite_insert
thf(fact_754_finite__insert,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
      = ( finite_finite_nat @ A ) ) ).

% finite_insert
thf(fact_755_Diff__empty,axiom,
    ! [A: set_a] :
      ( ( minus_minus_set_a @ A @ bot_bot_set_a )
      = A ) ).

% Diff_empty
thf(fact_756_Diff__empty,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Diff_empty
thf(fact_757_empty__Diff,axiom,
    ! [A: set_a] :
      ( ( minus_minus_set_a @ bot_bot_set_a @ A )
      = bot_bot_set_a ) ).

% empty_Diff
thf(fact_758_empty__Diff,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_759_Diff__cancel,axiom,
    ! [A: set_a] :
      ( ( minus_minus_set_a @ A @ A )
      = bot_bot_set_a ) ).

% Diff_cancel
thf(fact_760_Diff__cancel,axiom,
    ! [A: set_nat] :
      ( ( minus_minus_set_nat @ A @ A )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_761_Un__empty,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ( sup_sup_set_a @ A @ B3 )
        = bot_bot_set_a )
      = ( ( A = bot_bot_set_a )
        & ( B3 = bot_bot_set_a ) ) ) ).

% Un_empty
thf(fact_762_Un__empty,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B3 )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B3 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_763_Un__subset__iff,axiom,
    ! [A: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B3 ) @ C2 )
      = ( ( ord_less_eq_set_nat @ A @ C2 )
        & ( ord_less_eq_set_nat @ B3 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_764_Un__subset__iff,axiom,
    ! [A: set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B3 ) @ C2 )
      = ( ( ord_less_eq_set_a @ A @ C2 )
        & ( ord_less_eq_set_a @ B3 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_765_finite__Diff,axiom,
    ! [A: set_set_nat,B3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B3 ) ) ) ).

% finite_Diff
thf(fact_766_finite__Diff,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B3 ) ) ) ).

% finite_Diff
thf(fact_767_finite__Diff,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( finite_finite_a @ A )
     => ( finite_finite_a @ ( minus_minus_set_a @ A @ B3 ) ) ) ).

% finite_Diff
thf(fact_768_finite__Diff,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B3 ) ) ) ).

% finite_Diff
thf(fact_769_finite__Diff2,axiom,
    ! [B3: set_set_nat,A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B3 ) )
        = ( finite1152437895449049373et_nat @ A ) ) ) ).

% finite_Diff2
thf(fact_770_finite__Diff2,axiom,
    ! [B3: set_set_a,A: set_set_a] :
      ( ( finite_finite_set_a @ B3 )
     => ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B3 ) )
        = ( finite_finite_set_a @ A ) ) ) ).

% finite_Diff2
thf(fact_771_finite__Diff2,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( finite_finite_a @ B3 )
     => ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B3 ) )
        = ( finite_finite_a @ A ) ) ) ).

% finite_Diff2
thf(fact_772_finite__Diff2,axiom,
    ! [B3: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B3 ) )
        = ( finite_finite_nat @ A ) ) ) ).

% finite_Diff2
thf(fact_773_finite__Un,axiom,
    ! [F: set_set_nat,G: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F @ G ) )
      = ( ( finite1152437895449049373et_nat @ F )
        & ( finite1152437895449049373et_nat @ G ) ) ) ).

% finite_Un
thf(fact_774_finite__Un,axiom,
    ! [F: set_set_a,G: set_set_a] :
      ( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F @ G ) )
      = ( ( finite_finite_set_a @ F )
        & ( finite_finite_set_a @ G ) ) ) ).

% finite_Un
thf(fact_775_finite__Un,axiom,
    ! [F: set_a,G: set_a] :
      ( ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) )
      = ( ( finite_finite_a @ F )
        & ( finite_finite_a @ G ) ) ) ).

% finite_Un
thf(fact_776_finite__Un,axiom,
    ! [F: set_nat,G: set_nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) )
      = ( ( finite_finite_nat @ F )
        & ( finite_finite_nat @ G ) ) ) ).

% finite_Un
thf(fact_777_emptyE,axiom,
    ! [A2: set_a] :
      ~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).

% emptyE
thf(fact_778_emptyE,axiom,
    ! [A2: set_nat] :
      ~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_779_emptyE,axiom,
    ! [A2: a] :
      ~ ( member_a @ A2 @ bot_bot_set_a ) ).

% emptyE
thf(fact_780_emptyE,axiom,
    ! [A2: nat] :
      ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).

% emptyE
thf(fact_781_equals0D,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( A = bot_bot_set_set_a )
     => ~ ( member_set_a @ A2 @ A ) ) ).

% equals0D
thf(fact_782_equals0D,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( A = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_783_equals0D,axiom,
    ! [A: set_a,A2: a] :
      ( ( A = bot_bot_set_a )
     => ~ ( member_a @ A2 @ A ) ) ).

% equals0D
thf(fact_784_equals0D,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( A = bot_bot_set_nat )
     => ~ ( member_nat @ A2 @ A ) ) ).

% equals0D
thf(fact_785_equals0I,axiom,
    ! [A: set_set_a] :
      ( ! [Y3: set_a] :
          ~ ( member_set_a @ Y3 @ A )
     => ( A = bot_bot_set_set_a ) ) ).

% equals0I
thf(fact_786_equals0I,axiom,
    ! [A: set_set_nat] :
      ( ! [Y3: set_nat] :
          ~ ( member_set_nat @ Y3 @ A )
     => ( A = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_787_equals0I,axiom,
    ! [A: set_a] :
      ( ! [Y3: a] :
          ~ ( member_a @ Y3 @ A )
     => ( A = bot_bot_set_a ) ) ).

% equals0I
thf(fact_788_equals0I,axiom,
    ! [A: set_nat] :
      ( ! [Y3: nat] :
          ~ ( member_nat @ Y3 @ A )
     => ( A = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_789_ex__in__conv,axiom,
    ! [A: set_set_a] :
      ( ( ? [X3: set_a] : ( member_set_a @ X3 @ A ) )
      = ( A != bot_bot_set_set_a ) ) ).

% ex_in_conv
thf(fact_790_ex__in__conv,axiom,
    ! [A: set_set_nat] :
      ( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
      = ( A != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_791_ex__in__conv,axiom,
    ! [A: set_a] :
      ( ( ? [X3: a] : ( member_a @ X3 @ A ) )
      = ( A != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_792_ex__in__conv,axiom,
    ! [A: set_nat] :
      ( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
      = ( A != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_793_in__mono,axiom,
    ! [A: set_set_a,B3: set_set_a,X: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( member_set_a @ X @ A )
       => ( member_set_a @ X @ B3 ) ) ) ).

% in_mono
thf(fact_794_in__mono,axiom,
    ! [A: set_set_nat,B3: set_set_nat,X: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B3 )
     => ( ( member_set_nat @ X @ A )
       => ( member_set_nat @ X @ B3 ) ) ) ).

% in_mono
thf(fact_795_in__mono,axiom,
    ! [A: set_nat,B3: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( member_nat @ X @ A )
       => ( member_nat @ X @ B3 ) ) ) ).

% in_mono
thf(fact_796_in__mono,axiom,
    ! [A: set_a,B3: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( member_a @ X @ A )
       => ( member_a @ X @ B3 ) ) ) ).

% in_mono
thf(fact_797_subsetD,axiom,
    ! [A: set_set_a,B3: set_set_a,C: set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( member_set_a @ C @ A )
       => ( member_set_a @ C @ B3 ) ) ) ).

% subsetD
thf(fact_798_subsetD,axiom,
    ! [A: set_set_nat,B3: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B3 )
     => ( ( member_set_nat @ C @ A )
       => ( member_set_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_799_subsetD,axiom,
    ! [A: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( member_nat @ C @ A )
       => ( member_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_800_subsetD,axiom,
    ! [A: set_a,B3: set_a,C: a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( member_a @ C @ A )
       => ( member_a @ C @ B3 ) ) ) ).

% subsetD
thf(fact_801_equalityE,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( A = B3 )
     => ~ ( ( ord_less_eq_set_nat @ A @ B3 )
         => ~ ( ord_less_eq_set_nat @ B3 @ A ) ) ) ).

% equalityE
thf(fact_802_equalityE,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( A = B3 )
     => ~ ( ( ord_less_eq_set_a @ A @ B3 )
         => ~ ( ord_less_eq_set_a @ B3 @ A ) ) ) ).

% equalityE
thf(fact_803_subset__eq,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A6: set_set_a,B7: set_set_a] :
        ! [X3: set_a] :
          ( ( member_set_a @ X3 @ A6 )
         => ( member_set_a @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_804_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
        ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A6 )
         => ( member_set_nat @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_805_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ A6 )
         => ( member_nat @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_806_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B7: set_a] :
        ! [X3: a] :
          ( ( member_a @ X3 @ A6 )
         => ( member_a @ X3 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_807_equalityD1,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( A = B3 )
     => ( ord_less_eq_set_nat @ A @ B3 ) ) ).

% equalityD1
thf(fact_808_equalityD1,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( A = B3 )
     => ( ord_less_eq_set_a @ A @ B3 ) ) ).

% equalityD1
thf(fact_809_equalityD2,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( A = B3 )
     => ( ord_less_eq_set_nat @ B3 @ A ) ) ).

% equalityD2
thf(fact_810_equalityD2,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( A = B3 )
     => ( ord_less_eq_set_a @ B3 @ A ) ) ).

% equalityD2
thf(fact_811_subset__iff,axiom,
    ( ord_le3724670747650509150_set_a
    = ( ^ [A6: set_set_a,B7: set_set_a] :
        ! [T: set_a] :
          ( ( member_set_a @ T @ A6 )
         => ( member_set_a @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_812_subset__iff,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A6: set_set_nat,B7: set_set_nat] :
        ! [T: set_nat] :
          ( ( member_set_nat @ T @ A6 )
         => ( member_set_nat @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_813_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
        ! [T: nat] :
          ( ( member_nat @ T @ A6 )
         => ( member_nat @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_814_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B7: set_a] :
        ! [T: a] :
          ( ( member_a @ T @ A6 )
         => ( member_a @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_815_subset__refl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% subset_refl
thf(fact_816_subset__refl,axiom,
    ! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).

% subset_refl
thf(fact_817_Collect__mono,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P2 @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_818_Collect__mono,axiom,
    ! [P2: a > $o,Q: a > $o] :
      ( ! [X4: a] :
          ( ( P2 @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).

% Collect_mono
thf(fact_819_subset__trans,axiom,
    ! [A: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_820_subset__trans,axiom,
    ! [A: set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ord_less_eq_set_a @ A @ C2 ) ) ) ).

% subset_trans
thf(fact_821_set__eq__subset,axiom,
    ( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_822_set__eq__subset,axiom,
    ( ( ^ [Y5: set_a,Z: set_a] : ( Y5 = Z ) )
    = ( ^ [A6: set_a,B7: set_a] :
          ( ( ord_less_eq_set_a @ A6 @ B7 )
          & ( ord_less_eq_set_a @ B7 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_823_Collect__mono__iff,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
      = ( ! [X3: nat] :
            ( ( P2 @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_824_Collect__mono__iff,axiom,
    ! [P2: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
      = ( ! [X3: a] :
            ( ( P2 @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_825_finite__has__maximal2,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A2 @ A )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A )
            & ( ord_less_eq_set_nat @ A2 @ X4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_826_finite__has__maximal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ( ord_less_eq_set_a @ A2 @ X4 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_827_finite__has__maximal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ( ord_less_eq_nat @ A2 @ X4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_828_finite__has__minimal2,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( member_set_nat @ A2 @ A )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A )
            & ( ord_less_eq_set_nat @ X4 @ A2 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_829_finite__has__minimal2,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( member_set_a @ A2 @ A )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ( ord_less_eq_set_a @ X4 @ A2 )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_830_finite__has__minimal2,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( ( member_nat @ A2 @ A )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ( ord_less_eq_nat @ X4 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_831_finite_OemptyI,axiom,
    finite1152437895449049373et_nat @ bot_bot_set_set_nat ).

% finite.emptyI
thf(fact_832_finite_OemptyI,axiom,
    finite_finite_set_a @ bot_bot_set_set_a ).

% finite.emptyI
thf(fact_833_finite_OemptyI,axiom,
    finite_finite_a @ bot_bot_set_a ).

% finite.emptyI
thf(fact_834_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_835_infinite__imp__nonempty,axiom,
    ! [S: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ S )
     => ( S != bot_bot_set_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_836_infinite__imp__nonempty,axiom,
    ! [S: set_set_a] :
      ( ~ ( finite_finite_set_a @ S )
     => ( S != bot_bot_set_set_a ) ) ).

% infinite_imp_nonempty
thf(fact_837_infinite__imp__nonempty,axiom,
    ! [S: set_a] :
      ( ~ ( finite_finite_a @ S )
     => ( S != bot_bot_set_a ) ) ).

% infinite_imp_nonempty
thf(fact_838_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_839_finite__subset,axiom,
    ! [A: set_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ B3 )
     => ( ( finite1152437895449049373et_nat @ B3 )
       => ( finite1152437895449049373et_nat @ A ) ) ) ).

% finite_subset
thf(fact_840_finite__subset,axiom,
    ! [A: set_set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ B3 )
     => ( ( finite_finite_set_a @ B3 )
       => ( finite_finite_set_a @ A ) ) ) ).

% finite_subset
thf(fact_841_finite__subset,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( finite_finite_nat @ B3 )
       => ( finite_finite_nat @ A ) ) ) ).

% finite_subset
thf(fact_842_finite__subset,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( finite_finite_a @ B3 )
       => ( finite_finite_a @ A ) ) ) ).

% finite_subset
thf(fact_843_infinite__super,axiom,
    ! [S: set_set_nat,T2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ S @ T2 )
     => ( ~ ( finite1152437895449049373et_nat @ S )
       => ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_844_infinite__super,axiom,
    ! [S: set_set_a,T2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ S @ T2 )
     => ( ~ ( finite_finite_set_a @ S )
       => ~ ( finite_finite_set_a @ T2 ) ) ) ).

% infinite_super
thf(fact_845_infinite__super,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_super
thf(fact_846_infinite__super,axiom,
    ! [S: set_a,T2: set_a] :
      ( ( ord_less_eq_set_a @ S @ T2 )
     => ( ~ ( finite_finite_a @ S )
       => ~ ( finite_finite_a @ T2 ) ) ) ).

% infinite_super
thf(fact_847_rev__finite__subset,axiom,
    ! [B3: set_set_nat,A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ A @ B3 )
       => ( finite1152437895449049373et_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_848_rev__finite__subset,axiom,
    ! [B3: set_set_a,A: set_set_a] :
      ( ( finite_finite_set_a @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ A @ B3 )
       => ( finite_finite_set_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_849_rev__finite__subset,axiom,
    ! [B3: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A @ B3 )
       => ( finite_finite_nat @ A ) ) ) ).

% rev_finite_subset
thf(fact_850_rev__finite__subset,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( finite_finite_a @ B3 )
     => ( ( ord_less_eq_set_a @ A @ B3 )
       => ( finite_finite_a @ A ) ) ) ).

% rev_finite_subset
thf(fact_851_singletonD,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( member_set_a @ B @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_852_singletonD,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_853_singletonD,axiom,
    ! [B: a,A2: a] :
      ( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_854_singletonD,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_855_singleton__iff,axiom,
    ! [B: set_a,A2: set_a] :
      ( ( member_set_a @ B @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_856_singleton__iff,axiom,
    ! [B: set_nat,A2: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_857_singleton__iff,axiom,
    ! [B: a,A2: a] :
      ( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_858_singleton__iff,axiom,
    ! [B: nat,A2: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_859_doubleton__eq__iff,axiom,
    ! [A2: a,B: a,C: a,D: a] :
      ( ( ( insert_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) )
        = ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_860_doubleton__eq__iff,axiom,
    ! [A2: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A2 = C )
          & ( B = D ) )
        | ( ( A2 = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_861_insert__not__empty,axiom,
    ! [A2: a,A: set_a] :
      ( ( insert_a @ A2 @ A )
     != bot_bot_set_a ) ).

% insert_not_empty
thf(fact_862_insert__not__empty,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( insert_nat @ A2 @ A )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_863_singleton__inject,axiom,
    ! [A2: a,B: a] :
      ( ( ( insert_a @ A2 @ bot_bot_set_a )
        = ( insert_a @ B @ bot_bot_set_a ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_864_singleton__inject,axiom,
    ! [A2: nat,B: nat] :
      ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_865_insert__mono,axiom,
    ! [C2: set_nat,D2: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ C2 @ D2 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D2 ) ) ) ).

% insert_mono
thf(fact_866_insert__mono,axiom,
    ! [C2: set_a,D2: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ C2 @ D2 )
     => ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D2 ) ) ) ).

% insert_mono
thf(fact_867_subset__insert,axiom,
    ! [X: set_a,A: set_set_a,B3: set_set_a] :
      ( ~ ( member_set_a @ X @ A )
     => ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X @ B3 ) )
        = ( ord_le3724670747650509150_set_a @ A @ B3 ) ) ) ).

% subset_insert
thf(fact_868_subset__insert,axiom,
    ! [X: set_nat,A: set_set_nat,B3: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A )
     => ( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X @ B3 ) )
        = ( ord_le6893508408891458716et_nat @ A @ B3 ) ) ) ).

% subset_insert
thf(fact_869_subset__insert,axiom,
    ! [X: nat,A: set_nat,B3: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B3 ) )
        = ( ord_less_eq_set_nat @ A @ B3 ) ) ) ).

% subset_insert
thf(fact_870_subset__insert,axiom,
    ! [X: a,A: set_a,B3: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B3 ) )
        = ( ord_less_eq_set_a @ A @ B3 ) ) ) ).

% subset_insert
thf(fact_871_subset__insertI,axiom,
    ! [B3: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B3 @ ( insert_nat @ A2 @ B3 ) ) ).

% subset_insertI
thf(fact_872_subset__insertI,axiom,
    ! [B3: set_a,A2: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A2 @ B3 ) ) ).

% subset_insertI
thf(fact_873_subset__insertI2,axiom,
    ! [A: set_nat,B3: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_874_subset__insertI2,axiom,
    ! [A: set_a,B3: set_a,B: a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ord_less_eq_set_a @ A @ ( insert_a @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_875_finite_OinsertI,axiom,
    ! [A: set_set_nat,A2: set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( finite1152437895449049373et_nat @ ( insert_set_nat @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_876_finite_OinsertI,axiom,
    ! [A: set_a,A2: a] :
      ( ( finite_finite_a @ A )
     => ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_877_finite_OinsertI,axiom,
    ! [A: set_set_a,A2: set_a] :
      ( ( finite_finite_set_a @ A )
     => ( finite_finite_set_a @ ( insert_set_a @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_878_finite_OinsertI,axiom,
    ! [A: set_nat,A2: nat] :
      ( ( finite_finite_nat @ A )
     => ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).

% finite.insertI
thf(fact_879_double__diff,axiom,
    ! [A: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_880_double__diff,axiom,
    ! [A: set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C2 @ A ) )
          = A ) ) ) ).

% double_diff
thf(fact_881_Diff__subset,axiom,
    ! [A: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B3 ) @ A ) ).

% Diff_subset
thf(fact_882_Diff__subset,axiom,
    ! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B3 ) @ A ) ).

% Diff_subset
thf(fact_883_Diff__mono,axiom,
    ! [A: set_nat,C2: set_nat,D2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ D2 @ B3 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B3 ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_884_Diff__mono,axiom,
    ! [A: set_a,C2: set_a,D2: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ C2 )
     => ( ( ord_less_eq_set_a @ D2 @ B3 )
       => ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B3 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_885_Un__empty__right,axiom,
    ! [A: set_a] :
      ( ( sup_sup_set_a @ A @ bot_bot_set_a )
      = A ) ).

% Un_empty_right
thf(fact_886_Un__empty__right,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% Un_empty_right
thf(fact_887_Un__empty__left,axiom,
    ! [B3: set_a] :
      ( ( sup_sup_set_a @ bot_bot_set_a @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_888_Un__empty__left,axiom,
    ! [B3: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_889_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( sup_sup_set_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% subset_Un_eq
thf(fact_890_subset__Un__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B7: set_a] :
          ( ( sup_sup_set_a @ A6 @ B7 )
          = B7 ) ) ) ).

% subset_Un_eq
thf(fact_891_subset__UnE,axiom,
    ! [C2: set_nat,A: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B3 ) )
     => ~ ! [A7: set_nat] :
            ( ( ord_less_eq_set_nat @ A7 @ A )
           => ! [B8: set_nat] :
                ( ( ord_less_eq_set_nat @ B8 @ B3 )
               => ( C2
                 != ( sup_sup_set_nat @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_892_subset__UnE,axiom,
    ! [C2: set_a,A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B3 ) )
     => ~ ! [A7: set_a] :
            ( ( ord_less_eq_set_a @ A7 @ A )
           => ! [B8: set_a] :
                ( ( ord_less_eq_set_a @ B8 @ B3 )
               => ( C2
                 != ( sup_sup_set_a @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_893_Un__absorb2,axiom,
    ! [B3: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A )
     => ( ( sup_sup_set_nat @ A @ B3 )
        = A ) ) ).

% Un_absorb2
thf(fact_894_Un__absorb2,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A )
     => ( ( sup_sup_set_a @ A @ B3 )
        = A ) ) ).

% Un_absorb2
thf(fact_895_Un__absorb1,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( sup_sup_set_nat @ A @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_896_Un__absorb1,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( sup_sup_set_a @ A @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_897_Un__upper2,axiom,
    ! [B3: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B3 @ ( sup_sup_set_nat @ A @ B3 ) ) ).

% Un_upper2
thf(fact_898_Un__upper2,axiom,
    ! [B3: set_a,A: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A @ B3 ) ) ).

% Un_upper2
thf(fact_899_Un__upper1,axiom,
    ! [A: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B3 ) ) ).

% Un_upper1
thf(fact_900_Un__upper1,axiom,
    ! [A: set_a,B3: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B3 ) ) ).

% Un_upper1
thf(fact_901_Un__least,axiom,
    ! [A: set_nat,C2: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B3 @ C2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B3 ) @ C2 ) ) ) ).

% Un_least
thf(fact_902_Un__least,axiom,
    ! [A: set_a,C2: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ C2 )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B3 ) @ C2 ) ) ) ).

% Un_least
thf(fact_903_Un__mono,axiom,
    ! [A: set_nat,C2: set_nat,B3: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B3 @ D2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B3 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_904_Un__mono,axiom,
    ! [A: set_a,C2: set_a,B3: set_a,D2: set_a] :
      ( ( ord_less_eq_set_a @ A @ C2 )
     => ( ( ord_less_eq_set_a @ B3 @ D2 )
       => ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B3 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_905_Diff__infinite__finite,axiom,
    ! [T2: set_set_nat,S: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ T2 )
     => ( ~ ( finite1152437895449049373et_nat @ S )
       => ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_906_Diff__infinite__finite,axiom,
    ! [T2: set_set_a,S: set_set_a] :
      ( ( finite_finite_set_a @ T2 )
     => ( ~ ( finite_finite_set_a @ S )
       => ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_907_Diff__infinite__finite,axiom,
    ! [T2: set_a,S: set_a] :
      ( ( finite_finite_a @ T2 )
     => ( ~ ( finite_finite_a @ S )
       => ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_908_Diff__infinite__finite,axiom,
    ! [T2: set_nat,S: set_nat] :
      ( ( finite_finite_nat @ T2 )
     => ( ~ ( finite_finite_nat @ S )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_909_infinite__Un,axiom,
    ! [S: set_set_nat,T2: set_set_nat] :
      ( ( ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) )
      = ( ~ ( finite1152437895449049373et_nat @ S )
        | ~ ( finite1152437895449049373et_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_910_infinite__Un,axiom,
    ! [S: set_set_a,T2: set_set_a] :
      ( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T2 ) ) )
      = ( ~ ( finite_finite_set_a @ S )
        | ~ ( finite_finite_set_a @ T2 ) ) ) ).

% infinite_Un
thf(fact_911_infinite__Un,axiom,
    ! [S: set_a,T2: set_a] :
      ( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) )
      = ( ~ ( finite_finite_a @ S )
        | ~ ( finite_finite_a @ T2 ) ) ) ).

% infinite_Un
thf(fact_912_infinite__Un,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
      = ( ~ ( finite_finite_nat @ S )
        | ~ ( finite_finite_nat @ T2 ) ) ) ).

% infinite_Un
thf(fact_913_Un__infinite,axiom,
    ! [S: set_set_nat,T2: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ S )
     => ~ ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_914_Un__infinite,axiom,
    ! [S: set_set_a,T2: set_set_a] :
      ( ~ ( finite_finite_set_a @ S )
     => ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_915_Un__infinite,axiom,
    ! [S: set_a,T2: set_a] :
      ( ~ ( finite_finite_a @ S )
     => ~ ( finite_finite_a @ ( sup_sup_set_a @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_916_Un__infinite,axiom,
    ! [S: set_nat,T2: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).

% Un_infinite
thf(fact_917_finite__UnI,axiom,
    ! [F: set_set_nat,G: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ F )
     => ( ( finite1152437895449049373et_nat @ G )
       => ( finite1152437895449049373et_nat @ ( sup_sup_set_set_nat @ F @ G ) ) ) ) ).

% finite_UnI
thf(fact_918_finite__UnI,axiom,
    ! [F: set_set_a,G: set_set_a] :
      ( ( finite_finite_set_a @ F )
     => ( ( finite_finite_set_a @ G )
       => ( finite_finite_set_a @ ( sup_sup_set_set_a @ F @ G ) ) ) ) ).

% finite_UnI
thf(fact_919_finite__UnI,axiom,
    ! [F: set_a,G: set_a] :
      ( ( finite_finite_a @ F )
     => ( ( finite_finite_a @ G )
       => ( finite_finite_a @ ( sup_sup_set_a @ F @ G ) ) ) ) ).

% finite_UnI
thf(fact_920_finite__UnI,axiom,
    ! [F: set_nat,G: set_nat] :
      ( ( finite_finite_nat @ F )
     => ( ( finite_finite_nat @ G )
       => ( finite_finite_nat @ ( sup_sup_set_nat @ F @ G ) ) ) ) ).

% finite_UnI
thf(fact_921_is__singletonI_H,axiom,
    ! [A: set_set_a] :
      ( ( A != bot_bot_set_set_a )
     => ( ! [X4: set_a,Y3: set_a] :
            ( ( member_set_a @ X4 @ A )
           => ( ( member_set_a @ Y3 @ A )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_set_a @ A ) ) ) ).

% is_singletonI'
thf(fact_922_is__singletonI_H,axiom,
    ! [A: set_set_nat] :
      ( ( A != bot_bot_set_set_nat )
     => ( ! [X4: set_nat,Y3: set_nat] :
            ( ( member_set_nat @ X4 @ A )
           => ( ( member_set_nat @ Y3 @ A )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_set_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_923_is__singletonI_H,axiom,
    ! [A: set_a] :
      ( ( A != bot_bot_set_a )
     => ( ! [X4: a,Y3: a] :
            ( ( member_a @ X4 @ A )
           => ( ( member_a @ Y3 @ A )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_a @ A ) ) ) ).

% is_singletonI'
thf(fact_924_is__singletonI_H,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
     => ( ! [X4: nat,Y3: nat] :
            ( ( member_nat @ X4 @ A )
           => ( ( member_nat @ Y3 @ A )
             => ( X4 = Y3 ) ) )
       => ( is_singleton_nat @ A ) ) ) ).

% is_singletonI'
thf(fact_925_finite__has__maximal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_926_finite__has__maximal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_927_finite__has__maximal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_928_finite__has__minimal,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( A != bot_bot_set_set_nat )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_929_finite__has__minimal,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( A != bot_bot_set_set_a )
       => ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ! [Xa: set_a] :
                ( ( member_set_a @ Xa @ A )
               => ( ( ord_less_eq_set_a @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_930_finite__has__minimal,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_931_subset__singletonD,axiom,
    ! [A: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
     => ( ( A = bot_bot_set_nat )
        | ( A
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_932_subset__singletonD,axiom,
    ! [A: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) )
     => ( ( A = bot_bot_set_a )
        | ( A
          = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).

% subset_singletonD
thf(fact_933_subset__singleton__iff,axiom,
    ! [X2: set_nat,A2: nat] :
      ( ( ord_less_eq_set_nat @ X2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
      = ( ( X2 = bot_bot_set_nat )
        | ( X2
          = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_934_subset__singleton__iff,axiom,
    ! [X2: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ X2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
      = ( ( X2 = bot_bot_set_a )
        | ( X2
          = ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_935_finite_Ocases,axiom,
    ! [A2: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( A2 != bot_bot_set_set_nat )
       => ~ ! [A3: set_set_nat] :
              ( ? [A4: set_nat] :
                  ( A2
                  = ( insert_set_nat @ A4 @ A3 ) )
             => ~ ( finite1152437895449049373et_nat @ A3 ) ) ) ) ).

% finite.cases
thf(fact_936_finite_Ocases,axiom,
    ! [A2: set_set_a] :
      ( ( finite_finite_set_a @ A2 )
     => ( ( A2 != bot_bot_set_set_a )
       => ~ ! [A3: set_set_a] :
              ( ? [A4: set_a] :
                  ( A2
                  = ( insert_set_a @ A4 @ A3 ) )
             => ~ ( finite_finite_set_a @ A3 ) ) ) ) ).

% finite.cases
thf(fact_937_finite_Ocases,axiom,
    ! [A2: set_a] :
      ( ( finite_finite_a @ A2 )
     => ( ( A2 != bot_bot_set_a )
       => ~ ! [A3: set_a] :
              ( ? [A4: a] :
                  ( A2
                  = ( insert_a @ A4 @ A3 ) )
             => ~ ( finite_finite_a @ A3 ) ) ) ) ).

% finite.cases
thf(fact_938_finite_Ocases,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ~ ! [A3: set_nat] :
              ( ? [A4: nat] :
                  ( A2
                  = ( insert_nat @ A4 @ A3 ) )
             => ~ ( finite_finite_nat @ A3 ) ) ) ) ).

% finite.cases
thf(fact_939_finite_Osimps,axiom,
    ( finite1152437895449049373et_nat
    = ( ^ [A5: set_set_nat] :
          ( ( A5 = bot_bot_set_set_nat )
          | ? [A6: set_set_nat,B5: set_nat] :
              ( ( A5
                = ( insert_set_nat @ B5 @ A6 ) )
              & ( finite1152437895449049373et_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_940_finite_Osimps,axiom,
    ( finite_finite_set_a
    = ( ^ [A5: set_set_a] :
          ( ( A5 = bot_bot_set_set_a )
          | ? [A6: set_set_a,B5: set_a] :
              ( ( A5
                = ( insert_set_a @ B5 @ A6 ) )
              & ( finite_finite_set_a @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_941_finite_Osimps,axiom,
    ( finite_finite_a
    = ( ^ [A5: set_a] :
          ( ( A5 = bot_bot_set_a )
          | ? [A6: set_a,B5: a] :
              ( ( A5
                = ( insert_a @ B5 @ A6 ) )
              & ( finite_finite_a @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_942_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A5: set_nat] :
          ( ( A5 = bot_bot_set_nat )
          | ? [A6: set_nat,B5: nat] :
              ( ( A5
                = ( insert_nat @ B5 @ A6 ) )
              & ( finite_finite_nat @ A6 ) ) ) ) ) ).

% finite.simps
thf(fact_943_finite__induct,axiom,
    ! [F: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [X4: set_nat,F2: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F2 )
             => ( ~ ( member_set_nat @ X4 @ F2 )
               => ( ( P2 @ F2 )
                 => ( P2 @ ( insert_set_nat @ X4 @ F2 ) ) ) ) )
         => ( P2 @ F ) ) ) ) ).

% finite_induct
thf(fact_944_finite__induct,axiom,
    ! [F: set_set_a,P2: set_set_a > $o] :
      ( ( finite_finite_set_a @ F )
     => ( ( P2 @ bot_bot_set_set_a )
       => ( ! [X4: set_a,F2: set_set_a] :
              ( ( finite_finite_set_a @ F2 )
             => ( ~ ( member_set_a @ X4 @ F2 )
               => ( ( P2 @ F2 )
                 => ( P2 @ ( insert_set_a @ X4 @ F2 ) ) ) ) )
         => ( P2 @ F ) ) ) ) ).

% finite_induct
thf(fact_945_finite__induct,axiom,
    ! [F: set_a,P2: set_a > $o] :
      ( ( finite_finite_a @ F )
     => ( ( P2 @ bot_bot_set_a )
       => ( ! [X4: a,F2: set_a] :
              ( ( finite_finite_a @ F2 )
             => ( ~ ( member_a @ X4 @ F2 )
               => ( ( P2 @ F2 )
                 => ( P2 @ ( insert_a @ X4 @ F2 ) ) ) ) )
         => ( P2 @ F ) ) ) ) ).

% finite_induct
thf(fact_946_finite__induct,axiom,
    ! [F: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X4: nat,F2: set_nat] :
              ( ( finite_finite_nat @ F2 )
             => ( ~ ( member_nat @ X4 @ F2 )
               => ( ( P2 @ F2 )
                 => ( P2 @ ( insert_nat @ X4 @ F2 ) ) ) ) )
         => ( P2 @ F ) ) ) ) ).

% finite_induct
thf(fact_947_finite__ne__induct,axiom,
    ! [F: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F )
     => ( ( F != bot_bot_set_set_nat )
       => ( ! [X4: set_nat] : ( P2 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
         => ( ! [X4: set_nat,F2: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F2 )
               => ( ( F2 != bot_bot_set_set_nat )
                 => ( ~ ( member_set_nat @ X4 @ F2 )
                   => ( ( P2 @ F2 )
                     => ( P2 @ ( insert_set_nat @ X4 @ F2 ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_948_finite__ne__induct,axiom,
    ! [F: set_set_a,P2: set_set_a > $o] :
      ( ( finite_finite_set_a @ F )
     => ( ( F != bot_bot_set_set_a )
       => ( ! [X4: set_a] : ( P2 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) )
         => ( ! [X4: set_a,F2: set_set_a] :
                ( ( finite_finite_set_a @ F2 )
               => ( ( F2 != bot_bot_set_set_a )
                 => ( ~ ( member_set_a @ X4 @ F2 )
                   => ( ( P2 @ F2 )
                     => ( P2 @ ( insert_set_a @ X4 @ F2 ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_949_finite__ne__induct,axiom,
    ! [F: set_a,P2: set_a > $o] :
      ( ( finite_finite_a @ F )
     => ( ( F != bot_bot_set_a )
       => ( ! [X4: a] : ( P2 @ ( insert_a @ X4 @ bot_bot_set_a ) )
         => ( ! [X4: a,F2: set_a] :
                ( ( finite_finite_a @ F2 )
               => ( ( F2 != bot_bot_set_a )
                 => ( ~ ( member_a @ X4 @ F2 )
                   => ( ( P2 @ F2 )
                     => ( P2 @ ( insert_a @ X4 @ F2 ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_950_finite__ne__induct,axiom,
    ! [F: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( F != bot_bot_set_nat )
       => ( ! [X4: nat] : ( P2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
         => ( ! [X4: nat,F2: set_nat] :
                ( ( finite_finite_nat @ F2 )
               => ( ( F2 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X4 @ F2 )
                   => ( ( P2 @ F2 )
                     => ( P2 @ ( insert_nat @ X4 @ F2 ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_ne_induct
thf(fact_951_infinite__finite__induct,axiom,
    ! [P2: set_set_nat > $o,A: set_set_nat] :
      ( ! [A3: set_set_nat] :
          ( ~ ( finite1152437895449049373et_nat @ A3 )
         => ( P2 @ A3 ) )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [X4: set_nat,F2: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F2 )
             => ( ~ ( member_set_nat @ X4 @ F2 )
               => ( ( P2 @ F2 )
                 => ( P2 @ ( insert_set_nat @ X4 @ F2 ) ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_952_infinite__finite__induct,axiom,
    ! [P2: set_set_a > $o,A: set_set_a] :
      ( ! [A3: set_set_a] :
          ( ~ ( finite_finite_set_a @ A3 )
         => ( P2 @ A3 ) )
     => ( ( P2 @ bot_bot_set_set_a )
       => ( ! [X4: set_a,F2: set_set_a] :
              ( ( finite_finite_set_a @ F2 )
             => ( ~ ( member_set_a @ X4 @ F2 )
               => ( ( P2 @ F2 )
                 => ( P2 @ ( insert_set_a @ X4 @ F2 ) ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_953_infinite__finite__induct,axiom,
    ! [P2: set_a > $o,A: set_a] :
      ( ! [A3: set_a] :
          ( ~ ( finite_finite_a @ A3 )
         => ( P2 @ A3 ) )
     => ( ( P2 @ bot_bot_set_a )
       => ( ! [X4: a,F2: set_a] :
              ( ( finite_finite_a @ F2 )
             => ( ~ ( member_a @ X4 @ F2 )
               => ( ( P2 @ F2 )
                 => ( P2 @ ( insert_a @ X4 @ F2 ) ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_954_infinite__finite__induct,axiom,
    ! [P2: set_nat > $o,A: set_nat] :
      ( ! [A3: set_nat] :
          ( ~ ( finite_finite_nat @ A3 )
         => ( P2 @ A3 ) )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X4: nat,F2: set_nat] :
              ( ( finite_finite_nat @ F2 )
             => ( ~ ( member_nat @ X4 @ F2 )
               => ( ( P2 @ F2 )
                 => ( P2 @ ( insert_nat @ X4 @ F2 ) ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% infinite_finite_induct
thf(fact_955_Diff__insert,axiom,
    ! [A: set_a,A2: a,B3: set_a] :
      ( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B3 ) )
      = ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B3 ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).

% Diff_insert
thf(fact_956_Diff__insert,axiom,
    ! [A: set_nat,A2: nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B3 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_957_insert__Diff,axiom,
    ! [A2: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ A )
     => ( ( insert_set_a @ A2 @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_958_insert__Diff,axiom,
    ! [A2: set_nat,A: set_set_nat] :
      ( ( member_set_nat @ A2 @ A )
     => ( ( insert_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_959_insert__Diff,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( insert_a @ A2 @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_960_insert__Diff,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_961_Diff__insert2,axiom,
    ! [A: set_a,A2: a,B3: set_a] :
      ( ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B3 ) )
      = ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_962_Diff__insert2,axiom,
    ! [A: set_nat,A2: nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B3 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_963_Diff__insert__absorb,axiom,
    ! [X: set_a,A: set_set_a] :
      ( ~ ( member_set_a @ X @ A )
     => ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_964_Diff__insert__absorb,axiom,
    ! [X: set_nat,A: set_set_nat] :
      ( ~ ( member_set_nat @ X @ A )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X @ A ) @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_965_Diff__insert__absorb,axiom,
    ! [X: a,A: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ( minus_minus_set_a @ ( insert_a @ X @ A ) @ ( insert_a @ X @ bot_bot_set_a ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_966_Diff__insert__absorb,axiom,
    ! [X: nat,A: set_nat] :
      ( ~ ( member_nat @ X @ A )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_967_subset__Diff__insert,axiom,
    ! [A: set_set_a,B3: set_set_a,X: set_a,C2: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B3 @ ( insert_set_a @ X @ C2 ) ) )
      = ( ( ord_le3724670747650509150_set_a @ A @ ( minus_5736297505244876581_set_a @ B3 @ C2 ) )
        & ~ ( member_set_a @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_968_subset__Diff__insert,axiom,
    ! [A: set_set_nat,B3: set_set_nat,X: set_nat,C2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B3 @ ( insert_set_nat @ X @ C2 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A @ ( minus_2163939370556025621et_nat @ B3 @ C2 ) )
        & ~ ( member_set_nat @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_969_subset__Diff__insert,axiom,
    ! [A: set_nat,B3: set_nat,X: nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B3 @ ( insert_nat @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B3 @ C2 ) )
        & ~ ( member_nat @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_970_subset__Diff__insert,axiom,
    ! [A: set_a,B3: set_a,X: a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B3 @ ( insert_a @ X @ C2 ) ) )
      = ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B3 @ C2 ) )
        & ~ ( member_a @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_971_insert__is__Un,axiom,
    ( insert_a
    = ( ^ [A5: a] : ( sup_sup_set_a @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ).

% insert_is_Un
thf(fact_972_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_973_Un__singleton__iff,axiom,
    ! [A: set_a,B3: set_a,X: a] :
      ( ( ( sup_sup_set_a @ A @ B3 )
        = ( insert_a @ X @ bot_bot_set_a ) )
      = ( ( ( A = bot_bot_set_a )
          & ( B3
            = ( insert_a @ X @ bot_bot_set_a ) ) )
        | ( ( A
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B3 = bot_bot_set_a ) )
        | ( ( A
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B3
            = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_974_Un__singleton__iff,axiom,
    ! [A: set_nat,B3: set_nat,X: nat] :
      ( ( ( sup_sup_set_nat @ A @ B3 )
        = ( insert_nat @ X @ bot_bot_set_nat ) )
      = ( ( ( A = bot_bot_set_nat )
          & ( B3
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B3 = bot_bot_set_nat ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B3
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_975_singleton__Un__iff,axiom,
    ! [X: a,A: set_a,B3: set_a] :
      ( ( ( insert_a @ X @ bot_bot_set_a )
        = ( sup_sup_set_a @ A @ B3 ) )
      = ( ( ( A = bot_bot_set_a )
          & ( B3
            = ( insert_a @ X @ bot_bot_set_a ) ) )
        | ( ( A
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B3 = bot_bot_set_a ) )
        | ( ( A
            = ( insert_a @ X @ bot_bot_set_a ) )
          & ( B3
            = ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_976_singleton__Un__iff,axiom,
    ! [X: nat,A: set_nat,B3: set_nat] :
      ( ( ( insert_nat @ X @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A @ B3 ) )
      = ( ( ( A = bot_bot_set_nat )
          & ( B3
            = ( insert_nat @ X @ bot_bot_set_nat ) ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B3 = bot_bot_set_nat ) )
        | ( ( A
            = ( insert_nat @ X @ bot_bot_set_nat ) )
          & ( B3
            = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_977_Diff__partition,axiom,
    ! [A: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B3 )
     => ( ( sup_sup_set_nat @ A @ ( minus_minus_set_nat @ B3 @ A ) )
        = B3 ) ) ).

% Diff_partition
thf(fact_978_Diff__partition,axiom,
    ! [A: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ B3 )
     => ( ( sup_sup_set_a @ A @ ( minus_minus_set_a @ B3 @ A ) )
        = B3 ) ) ).

% Diff_partition
thf(fact_979_Diff__subset__conv,axiom,
    ! [A: set_nat,B3: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B3 ) @ C2 )
      = ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ B3 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_980_Diff__subset__conv,axiom,
    ! [A: set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B3 ) @ C2 )
      = ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ B3 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_981_is__singletonE,axiom,
    ! [A: set_a] :
      ( ( is_singleton_a @ A )
     => ~ ! [X4: a] :
            ( A
           != ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).

% is_singletonE
thf(fact_982_is__singletonE,axiom,
    ! [A: set_nat] :
      ( ( is_singleton_nat @ A )
     => ~ ! [X4: nat] :
            ( A
           != ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_983_is__singleton__def,axiom,
    ( is_singleton_a
    = ( ^ [A6: set_a] :
        ? [X3: a] :
          ( A6
          = ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).

% is_singleton_def
thf(fact_984_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A6: set_nat] :
        ? [X3: nat] :
          ( A6
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_985_finite__subset__induct,axiom,
    ! [F: set_set_nat,A: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F )
     => ( ( ord_le6893508408891458716et_nat @ F @ A )
       => ( ( P2 @ bot_bot_set_set_nat )
         => ( ! [A4: set_nat,F2: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F2 )
               => ( ( member_set_nat @ A4 @ A )
                 => ( ~ ( member_set_nat @ A4 @ F2 )
                   => ( ( P2 @ F2 )
                     => ( P2 @ ( insert_set_nat @ A4 @ F2 ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_986_finite__subset__induct,axiom,
    ! [F: set_set_a,A: set_set_a,P2: set_set_a > $o] :
      ( ( finite_finite_set_a @ F )
     => ( ( ord_le3724670747650509150_set_a @ F @ A )
       => ( ( P2 @ bot_bot_set_set_a )
         => ( ! [A4: set_a,F2: set_set_a] :
                ( ( finite_finite_set_a @ F2 )
               => ( ( member_set_a @ A4 @ A )
                 => ( ~ ( member_set_a @ A4 @ F2 )
                   => ( ( P2 @ F2 )
                     => ( P2 @ ( insert_set_a @ A4 @ F2 ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_987_finite__subset__induct,axiom,
    ! [F: set_nat,A: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( ord_less_eq_set_nat @ F @ A )
       => ( ( P2 @ bot_bot_set_nat )
         => ( ! [A4: nat,F2: set_nat] :
                ( ( finite_finite_nat @ F2 )
               => ( ( member_nat @ A4 @ A )
                 => ( ~ ( member_nat @ A4 @ F2 )
                   => ( ( P2 @ F2 )
                     => ( P2 @ ( insert_nat @ A4 @ F2 ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_988_finite__subset__induct,axiom,
    ! [F: set_a,A: set_a,P2: set_a > $o] :
      ( ( finite_finite_a @ F )
     => ( ( ord_less_eq_set_a @ F @ A )
       => ( ( P2 @ bot_bot_set_a )
         => ( ! [A4: a,F2: set_a] :
                ( ( finite_finite_a @ F2 )
               => ( ( member_a @ A4 @ A )
                 => ( ~ ( member_a @ A4 @ F2 )
                   => ( ( P2 @ F2 )
                     => ( P2 @ ( insert_a @ A4 @ F2 ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_subset_induct
thf(fact_989_finite__subset__induct_H,axiom,
    ! [F: set_set_nat,A: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F )
     => ( ( ord_le6893508408891458716et_nat @ F @ A )
       => ( ( P2 @ bot_bot_set_set_nat )
         => ( ! [A4: set_nat,F2: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F2 )
               => ( ( member_set_nat @ A4 @ A )
                 => ( ( ord_le6893508408891458716et_nat @ F2 @ A )
                   => ( ~ ( member_set_nat @ A4 @ F2 )
                     => ( ( P2 @ F2 )
                       => ( P2 @ ( insert_set_nat @ A4 @ F2 ) ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_990_finite__subset__induct_H,axiom,
    ! [F: set_set_a,A: set_set_a,P2: set_set_a > $o] :
      ( ( finite_finite_set_a @ F )
     => ( ( ord_le3724670747650509150_set_a @ F @ A )
       => ( ( P2 @ bot_bot_set_set_a )
         => ( ! [A4: set_a,F2: set_set_a] :
                ( ( finite_finite_set_a @ F2 )
               => ( ( member_set_a @ A4 @ A )
                 => ( ( ord_le3724670747650509150_set_a @ F2 @ A )
                   => ( ~ ( member_set_a @ A4 @ F2 )
                     => ( ( P2 @ F2 )
                       => ( P2 @ ( insert_set_a @ A4 @ F2 ) ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_991_finite__subset__induct_H,axiom,
    ! [F: set_nat,A: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ F )
     => ( ( ord_less_eq_set_nat @ F @ A )
       => ( ( P2 @ bot_bot_set_nat )
         => ( ! [A4: nat,F2: set_nat] :
                ( ( finite_finite_nat @ F2 )
               => ( ( member_nat @ A4 @ A )
                 => ( ( ord_less_eq_set_nat @ F2 @ A )
                   => ( ~ ( member_nat @ A4 @ F2 )
                     => ( ( P2 @ F2 )
                       => ( P2 @ ( insert_nat @ A4 @ F2 ) ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_992_finite__subset__induct_H,axiom,
    ! [F: set_a,A: set_a,P2: set_a > $o] :
      ( ( finite_finite_a @ F )
     => ( ( ord_less_eq_set_a @ F @ A )
       => ( ( P2 @ bot_bot_set_a )
         => ( ! [A4: a,F2: set_a] :
                ( ( finite_finite_a @ F2 )
               => ( ( member_a @ A4 @ A )
                 => ( ( ord_less_eq_set_a @ F2 @ A )
                   => ( ~ ( member_a @ A4 @ F2 )
                     => ( ( P2 @ F2 )
                       => ( P2 @ ( insert_a @ A4 @ F2 ) ) ) ) ) ) )
           => ( P2 @ F ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_993_subset__insert__iff,axiom,
    ! [A: set_set_a,X: set_a,B3: set_set_a] :
      ( ( ord_le3724670747650509150_set_a @ A @ ( insert_set_a @ X @ B3 ) )
      = ( ( ( member_set_a @ X @ A )
         => ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B3 ) )
        & ( ~ ( member_set_a @ X @ A )
         => ( ord_le3724670747650509150_set_a @ A @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_994_subset__insert__iff,axiom,
    ! [A: set_set_nat,X: set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A @ ( insert_set_nat @ X @ B3 ) )
      = ( ( ( member_set_nat @ X @ A )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) @ B3 ) )
        & ( ~ ( member_set_nat @ X @ A )
         => ( ord_le6893508408891458716et_nat @ A @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_995_subset__insert__iff,axiom,
    ! [A: set_nat,X: nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B3 ) )
      = ( ( ( member_nat @ X @ A )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B3 ) )
        & ( ~ ( member_nat @ X @ A )
         => ( ord_less_eq_set_nat @ A @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_996_subset__insert__iff,axiom,
    ! [A: set_a,X: a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B3 ) )
      = ( ( ( member_a @ X @ A )
         => ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) )
        & ( ~ ( member_a @ X @ A )
         => ( ord_less_eq_set_a @ A @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_997_Diff__single__insert,axiom,
    ! [A: set_nat,X: nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B3 )
     => ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B3 ) ) ) ).

% Diff_single_insert
thf(fact_998_Diff__single__insert,axiom,
    ! [A: set_a,X: a,B3: set_a] :
      ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 )
     => ( ord_less_eq_set_a @ A @ ( insert_a @ X @ B3 ) ) ) ).

% Diff_single_insert
thf(fact_999_infinite__remove,axiom,
    ! [S: set_set_nat,A2: set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ S )
     => ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ S @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_1000_infinite__remove,axiom,
    ! [S: set_set_a,A2: set_a] :
      ( ~ ( finite_finite_set_a @ S )
     => ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ) ) ).

% infinite_remove
thf(fact_1001_infinite__remove,axiom,
    ! [S: set_a,A2: a] :
      ( ~ ( finite_finite_a @ S )
     => ~ ( finite_finite_a @ ( minus_minus_set_a @ S @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).

% infinite_remove
thf(fact_1002_infinite__remove,axiom,
    ! [S: set_nat,A2: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_1003_infinite__coinduct,axiom,
    ! [X2: set_set_nat > $o,A: set_set_nat] :
      ( ( X2 @ A )
     => ( ! [A3: set_set_nat] :
            ( ( X2 @ A3 )
           => ? [X5: set_nat] :
                ( ( member_set_nat @ X5 @ A3 )
                & ( ( X2 @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) )
                  | ~ ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) ) ) )
       => ~ ( finite1152437895449049373et_nat @ A ) ) ) ).

% infinite_coinduct
thf(fact_1004_infinite__coinduct,axiom,
    ! [X2: set_set_a > $o,A: set_set_a] :
      ( ( X2 @ A )
     => ( ! [A3: set_set_a] :
            ( ( X2 @ A3 )
           => ? [X5: set_a] :
                ( ( member_set_a @ X5 @ A3 )
                & ( ( X2 @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) )
                  | ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) ) ) )
       => ~ ( finite_finite_set_a @ A ) ) ) ).

% infinite_coinduct
thf(fact_1005_infinite__coinduct,axiom,
    ! [X2: set_a > $o,A: set_a] :
      ( ( X2 @ A )
     => ( ! [A3: set_a] :
            ( ( X2 @ A3 )
           => ? [X5: a] :
                ( ( member_a @ X5 @ A3 )
                & ( ( X2 @ ( minus_minus_set_a @ A3 @ ( insert_a @ X5 @ bot_bot_set_a ) ) )
                  | ~ ( finite_finite_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) ) ) )
       => ~ ( finite_finite_a @ A ) ) ) ).

% infinite_coinduct
thf(fact_1006_infinite__coinduct,axiom,
    ! [X2: set_nat > $o,A: set_nat] :
      ( ( X2 @ A )
     => ( ! [A3: set_nat] :
            ( ( X2 @ A3 )
           => ? [X5: nat] :
                ( ( member_nat @ X5 @ A3 )
                & ( ( X2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A ) ) ) ).

% infinite_coinduct
thf(fact_1007_finite__empty__induct,axiom,
    ! [A: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( P2 @ A )
       => ( ! [A4: set_nat,A3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A3 )
             => ( ( member_set_nat @ A4 @ A3 )
               => ( ( P2 @ A3 )
                 => ( P2 @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P2 @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_1008_finite__empty__induct,axiom,
    ! [A: set_set_a,P2: set_set_a > $o] :
      ( ( finite_finite_set_a @ A )
     => ( ( P2 @ A )
       => ( ! [A4: set_a,A3: set_set_a] :
              ( ( finite_finite_set_a @ A3 )
             => ( ( member_set_a @ A4 @ A3 )
               => ( ( P2 @ A3 )
                 => ( P2 @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ A4 @ bot_bot_set_set_a ) ) ) ) ) )
         => ( P2 @ bot_bot_set_set_a ) ) ) ) ).

% finite_empty_induct
thf(fact_1009_finite__empty__induct,axiom,
    ! [A: set_a,P2: set_a > $o] :
      ( ( finite_finite_a @ A )
     => ( ( P2 @ A )
       => ( ! [A4: a,A3: set_a] :
              ( ( finite_finite_a @ A3 )
             => ( ( member_a @ A4 @ A3 )
               => ( ( P2 @ A3 )
                 => ( P2 @ ( minus_minus_set_a @ A3 @ ( insert_a @ A4 @ bot_bot_set_a ) ) ) ) ) )
         => ( P2 @ bot_bot_set_a ) ) ) ) ).

% finite_empty_induct
thf(fact_1010_finite__empty__induct,axiom,
    ! [A: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ A )
     => ( ( P2 @ A )
       => ( ! [A4: nat,A3: set_nat] :
              ( ( finite_finite_nat @ A3 )
             => ( ( member_nat @ A4 @ A3 )
               => ( ( P2 @ A3 )
                 => ( P2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P2 @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_1011_remove__induct,axiom,
    ! [P2: set_set_nat > $o,B3: set_set_nat] :
      ( ( P2 @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B3 )
         => ( P2 @ B3 ) )
       => ( ! [A3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A3 )
             => ( ( A3 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
                 => ( ! [X5: set_nat] :
                        ( ( member_set_nat @ X5 @ A3 )
                       => ( P2 @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) )
                   => ( P2 @ A3 ) ) ) ) )
         => ( P2 @ B3 ) ) ) ) ).

% remove_induct
thf(fact_1012_remove__induct,axiom,
    ! [P2: set_set_a > $o,B3: set_set_a] :
      ( ( P2 @ bot_bot_set_set_a )
     => ( ( ~ ( finite_finite_set_a @ B3 )
         => ( P2 @ B3 ) )
       => ( ! [A3: set_set_a] :
              ( ( finite_finite_set_a @ A3 )
             => ( ( A3 != bot_bot_set_set_a )
               => ( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
                 => ( ! [X5: set_a] :
                        ( ( member_set_a @ X5 @ A3 )
                       => ( P2 @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
                   => ( P2 @ A3 ) ) ) ) )
         => ( P2 @ B3 ) ) ) ) ).

% remove_induct
thf(fact_1013_remove__induct,axiom,
    ! [P2: set_nat > $o,B3: set_nat] :
      ( ( P2 @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B3 )
         => ( P2 @ B3 ) )
       => ( ! [A3: set_nat] :
              ( ( finite_finite_nat @ A3 )
             => ( ( A3 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A3 @ B3 )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A3 )
                       => ( P2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P2 @ A3 ) ) ) ) )
         => ( P2 @ B3 ) ) ) ) ).

% remove_induct
thf(fact_1014_remove__induct,axiom,
    ! [P2: set_a > $o,B3: set_a] :
      ( ( P2 @ bot_bot_set_a )
     => ( ( ~ ( finite_finite_a @ B3 )
         => ( P2 @ B3 ) )
       => ( ! [A3: set_a] :
              ( ( finite_finite_a @ A3 )
             => ( ( A3 != bot_bot_set_a )
               => ( ( ord_less_eq_set_a @ A3 @ B3 )
                 => ( ! [X5: a] :
                        ( ( member_a @ X5 @ A3 )
                       => ( P2 @ ( minus_minus_set_a @ A3 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
                   => ( P2 @ A3 ) ) ) ) )
         => ( P2 @ B3 ) ) ) ) ).

% remove_induct
thf(fact_1015_finite__remove__induct,axiom,
    ! [B3: set_set_nat,P2: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( P2 @ bot_bot_set_set_nat )
       => ( ! [A3: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A3 )
             => ( ( A3 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
                 => ( ! [X5: set_nat] :
                        ( ( member_set_nat @ X5 @ A3 )
                       => ( P2 @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) )
                   => ( P2 @ A3 ) ) ) ) )
         => ( P2 @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_1016_finite__remove__induct,axiom,
    ! [B3: set_set_a,P2: set_set_a > $o] :
      ( ( finite_finite_set_a @ B3 )
     => ( ( P2 @ bot_bot_set_set_a )
       => ( ! [A3: set_set_a] :
              ( ( finite_finite_set_a @ A3 )
             => ( ( A3 != bot_bot_set_set_a )
               => ( ( ord_le3724670747650509150_set_a @ A3 @ B3 )
                 => ( ! [X5: set_a] :
                        ( ( member_set_a @ X5 @ A3 )
                       => ( P2 @ ( minus_5736297505244876581_set_a @ A3 @ ( insert_set_a @ X5 @ bot_bot_set_set_a ) ) ) )
                   => ( P2 @ A3 ) ) ) ) )
         => ( P2 @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_1017_finite__remove__induct,axiom,
    ! [B3: set_nat,P2: set_nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [A3: set_nat] :
              ( ( finite_finite_nat @ A3 )
             => ( ( A3 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A3 @ B3 )
                 => ( ! [X5: nat] :
                        ( ( member_nat @ X5 @ A3 )
                       => ( P2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) )
                   => ( P2 @ A3 ) ) ) ) )
         => ( P2 @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_1018_finite__remove__induct,axiom,
    ! [B3: set_a,P2: set_a > $o] :
      ( ( finite_finite_a @ B3 )
     => ( ( P2 @ bot_bot_set_a )
       => ( ! [A3: set_a] :
              ( ( finite_finite_a @ A3 )
             => ( ( A3 != bot_bot_set_a )
               => ( ( ord_less_eq_set_a @ A3 @ B3 )
                 => ( ! [X5: a] :
                        ( ( member_a @ X5 @ A3 )
                       => ( P2 @ ( minus_minus_set_a @ A3 @ ( insert_a @ X5 @ bot_bot_set_a ) ) ) )
                   => ( P2 @ A3 ) ) ) ) )
         => ( P2 @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_1019_empty__inter__implies__b__lt__v,axiom,
    ( ( lambda = zero_zero_nat )
   => ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).

% empty_inter_implies_b_lt_v
thf(fact_1020_block__sizes__non__empty__set,axiom,
    ( ( ( mset_set_a @ b_s )
     != zero_z5079479921072680283_set_a )
   => ( ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) )
     != bot_bot_set_nat ) ) ).

% block_sizes_non_empty_set
thf(fact_1021_add__block__design,axiom,
    ! [Bl2: set_a] :
      ( ( finite_finite_a @ Bl2 )
     => ( ( Bl2 != bot_bot_set_a )
       => ( design_design_a @ ( sup_sup_set_a @ ( set_a2 @ v_s ) @ Bl2 ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ) ) ).

% add_block_design
thf(fact_1022_add__point__sys__rep__numbers,axiom,
    ! [P: a] :
      ( ( design8835372594653258411bers_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P ) @ ( mset_set_a @ b_s ) )
      = ( sup_sup_set_nat @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) @ ( insert_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ P ) @ bot_bot_set_nat ) ) ) ).

% add_point_sys_rep_numbers
thf(fact_1023_add__block__rep__number__not__in,axiom,
    ! [X: a,B: set_a] :
      ( ~ ( member_a @ X @ B )
     => ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) @ X )
        = ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) ) ) ).

% add_block_rep_number_not_in
thf(fact_1024_finite__block__sizes,axiom,
    finite_finite_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ).

% finite_block_sizes
thf(fact_1025_sys__block__sizes__uniform__single,axiom,
    is_singleton_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ).

% sys_block_sizes_uniform_single
thf(fact_1026_sys__block__sizes__in,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( member_nat @ ( finite_card_a @ Bl2 ) @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ) ) ).

% sys_block_sizes_in
thf(fact_1027_sys__block__sizes__obtain__bl,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) )
     => ? [X4: set_a] :
          ( ( member_set_a @ X4 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
          & ( ( finite_card_a @ X4 )
            = X ) ) ) ).

% sys_block_sizes_obtain_bl
thf(fact_1028_constant__rep__alt,axiom,
    ! [X1: a,R2: nat,X: a] :
      ( ( member_a @ X1 @ ( set_a2 @ v_s ) )
     => ( ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X1 )
          = R2 )
       => ( ( member_a @ X @ ( set_a2 @ v_s ) )
         => ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X )
            = R2 ) ) ) ) ).

% constant_rep_alt
thf(fact_1029_constant__rep__point__pair,axiom,
    ! [X1: a,X22: a] :
      ( ( member_a @ X1 @ ( set_a2 @ v_s ) )
     => ( ( member_a @ X22 @ ( set_a2 @ v_s ) )
       => ( ( X1 != X22 )
         => ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X1 )
            = ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X22 ) ) ) ) ) ).

% constant_rep_point_pair
thf(fact_1030_wf__design,axiom,
    design_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).

% wf_design
thf(fact_1031_del__point__order,axiom,
    ! [P: a] :
      ( ( member_a @ P @ ( set_a2 @ v_s ) )
     => ( ( finite_card_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P ) )
        = ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) ) ).

% del_point_order
thf(fact_1032_block__complement__size,axiom,
    ! [B: set_a] :
      ( ( ord_less_eq_set_a @ B @ ( set_a2 @ v_s ) )
     => ( ( finite_card_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ B ) )
        = ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( finite_card_a @ B ) ) ) ) ).

% block_complement_size
thf(fact_1033_add__block__rep__number__in,axiom,
    ! [X: a,B: set_a] :
      ( ( member_a @ X @ B )
     => ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ B ) @ X )
        = ( plus_plus_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) @ one_one_nat ) ) ) ).

% add_block_rep_number_in
thf(fact_1034_complete__block__size__eq__points,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( ( finite_card_a @ Bl2 )
          = ( finite_card_a @ ( set_a2 @ v_s ) ) )
       => ( Bl2
          = ( set_a2 @ v_s ) ) ) ) ).

% complete_block_size_eq_points
thf(fact_1035_local_Osymmetric,axiom,
    ( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
    = ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).

% local.symmetric
thf(fact_1036_constant__rep__point__not__0,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( set_a2 @ v_s ) )
     => ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X )
       != zero_zero_nat ) ) ).

% constant_rep_point_not_0
thf(fact_1037_dual__blocks__b,axiom,
    ( ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
    = ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).

% dual_blocks_b
thf(fact_1038_obtain__point__with__rep,axiom,
    ! [R: nat] :
      ( ( member_nat @ R @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) )
     => ? [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ v_s ) )
          & ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X4 )
            = R ) ) ) ).

% obtain_point_with_rep
thf(fact_1039_point__rep__number__in__set,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( set_a2 @ v_s ) )
     => ( member_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).

% point_rep_number_in_set
thf(fact_1040_delete__block__design,axiom,
    ! [Bl2: set_a] : ( design_design_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ).

% delete_block_design
thf(fact_1041_add__point__design,axiom,
    ! [P: a] : ( design_design_a @ ( design2964366272795260673oint_a @ ( set_a2 @ v_s ) @ P ) @ ( mset_set_a @ b_s ) ) ).

% add_point_design
thf(fact_1042_v__eq0__imp__b__eq__0,axiom,
    ( ( ( finite_card_a @ ( set_a2 @ v_s ) )
      = zero_zero_nat )
   => ( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
      = zero_zero_nat ) ) ).

% v_eq0_imp_b_eq_0
thf(fact_1043_block__size__lt__order,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ord_less_eq_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).

% block_size_lt_order
thf(fact_1044_complete__block__all__subsets,axiom,
    ! [Bl2: set_a,Ps: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( ( finite_card_a @ Bl2 )
          = ( finite_card_a @ ( set_a2 @ v_s ) ) )
       => ( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
         => ( ord_less_eq_set_a @ Ps @ Bl2 ) ) ) ) ).

% complete_block_all_subsets
thf(fact_1045_index__zero__iff,axiom,
    ( ( lambda = zero_zero_nat )
    = ( ! [X3: set_a] :
          ( ( member_set_a @ X3 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
         => ( ( finite_card_a @ X3 )
            = one_one_nat ) ) ) ) ).

% index_zero_iff
thf(fact_1046_const__index__lt__rep,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( set_a2 @ v_s ) )
     => ( ord_less_eq_nat @ lambda @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) ) ) ).

% const_index_lt_rep
thf(fact_1047_intersect__num__same__eq__size,axiom,
    ! [Bl2: set_set_a] :
      ( ( design3520961687418077020_set_a @ Bl2 @ Bl2 )
      = ( finite_card_set_a @ Bl2 ) ) ).

% intersect_num_same_eq_size
thf(fact_1048_intersect__num__same__eq__size,axiom,
    ! [Bl2: set_a] :
      ( ( design7842873109100088828mber_a @ Bl2 @ Bl2 )
      = ( finite_card_a @ Bl2 ) ) ).

% intersect_num_same_eq_size
thf(fact_1049_intersect__num__same__eq__size,axiom,
    ! [Bl2: set_nat] :
      ( ( design7485525362727208274er_nat @ Bl2 @ Bl2 )
      = ( finite_card_nat @ Bl2 ) ) ).

% intersect_num_same_eq_size
thf(fact_1050_strong__del__point__design,axiom,
    ! [P: a] : ( design_design_a @ ( design108908007054065099oint_a @ ( set_a2 @ v_s ) @ P ) @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) ) ).

% strong_del_point_design
thf(fact_1051_eq__index__rep__imp__complete,axiom,
    ! [X: a,Bl2: set_a] :
      ( ( lambda
        = ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) )
     => ( ( member_a @ X @ ( set_a2 @ v_s ) )
       => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
         => ( ( member_a @ X @ Bl2 )
           => ( ( finite_card_a @ Bl2 )
              = ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ) ) ) ).

% eq_index_rep_imp_complete
thf(fact_1052_add__block__design__cond,axiom,
    ! [Bl2: set_a] :
      ( ( ord_less_eq_set_a @ Bl2 @ ( set_a2 @ v_s ) )
     => ( ( Bl2 != bot_bot_set_a )
       => ( design_design_a @ ( set_a2 @ v_s ) @ ( design4001997691126659652lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ) ) ).

% add_block_design_cond
thf(fact_1053_empty__inter__implies__rep__one,axiom,
    ! [X: a] :
      ( ( lambda = zero_zero_nat )
     => ( ( member_a @ X @ ( set_a2 @ v_s ) )
       => ( ord_less_eq_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) @ one_one_nat ) ) ) ).

% empty_inter_implies_rep_one
thf(fact_1054_wf__design__iff,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( design_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
        = ( ( ord_less_eq_set_a @ Bl2 @ ( set_a2 @ v_s ) )
          & ( finite_finite_a @ ( set_a2 @ v_s ) )
          & ( Bl2 != bot_bot_set_a ) ) ) ) ).

% wf_design_iff
thf(fact_1055_card_Oempty,axiom,
    ( ( finite_card_set_a @ bot_bot_set_set_a )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1056_card_Oempty,axiom,
    ( ( finite_card_a @ bot_bot_set_a )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1057_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_1058_card_Oinfinite,axiom,
    ! [A: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ A )
     => ( ( finite_card_set_nat @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1059_card_Oinfinite,axiom,
    ! [A: set_a] :
      ( ~ ( finite_finite_a @ A )
     => ( ( finite_card_a @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1060_card_Oinfinite,axiom,
    ! [A: set_set_a] :
      ( ~ ( finite_finite_set_a @ A )
     => ( ( finite_card_set_a @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1061_card_Oinfinite,axiom,
    ! [A: set_nat] :
      ( ~ ( finite_finite_nat @ A )
     => ( ( finite_card_nat @ A )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_1062_card__0__eq,axiom,
    ! [A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A )
     => ( ( ( finite_card_set_nat @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
thf(fact_1063_card__0__eq,axiom,
    ! [A: set_set_a] :
      ( ( finite_finite_set_a @ A )
     => ( ( ( finite_card_set_a @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_set_a ) ) ) ).

% card_0_eq
thf(fact_1064_card__0__eq,axiom,
    ! [A: set_a] :
      ( ( finite_finite_a @ A )
     => ( ( ( finite_card_a @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_a ) ) ) ).

% card_0_eq
thf(fact_1065_card__0__eq,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( ( finite_card_nat @ A )
          = zero_zero_nat )
        = ( A = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_1066_point__rep__singleton__inval,axiom,
    ! [X: set_a,B: set_set_a] :
      ( ~ ( member_set_a @ X @ B )
     => ( ( design5008467512594872073_set_a @ ( add_mset_set_set_a @ B @ zero_z6396401802697562811_set_a ) @ X )
        = zero_zero_nat ) ) ).

% point_rep_singleton_inval
thf(fact_1067_point__rep__singleton__inval,axiom,
    ! [X: set_nat,B: set_set_nat] :
      ( ~ ( member_set_nat @ X @ B )
     => ( ( design7496494955846209563et_nat @ ( add_mset_set_set_nat @ B @ zero_z6127839489552129301et_nat ) @ X )
        = zero_zero_nat ) ) ).

% point_rep_singleton_inval
thf(fact_1068_point__rep__singleton__inval,axiom,
    ! [X: a,B: set_a] :
      ( ~ ( member_a @ X @ B )
     => ( ( design6637022207325878697mber_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) @ X )
        = zero_zero_nat ) ) ).

% point_rep_singleton_inval
thf(fact_1069_point__rep__singleton__inval,axiom,
    ! [X: nat,B: set_nat] :
      ( ~ ( member_nat @ X @ B )
     => ( ( design3571518413069006949er_nat @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) @ X )
        = zero_zero_nat ) ) ).

% point_rep_singleton_inval
thf(fact_1070_point__rep__singleton__val,axiom,
    ! [X: set_a,B: set_set_a] :
      ( ( member_set_a @ X @ B )
     => ( ( design5008467512594872073_set_a @ ( add_mset_set_set_a @ B @ zero_z6396401802697562811_set_a ) @ X )
        = one_one_nat ) ) ).

% point_rep_singleton_val
thf(fact_1071_point__rep__singleton__val,axiom,
    ! [X: set_nat,B: set_set_nat] :
      ( ( member_set_nat @ X @ B )
     => ( ( design7496494955846209563et_nat @ ( add_mset_set_set_nat @ B @ zero_z6127839489552129301et_nat ) @ X )
        = one_one_nat ) ) ).

% point_rep_singleton_val
thf(fact_1072_point__rep__singleton__val,axiom,
    ! [X: a,B: set_a] :
      ( ( member_a @ X @ B )
     => ( ( design6637022207325878697mber_a @ ( add_mset_set_a @ B @ zero_z5079479921072680283_set_a ) @ X )
        = one_one_nat ) ) ).

% point_rep_singleton_val
thf(fact_1073_point__rep__singleton__val,axiom,
    ! [X: nat,B: set_nat] :
      ( ( member_nat @ X @ B )
     => ( ( design3571518413069006949er_nat @ ( add_mset_set_nat @ B @ zero_z3157962936165190495et_nat ) @ X )
        = one_one_nat ) ) ).

% point_rep_singleton_val
thf(fact_1074_point__rep__non__existance,axiom,
    ! [X: a] :
      ( ~ ( member_a @ X @ ( set_a2 @ v_s ) )
     => ( ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X )
        = zero_zero_nat ) ) ).

% point_rep_non_existance
thf(fact_1075_sub__designI,axiom,
    ! [U: set_a,A8: multiset_set_a] :
      ( ( design_design_a @ U @ A8 )
     => ( ( sub_su7923802003039619913stem_a @ U @ A8 @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
       => ( sub_sub_design_a @ U @ A8 @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).

% sub_designI
thf(fact_1076_sub__designII,axiom,
    ! [U: set_a,A8: multiset_set_a] :
      ( ( design_design_a @ U @ A8 )
     => ( ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ U @ A8 )
       => ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ U @ A8 ) ) ) ).

% sub_designII
thf(fact_1077_designI,axiom,
    ( ! [B6: set_a] :
        ( ( member_set_a @ B6 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( B6 != bot_bot_set_a ) )
   => ( ( ( mset_set_a @ b_s )
       != zero_z5079479921072680283_set_a )
     => ( ( ( set_a2 @ v_s )
         != bot_bot_set_a )
       => ( design_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ) ).

% designI
thf(fact_1078_design_Oblock__size__lt__v,axiom,
    ! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Bl2: set_set_a] :
      ( ( design_design_set_a @ Point_set @ Block_collection )
     => ( ( member_set_set_a @ Bl2 @ ( set_mset_set_set_a @ Block_collection ) )
       => ( ord_less_eq_nat @ ( finite_card_set_a @ Bl2 ) @ ( finite_card_set_a @ Point_set ) ) ) ) ).

% design.block_size_lt_v
thf(fact_1079_design_Oblock__size__lt__v,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
      ( ( design_design_a @ Point_set @ Block_collection )
     => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
       => ( ord_less_eq_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ Point_set ) ) ) ) ).

% design.block_size_lt_v
thf(fact_1080_design_Oblock__size__lt__v,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
      ( ( design_design_nat @ Point_set @ Block_collection )
     => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ Point_set ) ) ) ) ).

% design.block_size_lt_v
thf(fact_1081_pairwise__balance_Oeq__index__rep__imp__complete,axiom,
    ! [Point_set: set_set_nat,Block_collection: multiset_set_set_nat,Index: nat,X: set_nat,Bl2: set_set_nat] :
      ( ( block_5429691286618170193et_nat @ Point_set @ Block_collection @ Index )
     => ( ( Index
          = ( design7496494955846209563et_nat @ Block_collection @ X ) )
       => ( ( member_set_nat @ X @ Point_set )
         => ( ( member_set_set_nat @ Bl2 @ ( set_mset_set_set_nat @ Block_collection ) )
           => ( ( member_set_nat @ X @ Bl2 )
             => ( ( finite_card_set_nat @ Bl2 )
                = ( finite_card_set_nat @ Point_set ) ) ) ) ) ) ) ).

% pairwise_balance.eq_index_rep_imp_complete
thf(fact_1082_pairwise__balance_Oeq__index__rep__imp__complete,axiom,
    ! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Index: nat,X: set_a,Bl2: set_set_a] :
      ( ( block_6207159848980890963_set_a @ Point_set @ Block_collection @ Index )
     => ( ( Index
          = ( design5008467512594872073_set_a @ Block_collection @ X ) )
       => ( ( member_set_a @ X @ Point_set )
         => ( ( member_set_set_a @ Bl2 @ ( set_mset_set_set_a @ Block_collection ) )
           => ( ( member_set_a @ X @ Bl2 )
             => ( ( finite_card_set_a @ Bl2 )
                = ( finite_card_set_a @ Point_set ) ) ) ) ) ) ) ).

% pairwise_balance.eq_index_rep_imp_complete
thf(fact_1083_pairwise__balance_Oeq__index__rep__imp__complete,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a,Index: nat,X: a,Bl2: set_a] :
      ( ( block_5355636846524985331ance_a @ Point_set @ Block_collection @ Index )
     => ( ( Index
          = ( design6637022207325878697mber_a @ Block_collection @ X ) )
       => ( ( member_a @ X @ Point_set )
         => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
           => ( ( member_a @ X @ Bl2 )
             => ( ( finite_card_a @ Bl2 )
                = ( finite_card_a @ Point_set ) ) ) ) ) ) ) ).

% pairwise_balance.eq_index_rep_imp_complete
thf(fact_1084_pairwise__balance_Oeq__index__rep__imp__complete,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat,Index: nat,X: nat,Bl2: set_nat] :
      ( ( block_1456364645985477531ce_nat @ Point_set @ Block_collection @ Index )
     => ( ( Index
          = ( design3571518413069006949er_nat @ Block_collection @ X ) )
       => ( ( member_nat @ X @ Point_set )
         => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
           => ( ( member_nat @ X @ Bl2 )
             => ( ( finite_card_nat @ Bl2 )
                = ( finite_card_nat @ Point_set ) ) ) ) ) ) ) ).

% pairwise_balance.eq_index_rep_imp_complete
thf(fact_1085_design_Oaxioms_I1_J,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a] :
      ( ( design_design_a @ Point_set @ Block_collection )
     => ( design9187838744727572296stem_a @ Point_set @ Block_collection ) ) ).

% design.axioms(1)
thf(fact_1086_design_Oaxioms_I1_J,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat] :
      ( ( design_design_nat @ Point_set @ Block_collection )
     => ( design5426232790142929158em_nat @ Point_set @ Block_collection ) ) ).

% design.axioms(1)
thf(fact_1087_inter__num__of__eq__blocks,axiom,
    ! [B1: set_set_a,B2: set_set_a] :
      ( ( B1 = B2 )
     => ( ( design3520961687418077020_set_a @ B1 @ B2 )
        = ( finite_card_set_a @ B1 ) ) ) ).

% inter_num_of_eq_blocks
thf(fact_1088_inter__num__of__eq__blocks,axiom,
    ! [B1: set_a,B2: set_a] :
      ( ( B1 = B2 )
     => ( ( design7842873109100088828mber_a @ B1 @ B2 )
        = ( finite_card_a @ B1 ) ) ) ).

% inter_num_of_eq_blocks
thf(fact_1089_inter__num__of__eq__blocks,axiom,
    ! [B1: set_nat,B2: set_nat] :
      ( ( B1 = B2 )
     => ( ( design7485525362727208274er_nat @ B1 @ B2 )
        = ( finite_card_nat @ B1 ) ) ) ).

% inter_num_of_eq_blocks
thf(fact_1090_simple__design_Oaxioms_I1_J,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat] :
      ( ( design7861764274488435984gn_nat @ Point_set @ Block_collection )
     => ( design_design_nat @ Point_set @ Block_collection ) ) ).

% simple_design.axioms(1)
thf(fact_1091_simple__design_Oaxioms_I1_J,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a] :
      ( ( design3982635895484621246sign_a @ Point_set @ Block_collection )
     => ( design_design_a @ Point_set @ Block_collection ) ) ).

% simple_design.axioms(1)
thf(fact_1092_design_Oblocks__nempty,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
      ( ( design_design_a @ Point_set @ Block_collection )
     => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
       => ( Bl2 != bot_bot_set_a ) ) ) ).

% design.blocks_nempty
thf(fact_1093_design_Oblocks__nempty,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
      ( ( design_design_nat @ Point_set @ Block_collection )
     => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
       => ( Bl2 != bot_bot_set_nat ) ) ) ).

% design.blocks_nempty
thf(fact_1094_design_Oblocks__nempty__alt,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a] :
      ( ( design_design_a @ Point_set @ Block_collection )
     => ! [X5: set_a] :
          ( ( member_set_a @ X5 @ ( set_mset_set_a @ Block_collection ) )
         => ( X5 != bot_bot_set_a ) ) ) ).

% design.blocks_nempty_alt
thf(fact_1095_design_Oblocks__nempty__alt,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat] :
      ( ( design_design_nat @ Point_set @ Block_collection )
     => ! [X5: set_nat] :
          ( ( member_set_nat @ X5 @ ( set_mset_set_nat @ Block_collection ) )
         => ( X5 != bot_bot_set_nat ) ) ) ).

% design.blocks_nempty_alt
thf(fact_1096_design_Oblock__set__nempty__imp__points,axiom,
    ! [Point_set: set_a,Block_collection: multiset_set_a] :
      ( ( design_design_a @ Point_set @ Block_collection )
     => ( ( Block_collection != zero_z5079479921072680283_set_a )
       => ( Point_set != bot_bot_set_a ) ) ) ).

% design.block_set_nempty_imp_points
thf(fact_1097_design_Oblock__set__nempty__imp__points,axiom,
    ! [Point_set: set_nat,Block_collection: multiset_set_nat] :
      ( ( design_design_nat @ Point_set @ Block_collection )
     => ( ( Block_collection != zero_z3157962936165190495et_nat )
       => ( Point_set != bot_bot_set_nat ) ) ) ).

% design.block_set_nempty_imp_points
thf(fact_1098_infinite__arbitrarily__large,axiom,
    ! [A: set_set_nat,N: nat] :
      ( ~ ( finite1152437895449049373et_nat @ A )
     => ? [B4: set_set_nat] :
          ( ( finite1152437895449049373et_nat @ B4 )
          & ( ( finite_card_set_nat @ B4 )
            = N )
          & ( ord_le6893508408891458716et_nat @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1099_infinite__arbitrarily__large,axiom,
    ! [A: set_set_a,N: nat] :
      ( ~ ( finite_finite_set_a @ A )
     => ? [B4: set_set_a] :
          ( ( finite_finite_set_a @ B4 )
          & ( ( finite_card_set_a @ B4 )
            = N )
          & ( ord_le3724670747650509150_set_a @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1100_infinite__arbitrarily__large,axiom,
    ! [A: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ A )
     => ? [B4: set_nat] :
          ( ( finite_finite_nat @ B4 )
          & ( ( finite_card_nat @ B4 )
            = N )
          & ( ord_less_eq_set_nat @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1101_infinite__arbitrarily__large,axiom,
    ! [A: set_a,N: nat] :
      ( ~ ( finite_finite_a @ A )
     => ? [B4: set_a] :
          ( ( finite_finite_a @ B4 )
          & ( ( finite_card_a @ B4 )
            = N )
          & ( ord_less_eq_set_a @ B4 @ A ) ) ) ).

% infinite_arbitrarily_large
thf(fact_1102_card__subset__eq,axiom,
    ! [B3: set_set_nat,A: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B3 )
     => ( ( ord_le6893508408891458716et_nat @ A @ B3 )
       => ( ( ( finite_card_set_nat @ A )
            = ( finite_card_set_nat @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_1103_card__subset__eq,axiom,
    ! [B3: set_set_a,A: set_set_a] :
      ( ( finite_finite_set_a @ B3 )
     => ( ( ord_le3724670747650509150_set_a @ A @ B3 )
       => ( ( ( finite_card_set_a @ A )
            = ( finite_card_set_a @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_1104_card__subset__eq,axiom,
    ! [B3: set_nat,A: set_nat] :
      ( ( finite_finite_nat @ B3 )
     => ( ( ord_less_eq_set_nat @ A @ B3 )
       => ( ( ( finite_card_nat @ A )
            = ( finite_card_nat @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_1105_card__subset__eq,axiom,
    ! [B3: set_a,A: set_a] :
      ( ( finite_finite_a @ B3 )
     => ( ( ord_less_eq_set_a @ A @ B3 )
       => ( ( ( finite_card_a @ A )
            = ( finite_card_a @ B3 ) )
         => ( A = B3 ) ) ) ) ).

% card_subset_eq
thf(fact_1106_card__insert__le,axiom,
    ! [A: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ ( insert_a @ X @ A ) ) ) ).

% card_insert_le
thf(fact_1107_card__insert__le,axiom,
    ! [A: set_set_a,X: set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ ( insert_set_a @ X @ A ) ) ) ).

% card_insert_le
thf(fact_1108_card__insert__le,axiom,
    ! [A: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ ( insert_nat @ X @ A ) ) ) ).

% card_insert_le
thf(fact_1109_max__point__rep,axiom,
    ! [B3: multiset_set_a,X: a] : ( ord_less_eq_nat @ ( design6637022207325878697mber_a @ B3 @ X ) @ ( size_s6566526139600085008_set_a @ B3 ) ) ).

% max_point_rep
thf(fact_1110_max__point__rep,axiom,
    ! [B3: multiset_set_nat,X: nat] : ( ord_less_eq_nat @ ( design3571518413069006949er_nat @ B3 @ X ) @ ( size_s7462436076474991978et_nat @ B3 ) ) ).

% max_point_rep
thf(fact_1111_point__indices__elem__in,axiom,
    ! [Ps: set_a,T3: nat] :
      ( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
     => ( ( ( finite_card_a @ Ps )
          = T3 )
       => ( member_nat @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) @ ( design328527185268214962ices_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ T3 ) ) ) ) ).

% point_indices_elem_in
thf(fact_1112_incomplete__index__strict__lt__rep,axiom,
    ! [X: a] :
      ( ! [Bl: set_a] :
          ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
         => ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
            & ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) )
     => ( ( member_a @ X @ ( set_a2 @ v_s ) )
       => ( ( ord_less_nat @ zero_zero_nat @ lambda )
         => ( ord_less_nat @ lambda @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) ) ) ) ) ).

% incomplete_index_strict_lt_rep
thf(fact_1113_simple__const__inter__block__size,axiom,
    ( ! [Bl: set_a] :
        ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ord_less_nat @ lambda @ ( finite_card_a @ Bl ) ) )
   => ( design3982635895484621246sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ).

% simple_const_inter_block_size
thf(fact_1114_dual__blocks__nempty,axiom,
    ! [Bl2: set_nat] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ v_s ) )
         => ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X4 ) ) )
     => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
       => ( Bl2 != bot_bot_set_nat ) ) ) ).

% dual_blocks_nempty
thf(fact_1115_simple__block__size__eq__card,axiom,
    ( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
    = ( finite_card_set_a @ ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) ) ) ) ).

% simple_block_size_eq_card
thf(fact_1116_v__non__zero,axiom,
    ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).

% v_non_zero
thf(fact_1117_b__positive,axiom,
    ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).

% b_positive
thf(fact_1118_t__design__min__v,axiom,
    ord_less_nat @ one_one_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).

% t_design_min_v
thf(fact_1119_dual__sys_Oadd__block__rep__number__not__in,axiom,
    ! [X: nat,B: set_nat] :
      ( ~ ( member_nat @ X @ B )
     => ( ( design3571518413069006949er_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) @ X )
        = ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) ) ) ).

% dual_sys.add_block_rep_number_not_in
thf(fact_1120_block__size__gt__0,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Bl2 ) ) ) ).

% block_size_gt_0
thf(fact_1121_valid__blocks__index__obtains,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ~ ! [J3: nat] :
            ~ ( ( ( nth_set_a @ b_s @ J3 )
                = Bl2 )
              & ( ord_less_nat @ J3 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).

% valid_blocks_index_obtains
thf(fact_1122_valid__blocks__index__cons,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ? [J3: nat] :
          ( ( ( nth_set_a @ b_s @ J3 )
            = Bl2 )
          & ( ord_less_nat @ J3 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).

% valid_blocks_index_cons
thf(fact_1123_valid__blocks__index,axiom,
    ! [J: nat] :
      ( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
     => ( member_set_a @ ( nth_set_a @ b_s @ J ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).

% valid_blocks_index
thf(fact_1124_inter__num__lt__block__size__strict_I1_J,axiom,
    ! [Bl1: set_a,Bl22: set_a] :
      ( ( member_set_a @ Bl1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( member_set_a @ Bl22 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( Bl1 != Bl22 )
         => ( ( ( finite_card_a @ Bl1 )
              = ( finite_card_a @ Bl22 ) )
           => ( ord_less_nat @ ( design7842873109100088828mber_a @ Bl1 @ Bl22 ) @ ( finite_card_a @ Bl1 ) ) ) ) ) ) ).

% inter_num_lt_block_size_strict(1)
thf(fact_1125_inter__num__lt__block__size__strict_I2_J,axiom,
    ! [Bl1: set_a,Bl22: set_a] :
      ( ( member_set_a @ Bl1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( member_set_a @ Bl22 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( Bl1 != Bl22 )
         => ( ( ( finite_card_a @ Bl1 )
              = ( finite_card_a @ Bl22 ) )
           => ( ord_less_nat @ ( design7842873109100088828mber_a @ Bl1 @ Bl22 ) @ ( finite_card_a @ Bl22 ) ) ) ) ) ) ).

% inter_num_lt_block_size_strict(2)
thf(fact_1126_lt,axiom,
    ord_less_nat @ i1 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).

% lt
thf(fact_1127_lt2,axiom,
    ord_less_nat @ i2 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).

% lt2
thf(fact_1128_block__sizes__non__empty,axiom,
    ( ( ( mset_set_a @ b_s )
     != zero_z5079479921072680283_set_a )
   => ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ) ) ) ).

% block_sizes_non_empty
thf(fact_1129_dual__sys_Osys__block__sizes__obtain__bl,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ? [X4: set_nat] :
          ( ( member_set_nat @ X4 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
          & ( ( finite_card_nat @ X4 )
            = X ) ) ) ).

% dual_sys.sys_block_sizes_obtain_bl
thf(fact_1130_dual__sys_Osys__block__sizes__in,axiom,
    ! [Bl2: set_nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( member_nat @ ( finite_card_nat @ Bl2 ) @ ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).

% dual_sys.sys_block_sizes_in
thf(fact_1131_incomplete__imp__incomp__block,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
        & ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ).

% incomplete_imp_incomp_block
thf(fact_1132_incomplete__alt__size,axiom,
    ! [Bl2: set_a] :
      ( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
        & ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
     => ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).

% incomplete_alt_size
thf(fact_1133_incomplete__alt__in,axiom,
    ! [Bl2: set_a] :
      ( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
        & ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
     => ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).

% incomplete_alt_in
thf(fact_1134_rep__number__non__zero__system__point,axiom,
    ! [X: a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) )
     => ( member_a @ X @ ( set_a2 @ v_s ) ) ) ).

% rep_number_non_zero_system_point
thf(fact_1135_dual__blocks__size__is__rep__obtain,axiom,
    ! [Bl2: set_nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ~ ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ v_s ) )
           => ( ( finite_card_nat @ Bl2 )
             != ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X4 ) ) ) ) ).

% dual_blocks_size_is_rep_obtain
thf(fact_1136_dual__sys_Oadd__block__rep__number__in,axiom,
    ! [X: nat,B: set_nat] :
      ( ( member_nat @ X @ B )
     => ( ( design3571518413069006949er_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) @ X )
        = ( plus_plus_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) @ one_one_nat ) ) ) ).

% dual_sys.add_block_rep_number_in
thf(fact_1137_b__non__zero__imp__v__non__zero,axiom,
    ( ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
   => ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ).

% b_non_zero_imp_v_non_zero
thf(fact_1138_points__index__zero,axiom,
    ! [Ps: set_a] :
      ( ( ord_less_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( finite_card_a @ Ps ) )
     => ( ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps )
        = zero_zero_nat ) ) ).

% points_index_zero
thf(fact_1139_point__in__block__rep__min__iff,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( set_a2 @ v_s ) )
     => ? [Bl: set_a] :
          ( ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
            & ( member_a @ X @ Bl ) )
          = ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X ) ) ) ) ).

% point_in_block_rep_min_iff
thf(fact_1140_dual__sys_Oblock__sizes__non__empty,axiom,
    ( ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
     != zero_z3157962936165190495et_nat )
   => ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ).

% dual_sys.block_sizes_non_empty
thf(fact_1141_block__comp__incomplete,axiom,
    ! [Bl2: set_a] :
      ( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
        & ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) ) ) ) ).

% block_comp_incomplete
thf(fact_1142_block__comp__incomplete__nempty,axiom,
    ! [Bl2: set_a] :
      ( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
        & ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
     => ( ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 )
       != bot_bot_set_a ) ) ).

% block_comp_incomplete_nempty
thf(fact_1143_complement__design,axiom,
    ( ! [Bl: set_a] :
        ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
          & ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) )
   => ( design_design_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).

% complement_design
thf(fact_1144_incomplete__alt__imp,axiom,
    ! [Bl2: set_a] :
      ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
     => ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
          & ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ) ).

% incomplete_alt_imp
thf(fact_1145_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M7: nat,N5: nat] :
          ( ( ord_less_eq_nat @ M7 @ N5 )
          & ( M7 != N5 ) ) ) ) ).

% nat_less_le
thf(fact_1146_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_1147_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M7: nat,N5: nat] :
          ( ( ord_less_nat @ M7 @ N5 )
          | ( M7 = N5 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1148_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_1149_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_1150_less__mono__imp__le__mono,axiom,
    ! [F3: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ord_less_nat @ ( F3 @ I3 ) @ ( F3 @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F3 @ I ) @ ( F3 @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1151_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K5: nat] :
            ( ( ord_less_eq_nat @ K5 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K5 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ K5 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1152_less__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1153_diff__less__mono,axiom,
    ! [A2: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B )
     => ( ( ord_less_eq_nat @ C @ A2 )
       => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_1154_mono__nat__linear__lb,axiom,
    ! [F3: nat > nat,M2: nat,K2: nat] :
      ( ! [M8: nat,N6: nat] :
          ( ( ord_less_nat @ M8 @ N6 )
         => ( ord_less_nat @ ( F3 @ M8 ) @ ( F3 @ N6 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F3 @ M2 ) @ K2 ) @ ( F3 @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1155_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).

% less_diff_conv2
thf(fact_1156_remove__complete__blocks__set__pbd,axiom,
    ! [X: nat,A: multiset_set_a] :
      ( ( ord_less_nat @ X @ lambda )
     => ( ( ( size_s6566526139600085008_set_a @ A )
          = X )
       => ( ( subset_mset_set_a @ A @ ( mset_set_a @ b_s ) )
         => ( ! [A4: set_a] :
                ( ( member_set_a @ A4 @ ( set_mset_set_a @ A ) )
               => ( A4
                  = ( set_a2 @ v_s ) ) )
           => ( block_5355636846524985331ance_a @ ( set_a2 @ v_s ) @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ A ) @ ( minus_minus_nat @ lambda @ X ) ) ) ) ) ) ).

% remove_complete_blocks_set_pbd
thf(fact_1157_dual__blocks__points__index__inter,axiom,
    ! [I12: nat,I23: nat] :
      ( ( ord_less_nat @ I12 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( ord_less_nat @ I23 @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( design6574611146354332593ex_nat @ ( mset_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) @ ( insert_nat @ I12 @ ( insert_nat @ I23 @ bot_bot_set_nat ) ) )
          = ( design7842873109100088828mber_a @ ( nth_set_a @ b_s @ I12 ) @ ( nth_set_a @ b_s @ I23 ) ) ) ) ) ).

% dual_blocks_points_index_inter
thf(fact_1158_strong__del__block__des,axiom,
    ! [B: set_a] :
      ( ! [Bl: set_a] :
          ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
         => ~ ( ord_less_set_a @ Bl @ B ) )
     => ( design_design_a @ ( minus_minus_set_a @ ( set_a2 @ v_s ) @ B ) @ ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ B ) ) ) ).

% strong_del_block_des
thf(fact_1159_dual__blocks__ordered__eq,axiom,
    ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
    = ( mset_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) ) ).

% dual_blocks_ordered_eq
thf(fact_1160_incomplete__imp__proper__subset,axiom,
    ! [Bl2: set_a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ord_less_set_a @ Bl2 @ ( set_a2 @ v_s ) ) ) ).

% incomplete_imp_proper_subset
thf(fact_1161_ord__dual__blocks__b,axiom,
    ( ( size_s7462436076474991978et_nat @ ( mset_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
    = ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).

% ord_dual_blocks_b
thf(fact_1162_incomplete__block__proper__subset,axiom,
    ! [Bl2: set_a] :
      ( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
        & ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) )
     => ( ord_less_set_a @ Bl2 @ ( set_a2 @ v_s ) ) ) ).

% incomplete_block_proper_subset
thf(fact_1163_dual__blocks__rep__is__size,axiom,
    ! [I: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_set_a @ b_s ) )
     => ( ( design3571518413069006949er_nat @ ( mset_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) @ I )
        = ( finite_card_a @ ( nth_set_a @ b_s @ I ) ) ) ) ).

% dual_blocks_rep_is_size
thf(fact_1164_blocks__index__ne__belong,axiom,
    ! [I12: nat,I23: nat] :
      ( ( ord_less_nat @ I12 @ ( size_size_list_set_a @ b_s ) )
     => ( ( ord_less_nat @ I23 @ ( size_size_list_set_a @ b_s ) )
       => ( ( I12 != I23 )
         => ( member_set_a @ ( nth_set_a @ b_s @ I23 ) @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ ( mset_set_a @ b_s ) @ ( add_mset_set_a @ ( nth_set_a @ b_s @ I12 ) @ zero_z5079479921072680283_set_a ) ) ) ) ) ) ) ).

% blocks_index_ne_belong
thf(fact_1165_simp__blocks__length__card,axiom,
    ( ( size_size_list_set_a @ b_s )
    = ( finite_card_set_a @ ( set_set_a2 @ b_s ) ) ) ).

% simp_blocks_length_card
thf(fact_1166_blocks__index__simp__unique,axiom,
    ! [I12: nat,I23: nat] :
      ( ( ord_less_nat @ I12 @ ( size_size_list_set_a @ b_s ) )
     => ( ( ord_less_nat @ I23 @ ( size_size_list_set_a @ b_s ) )
       => ( ( I12 != I23 )
         => ( ( nth_set_a @ b_s @ I12 )
           != ( nth_set_a @ b_s @ I23 ) ) ) ) ) ).

% blocks_index_simp_unique
thf(fact_1167_blocks__list__length,axiom,
    ( ( size_size_list_set_a @ b_s )
    = ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).

% blocks_list_length
thf(fact_1168_assms_I1_J,axiom,
    ord_less_eq_set_nat @ ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ).

% assms(1)
thf(fact_1169_dual__sys_Odelete__point__blocks__wf,axiom,
    ! [B: set_nat,P: nat] :
      ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) )
     => ( ord_less_eq_set_nat @ B @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( insert_nat @ P @ bot_bot_set_nat ) ) ) ) ).

% dual_sys.delete_point_blocks_wf
thf(fact_1170_dual__sys_Oblock__comp__incomplete__nempty,axiom,
    ! [Bl2: set_nat] :
      ( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
        & ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
     => ( ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 )
       != bot_bot_set_nat ) ) ).

% dual_sys.block_comp_incomplete_nempty
thf(fact_1171_points__list__length,axiom,
    ( ( size_size_list_a @ v_s )
    = ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).

% points_list_length
thf(fact_1172_dual__blocks__len,axiom,
    ( ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) )
    = ( size_size_list_a @ v_s ) ) ).

% dual_blocks_len
thf(fact_1173_valid__points__index__obtains,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( set_a2 @ v_s ) )
     => ~ ! [I3: nat] :
            ~ ( ( ( nth_a @ v_s @ I3 )
                = X )
              & ( ord_less_nat @ I3 @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ) ).

% valid_points_index_obtains
thf(fact_1174_valid__points__index__cons,axiom,
    ! [X: a] :
      ( ( member_a @ X @ ( set_a2 @ v_s ) )
     => ? [I3: nat] :
          ( ( ( nth_a @ v_s @ I3 )
            = X )
          & ( ord_less_nat @ I3 @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) ) ) ).

% valid_points_index_cons
thf(fact_1175_valid__points__index,axiom,
    ! [I: nat] :
      ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
     => ( member_a @ ( nth_a @ v_s @ I ) @ ( set_a2 @ v_s ) ) ) ).

% valid_points_index
thf(fact_1176_dual__blocks__ne__index__ne,axiom,
    ! [J1: nat,J22: nat] :
      ( ( ord_less_nat @ J1 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
     => ( ( ord_less_nat @ J22 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
       => ( ( ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J1 )
           != ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J22 ) )
         => ( J1 != J22 ) ) ) ) ).

% dual_blocks_ne_index_ne
thf(fact_1177_dual__sys_Ofinite__sets,axiom,
    finite_finite_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ).

% dual_sys.finite_sets
thf(fact_1178_block__points__valid__point__index,axiom,
    ! [Bl2: set_a,X: a] :
      ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
     => ( ( member_a @ X @ Bl2 )
       => ~ ! [I3: nat] :
              ~ ( ( ord_less_nat @ I3 @ ( size_size_list_a @ v_s ) )
                & ( ( nth_a @ v_s @ I3 )
                  = X ) ) ) ) ).

% block_points_valid_point_index
thf(fact_1179_dual__sys_Oblock__complement__subset__points,axiom,
    ! [Ps: set_nat,Bl2: set_nat] :
      ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) )
     => ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).

% dual_sys.block_complement_subset_points
thf(fact_1180_dual__sys_Oblock__complement__elem__iff,axiom,
    ! [Ps: set_nat,Bl2: set_nat] :
      ( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) )
        = ( ! [X3: nat] :
              ( ( member_nat @ X3 @ Ps )
             => ~ ( member_nat @ X3 @ Bl2 ) ) ) ) ) ).

% dual_sys.block_complement_elem_iff
thf(fact_1181_dual__sys_Oblock__comp__elem__alt__right,axiom,
    ! [Ps: set_nat,Bl2: set_nat] :
      ( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ Ps )
           => ~ ( member_nat @ X4 @ Bl2 ) )
       => ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) ) ) ) ).

% dual_sys.block_comp_elem_alt_right
thf(fact_1182_dual__sys_Oblock__comp__elem__alt__left,axiom,
    ! [X: nat,Bl2: set_nat,Ps: set_nat] :
      ( ( member_nat @ X @ Bl2 )
     => ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) )
       => ~ ( member_nat @ X @ Ps ) ) ) ).

% dual_sys.block_comp_elem_alt_left
thf(fact_1183_dual__sys_Oblock__complement__def,axiom,
    ! [B: set_nat] :
      ( ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B )
      = ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B ) ) ).

% dual_sys.block_complement_def
thf(fact_1184_dual__blocks__v,axiom,
    ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
    = ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ).

% dual_blocks_v
thf(fact_1185_dual__sys_Ofinite__incidence__system__axioms,axiom,
    design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ).

% dual_sys.finite_incidence_system_axioms
thf(fact_1186_dual__sys_Ocomplement__finite,axiom,
    design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.complement_finite
thf(fact_1187_dual__blocks__const__intersect,axiom,
    ! [J1: nat,J22: nat] :
      ( ( ord_less_nat @ J1 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
     => ( ( ord_less_nat @ J22 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
       => ( ( J1 != J22 )
         => ( ( design7485525362727208274er_nat @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J1 ) @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J22 ) )
            = lambda ) ) ) ) ).

% dual_blocks_const_intersect
thf(fact_1188_dual__blocks__size__is__rep,axiom,
    ! [J: nat] :
      ( ( ord_less_nat @ J @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
     => ( ( finite_card_nat @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J ) )
        = ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ ( nth_a @ v_s @ J ) ) ) ) ).

% dual_blocks_size_is_rep
thf(fact_1189_dual__sys_Oblock__complement__size,axiom,
    ! [B: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( finite_card_nat @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B ) )
        = ( minus_minus_nat @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) @ ( finite_card_nat @ B ) ) ) ) ).

% dual_sys.block_complement_size
thf(fact_1190_dual__sys_Oreplication__numbers__non__empty,axiom,
    ( ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) )
     != bot_bot_set_nat )
   => ( ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
     != bot_bot_set_nat ) ) ).

% dual_sys.replication_numbers_non_empty
thf(fact_1191_dual__sys_Oreplication__numbers__finite,axiom,
    finite_finite_nat @ ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.replication_numbers_finite
thf(fact_1192_dual__sys_Owf__invalid__point,axiom,
    ! [X: nat,B: set_nat] :
      ( ~ ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
       => ~ ( member_nat @ X @ B ) ) ) ).

% dual_sys.wf_invalid_point
thf(fact_1193_dual__sys_Oincidence__alt__def,axiom,
    ! [P: nat,B: set_nat] :
      ( ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
       => ( ( design8502206366797944887nt_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P @ B )
          = ( member_nat @ P @ B ) ) ) ) ).

% dual_sys.incidence_alt_def
thf(fact_1194_dual__sys_Oobtain__point__with__rep,axiom,
    ! [R: nat] :
      ( ( member_nat @ R @ ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ? [X4: nat] :
          ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
          & ( ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X4 )
            = R ) ) ) ).

% dual_sys.obtain_point_with_rep
thf(fact_1195_dual__sys_Opoint__rep__number__in__set,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( member_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) @ ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).

% dual_sys.point_rep_number_in_set
thf(fact_1196_dual__sys_Odelete__block__fin__incidence__sys,axiom,
    ! [B: set_nat] : ( design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) ) ).

% dual_sys.delete_block_fin_incidence_sys
thf(fact_1197_dual__sys_Odelete__invalid__pt__strong__eq,axiom,
    ! [P: nat] :
      ( ~ ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
        = ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) ) ).

% dual_sys.delete_invalid_pt_strong_eq
thf(fact_1198_dual__sys_Odel__invalid__point__blocks,axiom,
    ! [P: nat] :
      ( ~ ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P )
        = ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.del_invalid_point_blocks
thf(fact_1199_dual__sys_Orep__number__non__zero__system__point,axiom,
    ! [X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) )
     => ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).

% dual_sys.rep_number_non_zero_system_point
thf(fact_1200_dual__sys_Ocomplete__block__size__eq__points,axiom,
    ! [Bl2: set_nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( ( finite_card_nat @ Bl2 )
          = ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
       => ( Bl2
          = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ) ).

% dual_sys.complete_block_size_eq_points
thf(fact_1201_dual__sys_Owellformed,axiom,
    ! [B: set_nat] :
      ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ord_less_eq_set_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).

% dual_sys.wellformed
thf(fact_1202_dual__sys_Ocomplement__blocks__wf,axiom,
    ! [Bl2: set_nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
     => ( ord_less_eq_set_nat @ Bl2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).

% dual_sys.complement_blocks_wf
thf(fact_1203_dual__blocks__elem__iff,axiom,
    ! [J: nat,X: nat] :
      ( ( ord_less_nat @ J @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
     => ( ( member_nat @ X @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J ) )
        = ( ( member_a @ ( nth_a @ v_s @ J ) @ ( nth_set_a @ b_s @ X ) )
          & ( ord_less_nat @ X @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ) ) ).

% dual_blocks_elem_iff
thf(fact_1204_dual__incidence__iff,axiom,
    ! [I: nat,J: nat,Bl2: set_a,X: a] :
      ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
     => ( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( ( nth_set_a @ b_s @ J )
            = Bl2 )
         => ( ( ( nth_a @ v_s @ I )
              = X )
           => ( ( member_a @ X @ Bl2 )
              = ( member_nat @ J @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ I ) ) ) ) ) ) ) ).

% dual_incidence_iff
thf(fact_1205_dual__incidence__iff2,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
     => ( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) )
          = ( member_nat @ J @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ I ) ) ) ) ) ).

% dual_incidence_iff2
thf(fact_1206_dual__sys_Opoints__index__ps__nin,axiom,
    ! [Ps: set_nat] :
      ( ~ ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps )
        = zero_zero_nat ) ) ).

% dual_sys.points_index_ps_nin
thf(fact_1207_dual__sys_Oadd__block__fin__cond,axiom,
    ! [B: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) ) ) ).

% dual_sys.add_block_fin_cond
thf(fact_1208_dual__sys_Oblock__complement__inv,axiom,
    ! [Bl2: set_nat,Bl22: set_nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 )
          = Bl22 )
       => ( ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl22 )
          = Bl2 ) ) ) ).

% dual_sys.block_complement_inv
thf(fact_1209_dual__sys_Oobtain__comp__block__orig,axiom,
    ! [Bl1: set_nat] :
      ( ( member_set_nat @ Bl1 @ ( set_mset_set_nat @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
     => ~ ! [Bl23: set_nat] :
            ( ( member_set_nat @ Bl23 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
           => ( Bl1
             != ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl23 ) ) ) ) ).

% dual_sys.obtain_comp_block_orig
thf(fact_1210_dual__is__const__intersect__des,axiom,
    ( ( ord_less_nat @ zero_zero_nat @ lambda )
   => ( design137120128173859224gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ lambda ) ) ).

% dual_is_const_intersect_des
thf(fact_1211_dual__sys_Ostrong__del__block__fin,axiom,
    ! [B: set_nat] : ( design5426232790142929158em_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B ) @ ( design3550126062406151447ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) ) ).

% dual_sys.strong_del_block_fin
thf(fact_1212_dual__is__design,axiom,
    ( ! [X4: a] :
        ( ( member_a @ X4 @ ( set_a2 @ v_s ) )
       => ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X4 ) ) )
   => ( design_design_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_is_design
thf(fact_1213_dual__blocks__inter__index,axiom,
    ! [J1: nat,J22: nat] :
      ( ( ord_less_nat @ J1 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
     => ( ( ord_less_nat @ J22 @ ( size_s3254054031482475050et_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) ) )
       => ( ( design7485525362727208274er_nat @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J1 ) @ ( nth_set_nat @ ( dual_o8653602421375082564ered_a @ v_s @ b_s ) @ J22 ) )
          = ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ ( insert_a @ ( nth_a @ v_s @ J1 ) @ ( insert_a @ ( nth_a @ v_s @ J22 ) @ bot_bot_set_a ) ) ) ) ) ) ).

% dual_blocks_inter_index
thf(fact_1214_dual__sys_Oincomplete__block__proper__subset,axiom,
    ! [Bl2: set_nat] :
      ( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
        & ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
     => ( ord_less_set_nat @ Bl2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).

% dual_sys.incomplete_block_proper_subset
thf(fact_1215_dual__sys_Oincomplete__alt__in,axiom,
    ! [Bl2: set_nat] :
      ( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
        & ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
     => ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).

% dual_sys.incomplete_alt_in
thf(fact_1216_dual__sys_Oincomplete__alt__size,axiom,
    ! [Bl2: set_nat] :
      ( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
        & ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
     => ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ) ).

% dual_sys.incomplete_alt_size
thf(fact_1217_dual__sys_Oblock__size__lt__order,axiom,
    ! [Bl2: set_nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ) ).

% dual_sys.block_size_lt_order
thf(fact_1218_dual__sys_Opoints__index__zero,axiom,
    ! [Ps: set_nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) @ ( finite_card_nat @ Ps ) )
     => ( ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps )
        = zero_zero_nat ) ) ).

% dual_sys.points_index_zero
thf(fact_1219_dual__sys_Opoint__in__block__rep__min__iff,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ? [Bl: set_nat] :
          ( ( ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
            & ( member_nat @ X @ Bl ) )
          = ( ord_less_nat @ zero_zero_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X ) ) ) ) ).

% dual_sys.point_in_block_rep_min_iff
thf(fact_1220_dual__sys_Ocomplete__block__all__subsets,axiom,
    ! [Bl2: set_nat,Ps: set_nat] :
      ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( ( finite_card_nat @ Bl2 )
          = ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
       => ( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
         => ( ord_less_eq_set_nat @ Ps @ Bl2 ) ) ) ) ).

% dual_sys.complete_block_all_subsets
thf(fact_1221_dual__sys_Ocomplement__rep__number,axiom,
    ! [X: nat,R: nat] :
      ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X )
          = R )
       => ( ( design3571518413069006949er_nat @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ X )
          = ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ R ) ) ) ) ).

% dual_sys.complement_rep_number
thf(fact_1222_dual__sys_Oadd__del__block__inv,axiom,
    ! [Bl2: set_nat] :
      ( ( ord_less_eq_set_nat @ Bl2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( design755385109423264192ck_nat @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) @ Bl2 )
        = ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.add_del_block_inv
thf(fact_1223_dual__sys_Oadd__block__fin,axiom,
    ! [B: set_nat] :
      ( ( finite_finite_nat @ B )
     => ( design5426232790142929158em_nat @ ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) ) ) ).

% dual_sys.add_block_fin
thf(fact_1224_dual__sys_Oblock__comp__incomplete,axiom,
    ! [Bl2: set_nat] :
      ( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
        & ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design2875492832550762736nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ Bl2 ) ) ) ) ).

% dual_sys.block_comp_incomplete
thf(fact_1225_dual__sys_Oremove__invalid__point__block,axiom,
    ! [P: nat,Bl2: set_nat] :
      ( ~ ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
       => ( ( minus_minus_set_nat @ Bl2 @ ( insert_nat @ P @ bot_bot_set_nat ) )
          = Bl2 ) ) ) ).

% dual_sys.remove_invalid_point_block
thf(fact_1226_dual__sys_Ofinite__sysI,axiom,
    ( ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
   => ( design5426232790142929158em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.finite_sysI
thf(fact_1227_dual__sys_Ocomplement__same__b,axiom,
    ( ( size_s7462436076474991978et_nat @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
    = ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_sys.complement_same_b
thf(fact_1228_dual__sys_Opoint__rep__non__existance,axiom,
    ! [X: nat] :
      ( ~ ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X )
        = zero_zero_nat ) ) ).

% dual_sys.point_rep_non_existance
thf(fact_1229_incidence__cond__indexed,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
     => ( ( ord_less_nat @ J @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( design3210447939978979927dent_a @ ( mset_set_a @ b_s ) @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) )
          = ( member_a @ ( nth_a @ v_s @ I ) @ ( nth_set_a @ b_s @ J ) ) ) ) ) ).

% incidence_cond_indexed
thf(fact_1230_dual__sys_Oincomplete__alt__imp,axiom,
    ! [Bl2: set_nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
     => ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
       => ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
          & ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ) ).

% dual_sys.incomplete_alt_imp
thf(fact_1231_dual__sys_OdesignI,axiom,
    ( ! [B6: set_nat] :
        ( ( member_set_nat @ B6 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
       => ( B6 != bot_bot_set_nat ) )
   => ( ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
       != zero_z3157962936165190495et_nat )
     => ( ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) )
         != bot_bot_set_nat )
       => ( design_design_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ).

% dual_sys.designI
thf(fact_1232_dual__is__simp__const__inter__des,axiom,
    ( ( ord_less_nat @ zero_zero_nat @ lambda )
   => ( ! [Bl: set_a] :
          ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
         => ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
            & ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) )
     => ( design8545500683235687882gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ lambda ) ) ) ).

% dual_is_simp_const_inter_des
thf(fact_1233_dual__sys_Opoint__indices__elem__in,axiom,
    ! [Ps: set_nat,T3: nat] :
      ( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( ( finite_card_nat @ Ps )
          = T3 )
       => ( member_nat @ ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps ) @ ( design1227534709319296284es_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ T3 ) ) ) ) ).

% dual_sys.point_indices_elem_in
thf(fact_1234_dual__sys_Oadd__point__sys__rep__numbers,axiom,
    ! [P: nat] :
      ( ( design3853898657598026467rs_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
      = ( sup_sup_set_nat @ ( design3853898657598026467rs_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ ( insert_nat @ ( design3571518413069006949er_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) @ bot_bot_set_nat ) ) ) ).

% dual_sys.add_point_sys_rep_numbers
thf(fact_1235_dual__sys_Oadd__point__def,axiom,
    ! [P: nat] :
      ( ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P )
      = ( insert_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).

% dual_sys.add_point_def
thf(fact_1236_dual__sys_Oadd__point__finite,axiom,
    ! [P: nat] : ( design5426232790142929158em_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.add_point_finite
thf(fact_1237_dual__sys_Oadd__point__to__blocks__finite,axiom,
    ! [P: nat,Bs: set_set_nat] : ( design5426232790142929158em_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( design5698312687278145166ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P @ Bs ) ) ).

% dual_sys.add_point_to_blocks_finite
thf(fact_1238_dual__sys_Oadd__existing__point,axiom,
    ! [P: nat] :
      ( ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).

% dual_sys.add_existing_point
thf(fact_1239_dual__sys_Odelete__point__index__eq,axiom,
    ! [Ps: set_nat,P: nat] :
      ( ( ord_less_eq_set_nat @ Ps @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) )
     => ( ( design6574611146354332593ex_nat @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) @ Ps )
        = ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps ) ) ) ).

% dual_sys.delete_point_index_eq
thf(fact_1240_dual__sys_Oadd__block__sub__sys,axiom,
    ! [B: set_nat] : ( sub_su5953739893325741765em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) ) ).

% dual_sys.add_block_sub_sys
thf(fact_1241_dual__sys_Odel__invalid__point,axiom,
    ! [P: nat] :
      ( ~ ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).

% dual_sys.del_invalid_point
thf(fact_1242_dual__sys_Oadd__delete__point__inv,axiom,
    ! [P: nat] :
      ( ~ ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( design4269233978287968195nt_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ P )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ).

% dual_sys.add_delete_point_inv
thf(fact_1243_dual__sys_Odel__point__order,axiom,
    ! [P: nat] :
      ( ( member_nat @ P @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( ( finite_card_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) )
        = ( minus_minus_nat @ ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) @ one_one_nat ) ) ) ).

% dual_sys.del_point_order
thf(fact_1244_dual__sys_Odel__point__def,axiom,
    ! [P: nat] :
      ( ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P )
      = ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( insert_nat @ P @ bot_bot_set_nat ) ) ) ).

% dual_sys.del_point_def
thf(fact_1245_dual__sys_Odelete__block__sub__sys,axiom,
    ! [B: set_nat] : ( sub_su5953739893325741765em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.delete_block_sub_sys
thf(fact_1246_dual__sys_Oadd__point__sub__sys,axiom,
    ! [P: nat] : ( sub_su5953739893325741765em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.add_point_sub_sys
thf(fact_1247_dual__sys_Ostrong__del__point__finite,axiom,
    ! [P: nat] : ( design5426232790142929158em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) ).

% dual_sys.strong_del_point_finite
thf(fact_1248_dual__sys_Odelete__point__finite,axiom,
    ! [P: nat] : ( design5426232790142929158em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) ).

% dual_sys.delete_point_finite
thf(fact_1249_dual__sys_Ostrong__del__point__sub__sys,axiom,
    ! [P: nat] : ( sub_su5953739893325741765em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.strong_del_point_sub_sys
thf(fact_1250_dual__is__proper__design,axiom,
    ( ! [X4: a] :
        ( ( member_a @ X4 @ ( set_a2 @ v_s ) )
       => ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ ( mset_set_a @ b_s ) @ X4 ) ) )
   => ( design435815215503836206gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ).

% dual_is_proper_design
thf(fact_1251_proper__design__axioms,axiom,
    design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).

% proper_design_axioms
thf(fact_1252_finite__atLeastLessThan,axiom,
    ! [L4: nat,U2: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L4 @ U2 ) ) ).

% finite_atLeastLessThan
thf(fact_1253_del__block__proper,axiom,
    ! [Bl2: set_a] :
      ( ( ord_less_nat @ one_one_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
     => ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ Bl2 ) ) ) ).

% del_block_proper
thf(fact_1254_complement__proper__design,axiom,
    ( ! [Bl: set_a] :
        ( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
       => ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
          & ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) )
   => ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).

% complement_proper_design
thf(fact_1255_proper__designI,axiom,
    ( ( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
     != zero_zero_nat )
   => ( design7287791228148780576sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ).

% proper_designI
thf(fact_1256_bounded__Max__nat,axiom,
    ! [P2: nat > $o,X: nat,M: nat] :
      ( ( P2 @ X )
     => ( ! [X4: nat] :
            ( ( P2 @ X4 )
           => ( ord_less_eq_nat @ X4 @ M ) )
       => ~ ! [M8: nat] :
              ( ( P2 @ M8 )
             => ~ ! [X5: nat] :
                    ( ( P2 @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M8 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1257_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M7: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N4 )
         => ( ord_less_nat @ X3 @ M7 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1258_bounded__nat__set__is__finite,axiom,
    ! [N2: set_nat,N: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N2 )
         => ( ord_less_nat @ X4 @ N ) )
     => ( finite_finite_nat @ N2 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1259_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N4: set_nat] :
        ? [M7: nat] :
        ! [X3: nat] :
          ( ( member_nat @ X3 @ N4 )
         => ( ord_less_eq_nat @ X3 @ M7 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1260_atLeastLessThan0,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_1261_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_1262_subset__card__intvl__is__intvl,axiom,
    ! [A: set_nat,K2: nat] :
      ( ( ord_less_eq_set_nat @ A @ ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A ) ) ) )
     => ( A
        = ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_1263_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K2 ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_1264_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N2: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N2 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_1265_dual__sys_Osimple__not__multiplicity,axiom,
    ! [B: set_nat] :
      ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
     => ( ( ord_less_nat @ one_one_nat @ ( count_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) )
       => ~ ( design164292856788568387em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).

% dual_sys.simple_not_multiplicity
thf(fact_1266_dual__sys_Oadd__point__blocks__wf,axiom,
    ! [P: nat,Bs: set_set_nat] : ( design3753904077504641269em_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( design5698312687278145166ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P @ Bs ) ) ).

% dual_sys.add_point_blocks_wf
thf(fact_1267_dual__sys_Oincidence__system__axioms,axiom,
    design3753904077504641269em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ).

% dual_sys.incidence_system_axioms
thf(fact_1268_dual__sys_Odelete__block__wf,axiom,
    ! [B: set_nat] : ( design3753904077504641269em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design755385109423264192ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) ) ).

% dual_sys.delete_block_wf
thf(fact_1269_dual__sys_Oadd__point__wf,axiom,
    ! [P: nat] : ( design3753904077504641269em_nat @ ( design8239173135376323853nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.add_point_wf
thf(fact_1270_dual__sys_Oadd__block__wf__cond,axiom,
    ! [B: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
     => ( design3753904077504641269em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) ) ) ).

% dual_sys.add_block_wf_cond
thf(fact_1271_dual__sys_Oadd__block__wf,axiom,
    ! [B: set_nat] : ( design3753904077504641269em_nat @ ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B ) @ ( design4725324266511619850ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) ) ).

% dual_sys.add_block_wf
thf(fact_1272_dual__sys_Ostrong__del__point__incidence__wf,axiom,
    ! [P: nat] : ( design3753904077504641269em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( design3278834155446248416ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) ).

% dual_sys.strong_del_point_incidence_wf
thf(fact_1273_dual__sys_Odelete__point__wf,axiom,
    ! [P: nat] : ( design3753904077504641269em_nat @ ( design4269233978287968195nt_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ P ) @ ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ P ) ) ).

% dual_sys.delete_point_wf
thf(fact_1274_dual__sys_Ostrong__del__block__wf,axiom,
    ! [B: set_nat] : ( design3753904077504641269em_nat @ ( minus_minus_set_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ B ) @ ( design3550126062406151447ck_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) ) ).

% dual_sys.strong_del_block_wf
thf(fact_1275_dual__sys_Ocomplement__wf,axiom,
    design3753904077504641269em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( design5569578106646884273ks_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).

% dual_sys.complement_wf

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ps )
    = lambda ) ).

%------------------------------------------------------------------------------