TPTP Problem File: SLH0772^1.p
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%------------------------------------------------------------------------------
% 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_00488_022310__28111042_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1534 ( 518 unt; 256 typ; 0 def)
% Number of atoms : 3295 (1174 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 10628 ( 362 ~; 36 |; 190 &;8793 @)
% ( 0 <=>;1247 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 6 avg)
% Number of types : 26 ( 25 usr)
% Number of type conns : 442 ( 442 >; 0 *; 0 +; 0 <<)
% Number of symbols : 234 ( 231 usr; 28 con; 0-4 aty)
% Number of variables : 2580 ( 125 ^;2373 !; 82 ?;2580 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:49:11.600
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_multiset_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J_J,type,
set_multiset_set_a: $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__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
set_list_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
set_list_set_a: $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__List__Olist_Itf__a_J_J,type,
multiset_list_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
multiset_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $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__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $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 (231)
thf(sy_c_BIBD_Obibd_001tf__a,type,
bibd_a: set_a > multiset_set_a > nat > nat > $o ).
thf(sy_c_BIBD_Osymmetric__bibd_001t__Set__Oset_Itf__a_J,type,
symmetric_bibd_set_a: set_set_a > multiset_set_set_a > nat > nat > $o ).
thf(sy_c_BIBD_Osymmetric__bibd_001tf__a,type,
symmetric_bibd_a: set_a > multiset_set_a > nat > nat > $o ).
thf(sy_c_Block__Designs_OK__block__design_001tf__a,type,
block_4943671260865530057sign_a: set_a > multiset_set_a > set_nat > $o ).
thf(sy_c_Block__Designs_O_092_060Lambda_062__PBD_001tf__a,type,
block_Lambda_PBD_a: set_a > multiset_set_a > nat > set_nat > $o ).
thf(sy_c_Block__Designs_Oblock__design_001t__Nat__Onat,type,
block_625751327111516584gn_nat: set_nat > multiset_set_nat > nat > $o ).
thf(sy_c_Block__Designs_Oblock__design_001t__Set__Oset_Itf__a_J,type,
block_4731189848718898310_set_a: set_set_a > multiset_set_set_a > nat > $o ).
thf(sy_c_Block__Designs_Oblock__design_001tf__a,type,
block_block_design_a: set_a > multiset_set_a > nat > $o ).
thf(sy_c_Block__Designs_Oconstant__rep__design_001t__Set__Oset_Itf__a_J,type,
block_1497795055661977978_set_a: set_set_a > multiset_set_set_a > nat > $o ).
thf(sy_c_Block__Designs_Oconstant__rep__design_001tf__a,type,
block_6028206285060069402sign_a: set_a > multiset_set_a > nat > $o ).
thf(sy_c_Block__Designs_Oincomplete__design_001t__Set__Oset_Itf__a_J,type,
block_3501025933689710669_set_a: set_set_a > multiset_set_set_a > nat > $o ).
thf(sy_c_Block__Designs_Oincomplete__design_001tf__a,type,
block_1438872132225661677sign_a: set_a > multiset_set_a > nat > $o ).
thf(sy_c_Block__Designs_Ok__PBD_001tf__a,type,
block_k_PBD_a: set_a > multiset_set_a > nat > $o ).
thf(sy_c_Block__Designs_Ok___092_060Lambda_062__PBD_001tf__a,type,
block_k_Lambda_PBD_a: set_a > multiset_set_a > 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_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_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_On__intersect__number_001tf__a,type,
design735257067508376852mber_a: set_a > nat > set_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_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__Nat__Onat,type,
dual_dual_blocks_nat: set_nat > list_set_nat > multiset_set_nat ).
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_Finite__Set_Ocard_001t__List__Olist_Itf__a_J,type,
finite_card_list_a: set_list_a > 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__List__Olist_It__Nat__Onat_J,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
finite1091814263879798189et_nat: set_list_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
finite1971793804006318733_set_a: set_list_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_Itf__a_J,type,
finite_finite_list_a: set_list_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
finite5272093019064105709et_nat: set_multiset_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
finite2815193924343055693_set_a: set_multiset_set_a > $o ).
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__List__Olist_Itf__a_J_J,type,
minus_7431248565939055793list_a: multiset_list_a > multiset_list_a > multiset_list_a ).
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__List__Olist_Itf__a_J_J,type,
minus_646659088055828811list_a: set_list_a > set_list_a > set_list_a ).
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__List__Olist_Itf__a_J_J,type,
plus_p690419498615200257list_a: multiset_list_a > multiset_list_a > multiset_list_a ).
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_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__List__Olist_Itf__a_J_J,type,
zero_z4454100511807792257list_a: multiset_list_a ).
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_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__bibd_001t__Set__Oset_Itf__a_J,type,
incide532103555118123225_set_a: list_set_a > list_set_set_a > nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__bibd_001tf__a,type,
incide4817766913905363833bibd_a: list_a > list_set_a > nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__block__design_001t__Nat__Onat,type,
incide8589348598044109441gn_nat: list_nat > list_set_nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__block__design_001t__Set__Oset_Itf__a_J,type,
incide8999561475770975213_set_a: list_set_a > list_set_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__block__design_001tf__a,type,
incide5219153079875704461sign_a: list_a > list_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__constant__rep_001t__Set__Oset_Itf__a_J,type,
incide3293764431032925855_set_a: list_set_a > list_set_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__constant__rep_001tf__a,type,
incide6922509864216205631_rep_a: list_a > list_set_a > nat > $o ).
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__incidence__system_001t__Nat__Onat,type,
incide6998539924841383625em_nat: list_nat > list_set_nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incidence__system_001t__Set__Oset_Itf__a_J,type,
incide2166342132139297189_set_a: list_set_a > list_set_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incidence__system_001tf__a,type,
incide1624170830610365509stem_a: list_a > list_set_a > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incomplete__design_001t__Set__Oset_Itf__a_J,type,
incide4280386766309144998_set_a: list_set_a > list_set_set_a > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__incomplete__design_001tf__a,type,
incide1377962018248667206sign_a: list_a > list_set_a > nat > $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__Nat__Onat,type,
incide1001368407746664282gn_nat: list_nat > list_set_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__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_Incidence__Matrices_Oordered__sym__bibd_001t__Set__Oset_Itf__a_J,type,
incide3984264607369433128_set_a: list_set_a > list_set_set_a > nat > nat > $o ).
thf(sy_c_Incidence__Matrices_Oordered__sym__bibd_001tf__a,type,
incide4194285476649307848bibd_a: list_a > list_set_a > nat > nat > $o ).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
nil_nat: list_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Nat__Onat_J,type,
nil_set_nat: list_set_nat ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_Itf__a_J,type,
nil_set_a: list_set_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__a_J,type,
set_list_a2: list_list_a > set_list_a ).
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_Multiset_Omset_001t__List__Olist_Itf__a_J,type,
mset_list_a: list_list_a > multiset_list_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_Omultiset_Ocount_001t__Set__Oset_Itf__a_J,type,
count_set_a: multiset_set_a > set_a > nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_It__Nat__Onat_J,type,
repeat_mset_set_nat: nat > multiset_set_nat > multiset_set_nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Set__Oset_Itf__a_J,type,
repeat_mset_set_a: nat > multiset_set_a > multiset_set_a ).
thf(sy_c_Multiset_Oset__mset_001t__List__Olist_Itf__a_J,type,
set_mset_list_a: multiset_list_a > set_list_a ).
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_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_Multiset_Osubseteq__mset_001t__List__Olist_Itf__a_J,type,
subseteq_mset_list_a: multiset_list_a > multiset_list_a > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Nat__Onat,type,
subseteq_mset_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Set__Oset_It__Nat__Onat_J,type,
subset6078030600694693471et_nat: multiset_set_nat > multiset_set_nat > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001t__Set__Oset_Itf__a_J,type,
subseteq_mset_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Multiset_Osubseteq__mset_001tf__a,type,
subseteq_mset_a: multiset_a > multiset_a > $o ).
thf(sy_c_Multiset__Permutations_Opermutations__of__multiset_001t__Set__Oset_It__Nat__Onat_J,type,
multis124346860217030838et_nat: multiset_set_nat > set_list_set_nat ).
thf(sy_c_Multiset__Permutations_Opermutations__of__multiset_001t__Set__Oset_Itf__a_J,type,
multis5469701301851823918_set_a: multiset_set_a > set_list_set_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__multiset_001tf__a,type,
multis5886240593633752526iset_a: multiset_a > set_list_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001t__Nat__Onat,type,
multis1655833086286526861et_nat: set_nat > set_list_nat ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001t__Set__Oset_It__Nat__Onat_J,type,
multis4401431215724540227et_nat: set_set_nat > set_list_set_nat ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001t__Set__Oset_Itf__a_J,type,
multis2257881577744371681_set_a: set_set_a > set_list_set_a ).
thf(sy_c_Multiset__Permutations_Opermutations__of__set_001tf__a,type,
multis2428024204330136193_set_a: set_a > set_list_a ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
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_It__Set__Oset_Itf__a_J_J_J,type,
size_s7045802815848406192_set_a: list_set_set_a > 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__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__List__Olist_It__Nat__Onat_J_J,type,
bot_bot_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo2934890284768024416et_nat: set_list_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo4397488018069675312_set_a: set_list_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
bot_bot_set_list_a: set_list_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J_J,type,
bot_bo7575690689620579488et_nat: set_multiset_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J_J,type,
bot_bo9088538438451294192_set_a: set_multiset_set_a ).
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__List__Olist_Itf__a_J_J,type,
ord_less_set_list_a: set_list_a > set_list_a > $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_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_less_set_set_a: set_set_a > set_set_a > $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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
ord_le8861187494160871172list_a: set_list_a > set_list_a > $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_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
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__List__Olist_Itf__a_J,type,
insert_list_a: list_a > set_list_a > set_list_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__List__Olist_Itf__a_J,type,
is_singleton_list_a: set_list_a > $o ).
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__List__Olist_It__Nat__Onat_J,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
member_list_set_nat: list_set_nat > set_list_set_nat > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
member_list_set_a: list_set_a > set_list_set_a > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
member9214616765234488813et_nat: multiset_set_nat > set_multiset_set_nat > $o ).
thf(sy_c_member_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
member2747690772047059533_set_a: multiset_set_a > set_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_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__092_060k_062,type,
k: nat ).
thf(sy_v_ps,type,
ps: set_nat ).
% Relevant facts (1274)
thf(fact_0_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_1_ordered__pairwise__balance__axioms,axiom,
incide6880889959311561818ance_a @ v_s @ b_s @ lambda ).
% ordered_pairwise_balance_axioms
thf(fact_2_ordered__design__axioms,axiom,
incide2848671379600480836sign_a @ v_s @ b_s ).
% ordered_design_axioms
thf(fact_3_ordered__simple__design__axioms,axiom,
incide371748008924627346sign_a @ v_s @ b_s ).
% ordered_simple_design_axioms
thf(fact_4_ordered__proper__design__axioms,axiom,
incide3676903341588786676sign_a @ v_s @ b_s ).
% ordered_proper_design_axioms
thf(fact_5_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_6_points__indexing,axiom,
member_list_a @ v_s @ ( multis2428024204330136193_set_a @ ( set_a2 @ v_s ) ) ).
% points_indexing
thf(fact_7_design__points__nempty,axiom,
( ( set_a2 @ v_s )
!= bot_bot_set_a ) ).
% design_points_nempty
thf(fact_8_finite__sets,axiom,
finite_finite_a @ ( set_a2 @ v_s ) ).
% finite_sets
thf(fact_9_ordered__incidence__system__axioms,axiom,
incide1624170830610365509stem_a @ v_s @ b_s ).
% ordered_incidence_system_axioms
thf(fact_10_ordered__sym__bibd__axioms,axiom,
incide4194285476649307848bibd_a @ v_s @ b_s @ k @ lambda ).
% ordered_sym_bibd_axioms
thf(fact_11_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_12_ordered__incomplete__design__axioms,axiom,
incide1377962018248667206sign_a @ v_s @ b_s @ k ).
% ordered_incomplete_design_axioms
thf(fact_13_ordered__bibd__axioms,axiom,
incide4817766913905363833bibd_a @ v_s @ b_s @ k @ lambda ).
% ordered_bibd_axioms
thf(fact_14_ordered__block__design__axioms,axiom,
incide5219153079875704461sign_a @ v_s @ b_s @ k ).
% ordered_block_design_axioms
thf(fact_15_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_16_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_17_List_Ofinite__set,axiom,
! [Xs: list_a] : ( finite_finite_a @ ( set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_18_List_Ofinite__set,axiom,
! [Xs: list_set_nat] : ( finite1152437895449049373et_nat @ ( set_set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_19_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_20_List_Ofinite__set,axiom,
! [Xs: list_set_a] : ( finite_finite_set_a @ ( set_set_a2 @ Xs ) ) ).
% List.finite_set
thf(fact_21_points__list__empty__iff,axiom,
( ( v_s = nil_a )
= ( ( set_a2 @ v_s )
= bot_bot_set_a ) ) ).
% points_list_empty_iff
thf(fact_22_ordered__incidence__system_Opoints__indexing,axiom,
! [V_s: list_set_a,B_s: list_set_set_a] :
( ( incide2166342132139297189_set_a @ V_s @ B_s )
=> ( member_list_set_a @ V_s @ ( multis2257881577744371681_set_a @ ( set_set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_indexing
thf(fact_23_ordered__incidence__system_Opoints__indexing,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( member_list_a @ V_s @ ( multis2428024204330136193_set_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_indexing
thf(fact_24_ordered__incidence__system_Opoints__indexing,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( member_list_nat @ V_s @ ( multis1655833086286526861et_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_indexing
thf(fact_25_ordered__simple__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide371748008924627346sign_a @ V_s @ B_s )
=> ( incide2848671379600480836sign_a @ V_s @ B_s ) ) ).
% ordered_simple_design.axioms(1)
thf(fact_26_ordered__proper__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( incide2848671379600480836sign_a @ V_s @ B_s ) ) ).
% ordered_proper_design.axioms(1)
thf(fact_27_empty__iff,axiom,
! [C: list_a] :
~ ( member_list_a @ C @ bot_bot_set_list_a ) ).
% empty_iff
thf(fact_28_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_29_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_30_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_31_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_32_all__not__in__conv,axiom,
! [A: set_list_a] :
( ( ! [X: list_a] :
~ ( member_list_a @ X @ A ) )
= ( A = bot_bot_set_list_a ) ) ).
% all_not_in_conv
thf(fact_33_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X: set_nat] :
~ ( member_set_nat @ X @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_34_all__not__in__conv,axiom,
! [A: set_set_a] :
( ( ! [X: set_a] :
~ ( member_set_a @ X @ A ) )
= ( A = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_35_all__not__in__conv,axiom,
! [A: set_a] :
( ( ! [X: a] :
~ ( member_a @ X @ A ) )
= ( A = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_36_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_37_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_38_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P2 @ X ) ) ) ).
% Collect_empty_eq
thf(fact_39_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X: a] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_40_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X: nat] :
~ ( P2 @ X ) ) ) ).
% empty_Collect_eq
thf(fact_41_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_42_set__empty2,axiom,
! [Xs: list_set_a] :
( ( bot_bot_set_set_a
= ( set_set_a2 @ Xs ) )
= ( Xs = nil_set_a ) ) ).
% set_empty2
thf(fact_43_set__empty2,axiom,
! [Xs: list_a] :
( ( bot_bot_set_a
= ( set_a2 @ Xs ) )
= ( Xs = nil_a ) ) ).
% set_empty2
thf(fact_44_set__empty2,axiom,
! [Xs: list_nat] :
( ( bot_bot_set_nat
= ( set_nat2 @ Xs ) )
= ( Xs = nil_nat ) ) ).
% set_empty2
thf(fact_45_set__empty,axiom,
! [Xs: list_set_a] :
( ( ( set_set_a2 @ Xs )
= bot_bot_set_set_a )
= ( Xs = nil_set_a ) ) ).
% set_empty
thf(fact_46_set__empty,axiom,
! [Xs: list_a] :
( ( ( set_a2 @ Xs )
= bot_bot_set_a )
= ( Xs = nil_a ) ) ).
% set_empty
thf(fact_47_set__empty,axiom,
! [Xs: list_nat] :
( ( ( set_nat2 @ Xs )
= bot_bot_set_nat )
= ( Xs = nil_nat ) ) ).
% set_empty
thf(fact_48_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_49_ordered__incomplete__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat] :
( ( incide1377962018248667206sign_a @ V_s @ B_s @ K )
=> ( incide5219153079875704461sign_a @ V_s @ B_s @ K ) ) ).
% ordered_incomplete_design.axioms(1)
thf(fact_50_ordered__block__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat] :
( ( incide5219153079875704461sign_a @ V_s @ B_s @ K )
=> ( incide3676903341588786676sign_a @ V_s @ B_s ) ) ).
% ordered_block_design.axioms(1)
thf(fact_51_empty__set,axiom,
( bot_bot_set_set_a
= ( set_set_a2 @ nil_set_a ) ) ).
% empty_set
thf(fact_52_empty__set,axiom,
( bot_bot_set_a
= ( set_a2 @ nil_a ) ) ).
% empty_set
thf(fact_53_empty__set,axiom,
( bot_bot_set_nat
= ( set_nat2 @ nil_nat ) ) ).
% empty_set
thf(fact_54_ordered__bibd_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat,Lambda: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K @ Lambda )
=> ( incide3676903341588786676sign_a @ V_s @ B_s ) ) ).
% ordered_bibd.axioms(1)
thf(fact_55_ordered__sym__bibd_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat,Lambda: nat] :
( ( incide4194285476649307848bibd_a @ V_s @ B_s @ K @ Lambda )
=> ( incide4817766913905363833bibd_a @ V_s @ B_s @ K @ Lambda ) ) ).
% ordered_sym_bibd.axioms(1)
thf(fact_56_ordered__incidence__system_Opoints__list__empty__iff,axiom,
! [V_s: list_set_a,B_s: list_set_set_a] :
( ( incide2166342132139297189_set_a @ V_s @ B_s )
=> ( ( V_s = nil_set_a )
= ( ( set_set_a2 @ V_s )
= bot_bot_set_set_a ) ) ) ).
% ordered_incidence_system.points_list_empty_iff
thf(fact_57_ordered__incidence__system_Opoints__list__empty__iff,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( V_s = nil_a )
= ( ( set_a2 @ V_s )
= bot_bot_set_a ) ) ) ).
% ordered_incidence_system.points_list_empty_iff
thf(fact_58_ordered__incidence__system_Opoints__list__empty__iff,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( V_s = nil_nat )
= ( ( set_nat2 @ V_s )
= bot_bot_set_nat ) ) ) ).
% ordered_incidence_system.points_list_empty_iff
thf(fact_59_ex__in__conv,axiom,
! [A: set_list_a] :
( ( ? [X: list_a] : ( member_list_a @ X @ A ) )
= ( A != bot_bot_set_list_a ) ) ).
% ex_in_conv
thf(fact_60_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X: set_nat] : ( member_set_nat @ X @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_61_ex__in__conv,axiom,
! [A: set_set_a] :
( ( ? [X: set_a] : ( member_set_a @ X @ A ) )
= ( A != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_62_ex__in__conv,axiom,
! [A: set_a] :
( ( ? [X: a] : ( member_a @ X @ A ) )
= ( A != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_63_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_64_equals0I,axiom,
! [A: set_list_a] :
( ! [Y: list_a] :
~ ( member_list_a @ Y @ A )
=> ( A = bot_bot_set_list_a ) ) ).
% equals0I
thf(fact_65_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y: set_nat] :
~ ( member_set_nat @ Y @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_66_equals0I,axiom,
! [A: set_set_a] :
( ! [Y: set_a] :
~ ( member_set_a @ Y @ A )
=> ( A = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_67_equals0I,axiom,
! [A: set_a] :
( ! [Y: a] :
~ ( member_a @ Y @ A )
=> ( A = bot_bot_set_a ) ) ).
% equals0I
thf(fact_68_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_69_equals0D,axiom,
! [A: set_list_a,A2: list_a] :
( ( A = bot_bot_set_list_a )
=> ~ ( member_list_a @ A2 @ A ) ) ).
% equals0D
thf(fact_70_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_71_equals0D,axiom,
! [A: set_set_a,A2: set_a] :
( ( A = bot_bot_set_set_a )
=> ~ ( member_set_a @ A2 @ A ) ) ).
% equals0D
thf(fact_72_equals0D,axiom,
! [A: set_a,A2: a] :
( ( A = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A ) ) ).
% equals0D
thf(fact_73_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_74_emptyE,axiom,
! [A2: list_a] :
~ ( member_list_a @ A2 @ bot_bot_set_list_a ) ).
% emptyE
thf(fact_75_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_76_emptyE,axiom,
! [A2: set_a] :
~ ( member_set_a @ A2 @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_77_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_78_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_79_infinite__imp__nonempty,axiom,
! [S: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ S )
=> ( S != bot_bot_set_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_80_infinite__imp__nonempty,axiom,
! [S: set_set_a] :
( ~ ( finite_finite_set_a @ S )
=> ( S != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_81_infinite__imp__nonempty,axiom,
! [S: set_a] :
( ~ ( finite_finite_a @ S )
=> ( S != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_82_infinite__imp__nonempty,axiom,
! [S: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ( S != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_83_mem__Collect__eq,axiom,
! [A2: list_a,P2: list_a > $o] :
( ( member_list_a @ A2 @ ( collect_list_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_84_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_85_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_86_mem__Collect__eq,axiom,
! [A2: nat,P2: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_87_mem__Collect__eq,axiom,
! [A2: a,P2: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A: set_list_a] :
( ( collect_list_a
@ ^ [X: list_a] : ( member_list_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X: set_nat] : ( member_set_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
! [A: set_set_a] :
( ( collect_set_a
@ ^ [X: set_a] : ( member_set_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
! [A: set_a] :
( ( collect_a
@ ^ [X: a] : ( member_a @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_93_finite_OemptyI,axiom,
finite1152437895449049373et_nat @ bot_bot_set_set_nat ).
% finite.emptyI
thf(fact_94_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_95_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_96_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_97_finite__list,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ? [Xs2: list_a] :
( ( set_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_98_finite__list,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ? [Xs2: list_set_nat] :
( ( set_set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_99_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_100_finite__list,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ? [Xs2: list_set_a] :
( ( set_set_a2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_101_ordered__design_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide2848671379600480836sign_a @ V_s @ B_s )
=> ( incide1624170830610365509stem_a @ V_s @ B_s ) ) ).
% ordered_design.axioms(1)
thf(fact_102_ordered__design_Oaxioms_I1_J,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide8999572217031194378gn_nat @ V_s @ B_s )
=> ( incide6998539924841383625em_nat @ V_s @ B_s ) ) ).
% ordered_design.axioms(1)
thf(fact_103_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_104_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_105_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_106_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_107_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_108_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_109_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_110_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_111_permutations__of__set__empty__iff,axiom,
! [A: set_set_nat] :
( ( ( multis4401431215724540227et_nat @ A )
= bot_bo2934890284768024416et_nat )
= ( ~ ( finite1152437895449049373et_nat @ A ) ) ) ).
% permutations_of_set_empty_iff
thf(fact_112_permutations__of__set__empty__iff,axiom,
! [A: set_set_a] :
( ( ( multis2257881577744371681_set_a @ A )
= bot_bo4397488018069675312_set_a )
= ( ~ ( finite_finite_set_a @ A ) ) ) ).
% permutations_of_set_empty_iff
thf(fact_113_permutations__of__set__empty__iff,axiom,
! [A: set_a] :
( ( ( multis2428024204330136193_set_a @ A )
= bot_bot_set_list_a )
= ( ~ ( finite_finite_a @ A ) ) ) ).
% permutations_of_set_empty_iff
thf(fact_114_permutations__of__set__empty__iff,axiom,
! [A: set_nat] :
( ( ( multis1655833086286526861et_nat @ A )
= bot_bot_set_list_nat )
= ( ~ ( finite_finite_nat @ A ) ) ) ).
% permutations_of_set_empty_iff
thf(fact_115_finite__set__mset,axiom,
! [M: multiset_a] : ( finite_finite_a @ ( set_mset_a @ M ) ) ).
% finite_set_mset
thf(fact_116_finite__set__mset,axiom,
! [M: multiset_nat] : ( finite_finite_nat @ ( set_mset_nat @ M ) ) ).
% finite_set_mset
thf(fact_117_finite__set__mset,axiom,
! [M: multiset_set_nat] : ( finite1152437895449049373et_nat @ ( set_mset_set_nat @ M ) ) ).
% finite_set_mset
thf(fact_118_finite__set__mset,axiom,
! [M: multiset_set_a] : ( finite_finite_set_a @ ( set_mset_set_a @ M ) ) ).
% finite_set_mset
thf(fact_119_elem__permutation__of__set__empty__iff,axiom,
! [A: set_set_nat,Xs: list_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_list_set_nat @ Xs @ ( multis4401431215724540227et_nat @ A ) )
=> ( ( Xs = nil_set_nat )
= ( A = bot_bot_set_set_nat ) ) ) ) ).
% elem_permutation_of_set_empty_iff
thf(fact_120_elem__permutation__of__set__empty__iff,axiom,
! [A: set_set_a,Xs: list_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_list_set_a @ Xs @ ( multis2257881577744371681_set_a @ A ) )
=> ( ( Xs = nil_set_a )
= ( A = bot_bot_set_set_a ) ) ) ) ).
% elem_permutation_of_set_empty_iff
thf(fact_121_elem__permutation__of__set__empty__iff,axiom,
! [A: set_a,Xs: list_a] :
( ( finite_finite_a @ A )
=> ( ( member_list_a @ Xs @ ( multis2428024204330136193_set_a @ A ) )
=> ( ( Xs = nil_a )
= ( A = bot_bot_set_a ) ) ) ) ).
% elem_permutation_of_set_empty_iff
thf(fact_122_elem__permutation__of__set__empty__iff,axiom,
! [A: set_nat,Xs: list_nat] :
( ( finite_finite_nat @ A )
=> ( ( member_list_nat @ Xs @ ( multis1655833086286526861et_nat @ A ) )
=> ( ( Xs = nil_nat )
= ( A = bot_bot_set_nat ) ) ) ) ).
% elem_permutation_of_set_empty_iff
thf(fact_123_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_124_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_125_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_126_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_127_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_128_set__mset__empty,axiom,
( ( set_mset_set_nat @ zero_z3157962936165190495et_nat )
= bot_bot_set_set_nat ) ).
% set_mset_empty
thf(fact_129_set__mset__empty,axiom,
( ( set_mset_set_a @ zero_z5079479921072680283_set_a )
= bot_bot_set_set_a ) ).
% set_mset_empty
thf(fact_130_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_131_set__mset__empty,axiom,
( ( set_mset_nat @ zero_z7348594199698428585et_nat )
= bot_bot_set_nat ) ).
% set_mset_empty
thf(fact_132_multiset__nonemptyE,axiom,
! [A: multiset_list_a] :
( ( A != zero_z4454100511807792257list_a )
=> ~ ! [X2: list_a] :
~ ( member_list_a @ X2 @ ( set_mset_list_a @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_133_multiset__nonemptyE,axiom,
! [A: multiset_nat] :
( ( A != zero_z7348594199698428585et_nat )
=> ~ ! [X2: nat] :
~ ( member_nat @ X2 @ ( set_mset_nat @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_134_multiset__nonemptyE,axiom,
! [A: multiset_a] :
( ( A != zero_zero_multiset_a )
=> ~ ! [X2: a] :
~ ( member_a @ X2 @ ( set_mset_a @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_135_multiset__nonemptyE,axiom,
! [A: multiset_set_nat] :
( ( A != zero_z3157962936165190495et_nat )
=> ~ ! [X2: set_nat] :
~ ( member_set_nat @ X2 @ ( set_mset_set_nat @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_136_multiset__nonemptyE,axiom,
! [A: multiset_set_a] :
( ( A != zero_z5079479921072680283_set_a )
=> ~ ! [X2: set_a] :
~ ( member_set_a @ X2 @ ( set_mset_set_a @ A ) ) ) ).
% multiset_nonemptyE
thf(fact_137_finite__permutations__of__set,axiom,
! [A: set_a] : ( finite_finite_list_a @ ( multis2428024204330136193_set_a @ A ) ) ).
% finite_permutations_of_set
thf(fact_138_finite__permutations__of__set,axiom,
! [A: set_nat] : ( finite8100373058378681591st_nat @ ( multis1655833086286526861et_nat @ A ) ) ).
% finite_permutations_of_set
thf(fact_139_permutations__of__setD_I1_J,axiom,
! [Xs: list_set_a,A: set_set_a] :
( ( member_list_set_a @ Xs @ ( multis2257881577744371681_set_a @ A ) )
=> ( ( set_set_a2 @ Xs )
= A ) ) ).
% permutations_of_setD(1)
thf(fact_140_permutations__of__setD_I1_J,axiom,
! [Xs: list_a,A: set_a] :
( ( member_list_a @ Xs @ ( multis2428024204330136193_set_a @ A ) )
=> ( ( set_a2 @ Xs )
= A ) ) ).
% permutations_of_setD(1)
thf(fact_141_permutations__of__setD_I1_J,axiom,
! [Xs: list_nat,A: set_nat] :
( ( member_list_nat @ Xs @ ( multis1655833086286526861et_nat @ A ) )
=> ( ( set_nat2 @ Xs )
= A ) ) ).
% permutations_of_setD(1)
thf(fact_142_permutations__of__set__infinite,axiom,
! [A: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( multis4401431215724540227et_nat @ A )
= bot_bo2934890284768024416et_nat ) ) ).
% permutations_of_set_infinite
thf(fact_143_permutations__of__set__infinite,axiom,
! [A: set_set_a] :
( ~ ( finite_finite_set_a @ A )
=> ( ( multis2257881577744371681_set_a @ A )
= bot_bo4397488018069675312_set_a ) ) ).
% permutations_of_set_infinite
thf(fact_144_permutations__of__set__infinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( multis2428024204330136193_set_a @ A )
= bot_bot_set_list_a ) ) ).
% permutations_of_set_infinite
thf(fact_145_permutations__of__set__infinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( multis1655833086286526861et_nat @ A )
= bot_bot_set_list_nat ) ) ).
% permutations_of_set_infinite
thf(fact_146_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_147_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_148_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_149_dual__sys_Osys__block__sizes__obtain__bl,axiom,
! [X3: nat] :
( ( member_nat @ X3 @ ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
& ( ( finite_card_nat @ X2 )
= X3 ) ) ) ).
% dual_sys.sys_block_sizes_obtain_bl
thf(fact_150_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_151_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_152_dual__sys_Odelete__point__strong__block__subset,axiom,
! [P: nat] : ( subset6078030600694693471et_nat @ ( design3278834155446248416ks_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_strong_block_subset
thf(fact_153_dual__sys_Odelete__block__subset,axiom,
! [B: set_nat] : ( subset6078030600694693471et_nat @ ( 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_block_subset
thf(fact_154_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_155_dual__sys_Omultiple__block__in__original,axiom,
! [B: set_nat,N: nat] :
( ( member_set_nat @ B @ ( set_mset_set_nat @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
=> ( member_set_nat @ B @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.multiple_block_in_original
thf(fact_156_wf__invalid__point,axiom,
! [X3: a,B: set_a] :
( ~ ( member_a @ X3 @ ( set_a2 @ v_s ) )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ~ ( member_a @ X3 @ B ) ) ) ).
% wf_invalid_point
thf(fact_157_design__blocks__nempty,axiom,
( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a ) ).
% design_blocks_nempty
thf(fact_158_blocks__list__empty__iff,axiom,
( ( b_s = nil_set_a )
= ( ( mset_set_a @ b_s )
= zero_z5079479921072680283_set_a ) ) ).
% blocks_list_empty_iff
thf(fact_159_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_160_subsetI,axiom,
! [A: set_list_a,B2: set_list_a] :
( ! [X2: list_a] :
( ( member_list_a @ X2 @ A )
=> ( member_list_a @ X2 @ B2 ) )
=> ( ord_le8861187494160871172list_a @ A @ B2 ) ) ).
% subsetI
thf(fact_161_subsetI,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( member_set_nat @ X2 @ B2 ) )
=> ( ord_le6893508408891458716et_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_162_subsetI,axiom,
! [A: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A )
=> ( member_set_a @ X2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_163_subsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% subsetI
thf(fact_164_subsetI,axiom,
! [A: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% subsetI
thf(fact_165_subset__antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_166_subset__antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% subset_antisym
thf(fact_167_subset__mset_Oorder__refl,axiom,
! [X3: multiset_set_nat] : ( subset6078030600694693471et_nat @ X3 @ X3 ) ).
% subset_mset.order_refl
thf(fact_168_subset__mset_Oorder__refl,axiom,
! [X3: multiset_set_a] : ( subseteq_mset_set_a @ X3 @ X3 ) ).
% subset_mset.order_refl
thf(fact_169_subset__mset_Odual__order_Orefl,axiom,
! [A2: multiset_set_nat] : ( subset6078030600694693471et_nat @ A2 @ A2 ) ).
% subset_mset.dual_order.refl
thf(fact_170_subset__mset_Odual__order_Orefl,axiom,
! [A2: multiset_set_a] : ( subseteq_mset_set_a @ A2 @ A2 ) ).
% subset_mset.dual_order.refl
thf(fact_171_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_172_blocks__nempty__alt,axiom,
! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( X4 != bot_bot_set_a ) ) ).
% blocks_nempty_alt
thf(fact_173_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_174_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_175_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_176_empty__subsetI,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% empty_subsetI
thf(fact_177_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_178_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_179_subset__mset_Oextremum__unique,axiom,
! [A2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ zero_z3157962936165190495et_nat )
= ( A2 = zero_z3157962936165190495et_nat ) ) ).
% subset_mset.extremum_unique
thf(fact_180_subset__mset_Oextremum__unique,axiom,
! [A2: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ zero_z5079479921072680283_set_a )
= ( A2 = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.extremum_unique
thf(fact_181_subset__mset_Ole__zero__eq,axiom,
! [N: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ N @ zero_z3157962936165190495et_nat )
= ( N = zero_z3157962936165190495et_nat ) ) ).
% subset_mset.le_zero_eq
thf(fact_182_subset__mset_Ole__zero__eq,axiom,
! [N: multiset_set_a] :
( ( subseteq_mset_set_a @ N @ zero_z5079479921072680283_set_a )
= ( N = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.le_zero_eq
thf(fact_183_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_184_repeat__mset__empty,axiom,
! [N: nat] :
( ( repeat_mset_set_nat @ N @ zero_z3157962936165190495et_nat )
= zero_z3157962936165190495et_nat ) ).
% repeat_mset_empty
thf(fact_185_repeat__mset__empty,axiom,
! [N: nat] :
( ( repeat_mset_set_a @ N @ zero_z5079479921072680283_set_a )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_empty
thf(fact_186_in__multiset__in__set,axiom,
! [X3: list_a,Xs: list_list_a] :
( ( member_list_a @ X3 @ ( set_mset_list_a @ ( mset_list_a @ Xs ) ) )
= ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_187_in__multiset__in__set,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
= ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_188_in__multiset__in__set,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_mset_a @ ( mset_a @ Xs ) ) )
= ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_189_in__multiset__in__set,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X3 @ ( set_mset_set_nat @ ( mset_set_nat @ Xs ) ) )
= ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_190_in__multiset__in__set,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( member_set_a @ X3 @ ( set_mset_set_a @ ( mset_set_a @ Xs ) ) )
= ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) ) ) ).
% in_multiset_in_set
thf(fact_191_set__mset__mset,axiom,
! [Xs: list_a] :
( ( set_mset_a @ ( mset_a @ Xs ) )
= ( set_a2 @ Xs ) ) ).
% set_mset_mset
thf(fact_192_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_193_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_194_mset__zero__iff,axiom,
! [X3: list_a] :
( ( ( mset_a @ X3 )
= zero_zero_multiset_a )
= ( X3 = nil_a ) ) ).
% mset_zero_iff
thf(fact_195_mset__zero__iff,axiom,
! [X3: list_set_nat] :
( ( ( mset_set_nat @ X3 )
= zero_z3157962936165190495et_nat )
= ( X3 = nil_set_nat ) ) ).
% mset_zero_iff
thf(fact_196_mset__zero__iff,axiom,
! [X3: list_set_a] :
( ( ( mset_set_a @ X3 )
= zero_z5079479921072680283_set_a )
= ( X3 = nil_set_a ) ) ).
% mset_zero_iff
thf(fact_197_mset__zero__iff__right,axiom,
! [X3: list_a] :
( ( zero_zero_multiset_a
= ( mset_a @ X3 ) )
= ( X3 = nil_a ) ) ).
% mset_zero_iff_right
thf(fact_198_mset__zero__iff__right,axiom,
! [X3: list_set_nat] :
( ( zero_z3157962936165190495et_nat
= ( mset_set_nat @ X3 ) )
= ( X3 = nil_set_nat ) ) ).
% mset_zero_iff_right
thf(fact_199_mset__zero__iff__right,axiom,
! [X3: list_set_a] :
( ( zero_z5079479921072680283_set_a
= ( mset_set_a @ X3 ) )
= ( X3 = nil_set_a ) ) ).
% mset_zero_iff_right
thf(fact_200_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_201_subset__mset_Ofinite__has__maximal,axiom,
! [A: set_multiset_set_nat] :
( ( finite5272093019064105709et_nat @ A )
=> ( ( A != bot_bo7575690689620579488et_nat )
=> ? [X2: multiset_set_nat] :
( ( member9214616765234488813et_nat @ X2 @ A )
& ! [Xa: multiset_set_nat] :
( ( member9214616765234488813et_nat @ Xa @ A )
=> ( ( subset6078030600694693471et_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
thf(fact_202_subset__mset_Ofinite__has__maximal,axiom,
! [A: set_multiset_set_a] :
( ( finite2815193924343055693_set_a @ A )
=> ( ( A != bot_bo9088538438451294192_set_a )
=> ? [X2: multiset_set_a] :
( ( member2747690772047059533_set_a @ X2 @ A )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A )
=> ( ( subseteq_mset_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal
thf(fact_203_subset__mset_Ofinite__has__minimal,axiom,
! [A: set_multiset_set_nat] :
( ( finite5272093019064105709et_nat @ A )
=> ( ( A != bot_bo7575690689620579488et_nat )
=> ? [X2: multiset_set_nat] :
( ( member9214616765234488813et_nat @ X2 @ A )
& ! [Xa: multiset_set_nat] :
( ( member9214616765234488813et_nat @ Xa @ A )
=> ( ( subset6078030600694693471et_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
thf(fact_204_subset__mset_Ofinite__has__minimal,axiom,
! [A: set_multiset_set_a] :
( ( finite2815193924343055693_set_a @ A )
=> ( ( A != bot_bo9088538438451294192_set_a )
=> ? [X2: multiset_set_a] :
( ( member2747690772047059533_set_a @ X2 @ A )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A )
=> ( ( subseteq_mset_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal
thf(fact_205_subset__mset_Ofinite__has__maximal2,axiom,
! [A: set_multiset_set_nat,A2: multiset_set_nat] :
( ( finite5272093019064105709et_nat @ A )
=> ( ( member9214616765234488813et_nat @ A2 @ A )
=> ? [X2: multiset_set_nat] :
( ( member9214616765234488813et_nat @ X2 @ A )
& ( subset6078030600694693471et_nat @ A2 @ X2 )
& ! [Xa: multiset_set_nat] :
( ( member9214616765234488813et_nat @ Xa @ A )
=> ( ( subset6078030600694693471et_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal2
thf(fact_206_subset__mset_Ofinite__has__maximal2,axiom,
! [A: set_multiset_set_a,A2: multiset_set_a] :
( ( finite2815193924343055693_set_a @ A )
=> ( ( member2747690772047059533_set_a @ A2 @ A )
=> ? [X2: multiset_set_a] :
( ( member2747690772047059533_set_a @ X2 @ A )
& ( subseteq_mset_set_a @ A2 @ X2 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A )
=> ( ( subseteq_mset_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_maximal2
thf(fact_207_subset__mset_Ofinite__has__minimal2,axiom,
! [A: set_multiset_set_nat,A2: multiset_set_nat] :
( ( finite5272093019064105709et_nat @ A )
=> ( ( member9214616765234488813et_nat @ A2 @ A )
=> ? [X2: multiset_set_nat] :
( ( member9214616765234488813et_nat @ X2 @ A )
& ( subset6078030600694693471et_nat @ X2 @ A2 )
& ! [Xa: multiset_set_nat] :
( ( member9214616765234488813et_nat @ Xa @ A )
=> ( ( subset6078030600694693471et_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal2
thf(fact_208_subset__mset_Ofinite__has__minimal2,axiom,
! [A: set_multiset_set_a,A2: multiset_set_a] :
( ( finite2815193924343055693_set_a @ A )
=> ( ( member2747690772047059533_set_a @ A2 @ A )
=> ? [X2: multiset_set_a] :
( ( member2747690772047059533_set_a @ X2 @ A )
& ( subseteq_mset_set_a @ X2 @ A2 )
& ! [Xa: multiset_set_a] :
( ( member2747690772047059533_set_a @ Xa @ A )
=> ( ( subseteq_mset_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% subset_mset.finite_has_minimal2
thf(fact_209_subset__mset_Otrans,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( subset6078030600694693471et_nat @ B @ C )
=> ( subset6078030600694693471et_nat @ A2 @ C ) ) ) ).
% subset_mset.trans
thf(fact_210_subset__mset_Otrans,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ A2 @ C ) ) ) ).
% subset_mset.trans
thf(fact_211_subset__mset_Oeq__iff,axiom,
( ( ^ [Y2: multiset_set_nat,Z: multiset_set_nat] : ( Y2 = Z ) )
= ( ^ [A3: multiset_set_nat,B3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A3 @ B3 )
& ( subset6078030600694693471et_nat @ B3 @ A3 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_212_subset__mset_Oeq__iff,axiom,
( ( ^ [Y2: multiset_set_a,Z: multiset_set_a] : ( Y2 = Z ) )
= ( ^ [A3: multiset_set_a,B3: multiset_set_a] :
( ( subseteq_mset_set_a @ A3 @ B3 )
& ( subseteq_mset_set_a @ B3 @ A3 ) ) ) ) ).
% subset_mset.eq_iff
thf(fact_213_subset__mset_Oantisym,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( subset6078030600694693471et_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_mset.antisym
thf(fact_214_subset__mset_Oantisym,axiom,
! [A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( subseteq_mset_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_mset.antisym
thf(fact_215_subset__mset_Oeq__refl,axiom,
! [X3: multiset_set_nat,Y3: multiset_set_nat] :
( ( X3 = Y3 )
=> ( subset6078030600694693471et_nat @ X3 @ Y3 ) ) ).
% subset_mset.eq_refl
thf(fact_216_subset__mset_Oeq__refl,axiom,
! [X3: multiset_set_a,Y3: multiset_set_a] :
( ( X3 = Y3 )
=> ( subseteq_mset_set_a @ X3 @ Y3 ) ) ).
% subset_mset.eq_refl
thf(fact_217_subset__mset_Oorder__trans,axiom,
! [X3: multiset_set_nat,Y3: multiset_set_nat,Z2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ X3 @ Y3 )
=> ( ( subset6078030600694693471et_nat @ Y3 @ Z2 )
=> ( subset6078030600694693471et_nat @ X3 @ Z2 ) ) ) ).
% subset_mset.order_trans
thf(fact_218_subset__mset_Oorder__trans,axiom,
! [X3: multiset_set_a,Y3: multiset_set_a,Z2: multiset_set_a] :
( ( subseteq_mset_set_a @ X3 @ Y3 )
=> ( ( subseteq_mset_set_a @ Y3 @ Z2 )
=> ( subseteq_mset_set_a @ X3 @ Z2 ) ) ) ).
% subset_mset.order_trans
thf(fact_219_subset__mset_Oantisym__conv,axiom,
! [Y3: multiset_set_nat,X3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ Y3 @ X3 )
=> ( ( subset6078030600694693471et_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% subset_mset.antisym_conv
thf(fact_220_subset__mset_Oantisym__conv,axiom,
! [Y3: multiset_set_a,X3: multiset_set_a] :
( ( subseteq_mset_set_a @ Y3 @ X3 )
=> ( ( subseteq_mset_set_a @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% subset_mset.antisym_conv
thf(fact_221_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y2: multiset_set_nat,Z: multiset_set_nat] : ( Y2 = Z ) )
= ( ^ [X: multiset_set_nat,Y4: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ X @ Y4 )
& ( subset6078030600694693471et_nat @ Y4 @ X ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_222_subset__mset_Oorder__eq__iff,axiom,
( ( ^ [Y2: multiset_set_a,Z: multiset_set_a] : ( Y2 = Z ) )
= ( ^ [X: multiset_set_a,Y4: multiset_set_a] :
( ( subseteq_mset_set_a @ X @ Y4 )
& ( subseteq_mset_set_a @ Y4 @ X ) ) ) ) ).
% subset_mset.order_eq_iff
thf(fact_223_subset__mset_Oorder__antisym,axiom,
! [X3: multiset_set_nat,Y3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ X3 @ Y3 )
=> ( ( subset6078030600694693471et_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% subset_mset.order_antisym
thf(fact_224_subset__mset_Oorder__antisym,axiom,
! [X3: multiset_set_a,Y3: multiset_set_a] :
( ( subseteq_mset_set_a @ X3 @ Y3 )
=> ( ( subseteq_mset_set_a @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% subset_mset.order_antisym
thf(fact_225_subset__mset_Oord__eq__le__trans,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( A2 = B )
=> ( ( subset6078030600694693471et_nat @ B @ C )
=> ( subset6078030600694693471et_nat @ A2 @ C ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_226_subset__mset_Oord__eq__le__trans,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( A2 = B )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ A2 @ C ) ) ) ).
% subset_mset.ord_eq_le_trans
thf(fact_227_subset__mset_Oord__le__eq__trans,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( B = C )
=> ( subset6078030600694693471et_nat @ A2 @ C ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_228_subset__mset_Oord__le__eq__trans,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( B = C )
=> ( subseteq_mset_set_a @ A2 @ C ) ) ) ).
% subset_mset.ord_le_eq_trans
thf(fact_229_subset__mset_Odual__order_Otrans,axiom,
! [B: multiset_set_nat,A2: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ B @ A2 )
=> ( ( subset6078030600694693471et_nat @ C @ B )
=> ( subset6078030600694693471et_nat @ C @ A2 ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_230_subset__mset_Odual__order_Otrans,axiom,
! [B: multiset_set_a,A2: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ B @ A2 )
=> ( ( subseteq_mset_set_a @ C @ B )
=> ( subseteq_mset_set_a @ C @ A2 ) ) ) ).
% subset_mset.dual_order.trans
thf(fact_231_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y2: multiset_set_nat,Z: multiset_set_nat] : ( Y2 = Z ) )
= ( ^ [A3: multiset_set_nat,B3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ B3 @ A3 )
& ( subset6078030600694693471et_nat @ A3 @ B3 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_232_subset__mset_Odual__order_Oeq__iff,axiom,
( ( ^ [Y2: multiset_set_a,Z: multiset_set_a] : ( Y2 = Z ) )
= ( ^ [A3: multiset_set_a,B3: multiset_set_a] :
( ( subseteq_mset_set_a @ B3 @ A3 )
& ( subseteq_mset_set_a @ A3 @ B3 ) ) ) ) ).
% subset_mset.dual_order.eq_iff
thf(fact_233_subset__mset_Odual__order_Oantisym,axiom,
! [B: multiset_set_nat,A2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ B @ A2 )
=> ( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( A2 = B ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_234_subset__mset_Odual__order_Oantisym,axiom,
! [B: multiset_set_a,A2: multiset_set_a] :
( ( subseteq_mset_set_a @ B @ A2 )
=> ( ( subseteq_mset_set_a @ A2 @ B )
=> ( A2 = B ) ) ) ).
% subset_mset.dual_order.antisym
thf(fact_235_ex__mset,axiom,
! [X5: multiset_set_a] :
? [Xs2: list_set_a] :
( ( mset_set_a @ Xs2 )
= X5 ) ).
% ex_mset
thf(fact_236_in__mono,axiom,
! [A: set_list_a,B2: set_list_a,X3: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_list_a @ X3 @ A )
=> ( member_list_a @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_237_in__mono,axiom,
! [A: set_set_nat,B2: set_set_nat,X3: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( member_set_nat @ X3 @ A )
=> ( member_set_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_238_in__mono,axiom,
! [A: set_set_a,B2: set_set_a,X3: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( member_set_a @ X3 @ A )
=> ( member_set_a @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_239_in__mono,axiom,
! [A: set_nat,B2: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_240_in__mono,axiom,
! [A: set_a,B2: set_a,X3: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ X3 @ A )
=> ( member_a @ X3 @ B2 ) ) ) ).
% in_mono
thf(fact_241_subsetD,axiom,
! [A: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_le8861187494160871172list_a @ A @ B2 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_242_subsetD,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_243_subsetD,axiom,
! [A: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_244_subsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_245_subsetD,axiom,
! [A: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_246_equalityE,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B2 )
=> ~ ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% equalityE
thf(fact_247_equalityE,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% equalityE
thf(fact_248_subset__eq,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
! [X: list_a] :
( ( member_list_a @ X @ A4 )
=> ( member_list_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_249_subset__eq,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [X: set_nat] :
( ( member_set_nat @ X @ A4 )
=> ( member_set_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_250_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [X: set_a] :
( ( member_set_a @ X @ A4 )
=> ( member_set_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_251_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A4 )
=> ( member_nat @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_252_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X: a] :
( ( member_a @ X @ A4 )
=> ( member_a @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_253_equalityD1,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% equalityD1
thf(fact_254_equalityD1,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% equalityD1
thf(fact_255_equalityD2,axiom,
! [A: set_nat,B2: set_nat] :
( ( A = B2 )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% equalityD2
thf(fact_256_equalityD2,axiom,
! [A: set_a,B2: set_a] :
( ( A = B2 )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% equalityD2
thf(fact_257_subset__iff,axiom,
( ord_le8861187494160871172list_a
= ( ^ [A4: set_list_a,B4: set_list_a] :
! [T: list_a] :
( ( member_list_a @ T @ A4 )
=> ( member_list_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_258_subset__iff,axiom,
( ord_le6893508408891458716et_nat
= ( ^ [A4: set_set_nat,B4: set_set_nat] :
! [T: set_nat] :
( ( member_set_nat @ T @ A4 )
=> ( member_set_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_259_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A4: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A4 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_260_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
! [T: nat] :
( ( member_nat @ T @ A4 )
=> ( member_nat @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_261_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A4 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_262_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_263_subset__refl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% subset_refl
thf(fact_264_Collect__mono,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_265_Collect__mono,axiom,
! [P2: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P2 @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_266_subset__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_267_subset__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% subset_trans
thf(fact_268_set__eq__subset,axiom,
( ( ^ [Y2: set_nat,Z: set_nat] : ( Y2 = Z ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_269_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z: set_a] : ( Y2 = Z ) )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_270_Collect__mono__iff,axiom,
! [P2: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P2 ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_271_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X: a] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_272_card__subset__eq,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( ( finite_card_set_nat @ A )
= ( finite_card_set_nat @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_273_card__subset__eq,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ( finite_card_set_a @ A )
= ( finite_card_set_a @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_274_card__subset__eq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_275_card__subset__eq,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( finite_card_a @ A )
= ( finite_card_a @ B2 ) )
=> ( A = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_276_infinite__arbitrarily__large,axiom,
! [A: set_set_nat,N: nat] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ? [B5: set_set_nat] :
( ( finite1152437895449049373et_nat @ B5 )
& ( ( finite_card_set_nat @ B5 )
= N )
& ( ord_le6893508408891458716et_nat @ B5 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_277_infinite__arbitrarily__large,axiom,
! [A: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A )
=> ? [B5: set_set_a] :
( ( finite_finite_set_a @ B5 )
& ( ( finite_card_set_a @ B5 )
= N )
& ( ord_le3724670747650509150_set_a @ B5 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_278_infinite__arbitrarily__large,axiom,
! [A: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ( finite_card_nat @ B5 )
= N )
& ( ord_less_eq_set_nat @ B5 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_279_infinite__arbitrarily__large,axiom,
! [A: set_a,N: nat] :
( ~ ( finite_finite_a @ A )
=> ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ( finite_card_a @ B5 )
= N )
& ( ord_less_eq_set_a @ B5 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_280_set__mset__mono,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ord_le6893508408891458716et_nat @ ( set_mset_set_nat @ A ) @ ( set_mset_set_nat @ B2 ) ) ) ).
% set_mset_mono
thf(fact_281_set__mset__mono,axiom,
! [A: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B2 )
=> ( ord_le3724670747650509150_set_a @ ( set_mset_set_a @ A ) @ ( set_mset_set_a @ B2 ) ) ) ).
% set_mset_mono
thf(fact_282_set__mset__mono,axiom,
! [A: multiset_nat,B2: multiset_nat] :
( ( subseteq_mset_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ ( set_mset_nat @ A ) @ ( set_mset_nat @ B2 ) ) ) ).
% set_mset_mono
thf(fact_283_set__mset__mono,axiom,
! [A: multiset_a,B2: multiset_a] :
( ( subseteq_mset_a @ A @ B2 )
=> ( ord_less_eq_set_a @ ( set_mset_a @ A ) @ ( set_mset_a @ B2 ) ) ) ).
% set_mset_mono
thf(fact_284_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_285_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_286_mset__subset__eqD,axiom,
! [A: multiset_list_a,B2: multiset_list_a,X3: list_a] :
( ( subseteq_mset_list_a @ A @ B2 )
=> ( ( member_list_a @ X3 @ ( set_mset_list_a @ A ) )
=> ( member_list_a @ X3 @ ( set_mset_list_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_287_mset__subset__eqD,axiom,
! [A: multiset_nat,B2: multiset_nat,X3: nat] :
( ( subseteq_mset_nat @ A @ B2 )
=> ( ( member_nat @ X3 @ ( set_mset_nat @ A ) )
=> ( member_nat @ X3 @ ( set_mset_nat @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_288_mset__subset__eqD,axiom,
! [A: multiset_a,B2: multiset_a,X3: a] :
( ( subseteq_mset_a @ A @ B2 )
=> ( ( member_a @ X3 @ ( set_mset_a @ A ) )
=> ( member_a @ X3 @ ( set_mset_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_289_mset__subset__eqD,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,X3: set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( member_set_nat @ X3 @ ( set_mset_set_nat @ A ) )
=> ( member_set_nat @ X3 @ ( set_mset_set_nat @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_290_mset__subset__eqD,axiom,
! [A: multiset_set_a,B2: multiset_set_a,X3: set_a] :
( ( subseteq_mset_set_a @ A @ B2 )
=> ( ( member_set_a @ X3 @ ( set_mset_set_a @ A ) )
=> ( member_set_a @ X3 @ ( set_mset_set_a @ B2 ) ) ) ) ).
% mset_subset_eqD
thf(fact_291_empty__le,axiom,
! [A: multiset_set_nat] : ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ A ) ).
% empty_le
thf(fact_292_empty__le,axiom,
! [A: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A ) ).
% empty_le
thf(fact_293_subset__mset_Oextremum__uniqueI,axiom,
! [A2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ zero_z3157962936165190495et_nat )
=> ( A2 = zero_z3157962936165190495et_nat ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_294_subset__mset_Oextremum__uniqueI,axiom,
! [A2: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ zero_z5079479921072680283_set_a )
=> ( A2 = zero_z5079479921072680283_set_a ) ) ).
% subset_mset.extremum_uniqueI
thf(fact_295_subset__mset_Obot__least,axiom,
! [A2: multiset_set_nat] : ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ A2 ) ).
% subset_mset.bot_least
thf(fact_296_subset__mset_Obot__least,axiom,
! [A2: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A2 ) ).
% subset_mset.bot_least
thf(fact_297_subset__mset_Ozero__le,axiom,
! [X3: multiset_set_nat] : ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ X3 ) ).
% subset_mset.zero_le
thf(fact_298_subset__mset_Ozero__le,axiom,
! [X3: multiset_set_a] : ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ X3 ) ).
% subset_mset.zero_le
thf(fact_299_finite__has__minimal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ X2 @ A2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_300_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_301_finite__has__minimal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_302_finite__has__maximal2,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( member_set_nat @ A2 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ A2 @ X2 )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_303_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_304_finite__has__maximal2,axiom,
! [A: set_set_a,A2: set_a] :
( ( finite_finite_set_a @ A )
=> ( ( member_set_a @ A2 @ A )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ( ord_less_eq_set_a @ A2 @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_305_repeat__mset__cancel2,axiom,
! [A2: nat,A: multiset_set_nat,B: nat] :
( ( ( repeat_mset_set_nat @ A2 @ A )
= ( repeat_mset_set_nat @ B @ A ) )
= ( ( A2 = B )
| ( A = zero_z3157962936165190495et_nat ) ) ) ).
% repeat_mset_cancel2
thf(fact_306_repeat__mset__cancel2,axiom,
! [A2: nat,A: multiset_set_a,B: nat] :
( ( ( repeat_mset_set_a @ A2 @ A )
= ( repeat_mset_set_a @ B @ A ) )
= ( ( A2 = B )
| ( A = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_cancel2
thf(fact_307_subset__code_I1_J,axiom,
! [Xs: list_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( set_list_a2 @ Xs ) @ B2 )
= ( ! [X: list_a] :
( ( member_list_a @ X @ ( set_list_a2 @ Xs ) )
=> ( member_list_a @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_308_subset__code_I1_J,axiom,
! [Xs: list_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
=> ( member_set_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_309_subset__code_I1_J,axiom,
! [Xs: list_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( set_set_a2 @ Xs ) @ B2 )
= ( ! [X: set_a] :
( ( member_set_a @ X @ ( set_set_a2 @ Xs ) )
=> ( member_set_a @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_310_subset__code_I1_J,axiom,
! [Xs: list_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_311_subset__code_I1_J,axiom,
! [Xs: list_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B2 )
= ( ! [X: a] :
( ( member_a @ X @ ( set_a2 @ Xs ) )
=> ( member_a @ X @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_312_rev__finite__subset,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_313_rev__finite__subset,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( finite_finite_set_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_314_rev__finite__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_315_rev__finite__subset,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% rev_finite_subset
thf(fact_316_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_317_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_318_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_319_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_320_finite__subset,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( finite1152437895449049373et_nat @ A ) ) ) ).
% finite_subset
thf(fact_321_finite__subset,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( finite_finite_set_a @ A ) ) ) ).
% finite_subset
thf(fact_322_finite__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_323_finite__subset,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A ) ) ) ).
% finite_subset
thf(fact_324_ordered__incidence__system_Odual__blocks__b,axiom,
! [V_s: list_set_a,B_s: list_set_set_a] :
( ( incide2166342132139297189_set_a @ V_s @ B_s )
=> ( ( size_s7462436076474991978et_nat @ ( dual_d359914979145368543_set_a @ ( set_set_a2 @ V_s ) @ B_s ) )
= ( finite_card_set_a @ ( set_set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.dual_blocks_b
thf(fact_325_ordered__incidence__system_Odual__blocks__b,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ V_s ) @ B_s ) )
= ( finite_card_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.dual_blocks_b
thf(fact_326_ordered__incidence__system_Odual__blocks__b,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_nat @ ( set_nat2 @ V_s ) @ B_s ) )
= ( finite_card_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.dual_blocks_b
thf(fact_327_ordered__incidence__system_Owf__list,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,B: set_set_a] :
( ( incide2166342132139297189_set_a @ V_s @ B_s )
=> ( ( member_set_set_a @ B @ ( set_mset_set_set_a @ ( mset_set_set_a @ B_s ) ) )
=> ( ord_le3724670747650509150_set_a @ B @ ( set_set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.wf_list
thf(fact_328_ordered__incidence__system_Owf__list,axiom,
! [V_s: list_a,B_s: list_set_a,B: set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ B_s ) ) )
=> ( ord_less_eq_set_a @ B @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.wf_list
thf(fact_329_ordered__incidence__system_Owf__list,axiom,
! [V_s: list_nat,B_s: list_set_nat,B: set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( mset_set_nat @ B_s ) ) )
=> ( ord_less_eq_set_nat @ B @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.wf_list
thf(fact_330_mset_Osimps_I1_J,axiom,
( ( mset_a @ nil_a )
= zero_zero_multiset_a ) ).
% mset.simps(1)
thf(fact_331_mset_Osimps_I1_J,axiom,
( ( mset_set_nat @ nil_set_nat )
= zero_z3157962936165190495et_nat ) ).
% mset.simps(1)
thf(fact_332_mset_Osimps_I1_J,axiom,
( ( mset_set_a @ nil_set_a )
= zero_z5079479921072680283_set_a ) ).
% mset.simps(1)
thf(fact_333_finite__has__minimal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_334_finite__has__minimal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_335_finite__has__minimal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_336_finite__has__maximal,axiom,
! [A: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( A != bot_bot_set_set_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ A )
& ! [Xa: set_nat] :
( ( member_set_nat @ Xa @ A )
=> ( ( ord_less_eq_set_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_337_finite__has__maximal,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_338_finite__has__maximal,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_339_ordered__incidence__system_Oblocks__list__empty__iff,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( B_s = nil_set_a )
= ( ( mset_set_a @ B_s )
= zero_z5079479921072680283_set_a ) ) ) ).
% ordered_incidence_system.blocks_list_empty_iff
thf(fact_340_ordered__incidence__system_Oblocks__list__empty__iff,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( B_s = nil_set_nat )
= ( ( mset_set_nat @ B_s )
= zero_z3157962936165190495et_nat ) ) ) ).
% ordered_incidence_system.blocks_list_empty_iff
thf(fact_341_k___092_060Lambda_062__PBD__axioms,axiom,
block_k_Lambda_PBD_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ lambda @ k ).
% k_\<Lambda>_PBD_axioms
thf(fact_342_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_343_finite__design__support,axiom,
finite_finite_set_a @ ( design5397942185814921632port_a @ ( mset_set_a @ b_s ) ) ).
% finite_design_support
thf(fact_344_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_345_const__intersect__design__axioms,axiom,
design9190424834980853558sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ lambda ).
% const_intersect_design_axioms
thf(fact_346_simple__design__axioms,axiom,
design3982635895484621246sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).
% simple_design_axioms
thf(fact_347_simple__incidence__system__axioms,axiom,
design1338723777345758283stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ).
% simple_incidence_system_axioms
thf(fact_348_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_349_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_350_multiple__block__in__original,axiom,
! [B: set_a,N: nat] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_block_in_original
thf(fact_351_block__size__lt__v,axiom,
ord_less_eq_nat @ k @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% block_size_lt_v
thf(fact_352_uniform__alt__def__all,axiom,
! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( finite_card_a @ X4 )
= k ) ) ).
% uniform_alt_def_all
thf(fact_353_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_354_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_355_repeat__mset__block__point__rel,axiom,
! [B: set_a,N: nat,X3: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) )
=> ( ( member_a @ X3 @ B )
=> ( member_a @ X3 @ ( set_a2 @ v_s ) ) ) ) ).
% repeat_mset_block_point_rel
thf(fact_356_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_357_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_358_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_359_replication__numbers__finite,axiom,
finite_finite_nat @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% replication_numbers_finite
thf(fact_360_uniform,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( finite_card_a @ Bl2 )
= k ) ) ).
% uniform
thf(fact_361_size__mset__mono,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( size_s7462436076474991978et_nat @ A ) @ ( size_s7462436076474991978et_nat @ B2 ) ) ) ).
% size_mset_mono
thf(fact_362_size__mset__mono,axiom,
! [A: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B2 )
=> ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ A ) @ ( size_s6566526139600085008_set_a @ B2 ) ) ) ).
% size_mset_mono
thf(fact_363_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_nat,C2: nat] :
( ! [G: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ G @ F )
=> ( ( finite1152437895449049373et_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ G ) @ C2 ) ) )
=> ( ( finite1152437895449049373et_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_set_nat @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_364_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_set_a,C2: nat] :
( ! [G: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G @ F )
=> ( ( finite_finite_set_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_set_a @ F )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_365_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_nat,C2: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F )
& ( ord_less_eq_nat @ ( finite_card_nat @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_366_finite__if__finite__subsets__card__bdd,axiom,
! [F: set_a,C2: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_a @ F )
& ( ord_less_eq_nat @ ( finite_card_a @ F ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_367_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S ) )
=> ~ ! [T3: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ T3 @ S )
=> ( ( ( finite_card_set_nat @ T3 )
= N )
=> ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_368_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S ) )
=> ~ ! [T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T3 @ S )
=> ( ( ( finite_card_set_a @ T3 )
= N )
=> ~ ( finite_finite_set_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_369_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_370_obtain__subset__with__card__n,axiom,
! [N: nat,S: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_371_exists__subset__between,axiom,
! [A: set_set_nat,N: nat,C2: set_set_nat] :
( ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ C2 ) )
=> ( ( ord_le6893508408891458716et_nat @ A @ C2 )
=> ( ( finite1152437895449049373et_nat @ C2 )
=> ? [B5: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B5 )
& ( ord_le6893508408891458716et_nat @ B5 @ C2 )
& ( ( finite_card_set_nat @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_372_exists__subset__between,axiom,
! [A: set_set_a,N: nat,C2: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A @ C2 )
=> ( ( finite_finite_set_a @ C2 )
=> ? [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ C2 )
& ( ( finite_card_set_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_373_exists__subset__between,axiom,
! [A: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B5: set_nat] :
( ( ord_less_eq_set_nat @ A @ B5 )
& ( ord_less_eq_set_nat @ B5 @ C2 )
& ( ( finite_card_nat @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_374_exists__subset__between,axiom,
! [A: set_a,N: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B5: set_a] :
( ( ord_less_eq_set_a @ A @ B5 )
& ( ord_less_eq_set_a @ B5 @ C2 )
& ( ( finite_card_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_375_card__seteq,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_376_card__seteq,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B2 ) @ ( finite_card_set_a @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_377_card__seteq,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_378_card__seteq,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A ) )
=> ( A = B2 ) ) ) ) ).
% card_seteq
thf(fact_379_card__mono,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_380_card__mono,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_381_card__mono,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_382_card__mono,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_383_ordered__simple__design_Oaxioms_I2_J,axiom,
! [V_s: list_set_a,B_s: list_set_set_a] :
( ( incide5137607047756421874_set_a @ V_s @ B_s )
=> ( design1835266114905787166_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) ) ) ).
% ordered_simple_design.axioms(2)
thf(fact_384_ordered__simple__design_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide371748008924627346sign_a @ V_s @ B_s )
=> ( design3982635895484621246sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) ) ) ).
% ordered_simple_design.axioms(2)
thf(fact_385_ordered__simple__design_Ointro,axiom,
! [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 ) )
=> ( incide5137607047756421874_set_a @ V_s @ B_s ) ) ) ).
% ordered_simple_design.intro
thf(fact_386_ordered__simple__design_Ointro,axiom,
! [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 ) )
=> ( incide371748008924627346sign_a @ V_s @ B_s ) ) ) ).
% ordered_simple_design.intro
thf(fact_387_ordered__simple__design__def,axiom,
( incide5137607047756421874_set_a
= ( ^ [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 ) ) ) ) ) ).
% ordered_simple_design_def
thf(fact_388_ordered__simple__design__def,axiom,
( incide371748008924627346sign_a
= ( ^ [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 ) ) ) ) ) ).
% ordered_simple_design_def
thf(fact_389_replication__number__single,axiom,
is_singleton_nat @ ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ).
% replication_number_single
thf(fact_390_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_391_symmetric__bibd__axioms,axiom,
symmetric_bibd_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k @ lambda ).
% symmetric_bibd_axioms
thf(fact_392_exists__ordering__sysI,axiom,
? [Vs: list_a,Bs2: list_set_a] :
( ( member_list_a @ Vs @ ( multis2428024204330136193_set_a @ ( set_a2 @ v_s ) ) )
& ( member_list_set_a @ Bs2 @ ( multis5469701301851823918_set_a @ ( mset_set_a @ b_s ) ) )
& ( incide1624170830610365509stem_a @ Vs @ Bs2 ) ) ).
% exists_ordering_sysI
thf(fact_393_alt__ordering__sysI,axiom,
! [Vs2: list_a,Bs3: list_set_a] :
( ( member_list_a @ Vs2 @ ( multis2428024204330136193_set_a @ ( set_a2 @ v_s ) ) )
=> ( ( member_list_set_a @ Bs3 @ ( multis5469701301851823918_set_a @ ( mset_set_a @ b_s ) ) )
=> ( incide1624170830610365509stem_a @ Vs2 @ Bs3 ) ) ) ).
% alt_ordering_sysI
thf(fact_394_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_395_incomplete__design__axioms,axiom,
block_1438872132225661677sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k ).
% incomplete_design_axioms
thf(fact_396_pairwise__balance__axioms,axiom,
block_5355636846524985331ance_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ lambda ).
% pairwise_balance_axioms
thf(fact_397_delete__point__strong__block__subset,axiom,
! [P: a] : ( subseteq_mset_set_a @ ( design5657747894866638574ocks_a @ ( mset_set_a @ b_s ) @ P ) @ ( mset_set_a @ b_s ) ) ).
% delete_point_strong_block_subset
thf(fact_398_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_399_blocks__indexing,axiom,
member_list_set_a @ b_s @ ( multis5469701301851823918_set_a @ ( mset_set_a @ b_s ) ) ).
% blocks_indexing
thf(fact_400_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_401_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_402_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_403_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_404_b__gt__index,axiom,
ord_less_eq_nat @ lambda @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% b_gt_index
thf(fact_405_local_Osymmetric,axiom,
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% local.symmetric
thf(fact_406_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_407_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_408_symmetric__bibdI,axiom,
( ( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( symmetric_bibd_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k @ lambda ) ) ).
% symmetric_bibdI
thf(fact_409_permutations__of__multisetD,axiom,
! [Xs: list_a,A: multiset_a] :
( ( member_list_a @ Xs @ ( multis5886240593633752526iset_a @ A ) )
=> ( ( mset_a @ Xs )
= A ) ) ).
% permutations_of_multisetD
thf(fact_410_permutations__of__multisetD,axiom,
! [Xs: list_set_a,A: multiset_set_a] :
( ( member_list_set_a @ Xs @ ( multis5469701301851823918_set_a @ A ) )
=> ( ( mset_set_a @ Xs )
= A ) ) ).
% permutations_of_multisetD
thf(fact_411_permutations__of__multisetD,axiom,
! [Xs: list_set_nat,A: multiset_set_nat] :
( ( member_list_set_nat @ Xs @ ( multis124346860217030838et_nat @ A ) )
=> ( ( mset_set_nat @ Xs )
= A ) ) ).
% permutations_of_multisetD
thf(fact_412_permutations__of__multisetI,axiom,
! [Xs: list_a,A: multiset_a] :
( ( ( mset_a @ Xs )
= A )
=> ( member_list_a @ Xs @ ( multis5886240593633752526iset_a @ A ) ) ) ).
% permutations_of_multisetI
thf(fact_413_permutations__of__multisetI,axiom,
! [Xs: list_set_a,A: multiset_set_a] :
( ( ( mset_set_a @ Xs )
= A )
=> ( member_list_set_a @ Xs @ ( multis5469701301851823918_set_a @ A ) ) ) ).
% permutations_of_multisetI
thf(fact_414_permutations__of__multisetI,axiom,
! [Xs: list_set_nat,A: multiset_set_nat] :
( ( ( mset_set_nat @ Xs )
= A )
=> ( member_list_set_nat @ Xs @ ( multis124346860217030838et_nat @ A ) ) ) ).
% permutations_of_multisetI
thf(fact_415_permutations__of__multiset__not__empty,axiom,
! [A: multiset_set_a] :
( ( multis5469701301851823918_set_a @ A )
!= bot_bo4397488018069675312_set_a ) ).
% permutations_of_multiset_not_empty
thf(fact_416_permutations__of__multiset__not__empty,axiom,
! [A: multiset_set_nat] :
( ( multis124346860217030838et_nat @ A )
!= bot_bo2934890284768024416et_nat ) ).
% permutations_of_multiset_not_empty
thf(fact_417_finite__permutations__of__multiset,axiom,
! [A: multiset_set_a] : ( finite1971793804006318733_set_a @ ( multis5469701301851823918_set_a @ A ) ) ).
% finite_permutations_of_multiset
thf(fact_418_finite__permutations__of__multiset,axiom,
! [A: multiset_set_nat] : ( finite1091814263879798189et_nat @ ( multis124346860217030838et_nat @ A ) ) ).
% finite_permutations_of_multiset
thf(fact_419_ordered__incidence__system_Oblocks__indexing,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( member_list_set_a @ B_s @ ( multis5469701301851823918_set_a @ ( mset_set_a @ B_s ) ) ) ) ).
% ordered_incidence_system.blocks_indexing
thf(fact_420_ordered__incidence__system_Oblocks__indexing,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( member_list_set_nat @ B_s @ ( multis124346860217030838et_nat @ ( mset_set_nat @ B_s ) ) ) ) ).
% ordered_incidence_system.blocks_indexing
thf(fact_421_elem__permutation__of__mset__empty__iff,axiom,
! [Xs: list_a,A: multiset_a] :
( ( member_list_a @ Xs @ ( multis5886240593633752526iset_a @ A ) )
=> ( ( Xs = nil_a )
= ( A = zero_zero_multiset_a ) ) ) ).
% elem_permutation_of_mset_empty_iff
thf(fact_422_elem__permutation__of__mset__empty__iff,axiom,
! [Xs: list_set_a,A: multiset_set_a] :
( ( member_list_set_a @ Xs @ ( multis5469701301851823918_set_a @ A ) )
=> ( ( Xs = nil_set_a )
= ( A = zero_z5079479921072680283_set_a ) ) ) ).
% elem_permutation_of_mset_empty_iff
thf(fact_423_elem__permutation__of__mset__empty__iff,axiom,
! [Xs: list_set_nat,A: multiset_set_nat] :
( ( member_list_set_nat @ Xs @ ( multis124346860217030838et_nat @ A ) )
=> ( ( Xs = nil_set_nat )
= ( A = zero_z3157962936165190495et_nat ) ) ) ).
% elem_permutation_of_mset_empty_iff
thf(fact_424_is__singletonI_H,axiom,
! [A: set_list_a] :
( ( A != bot_bot_set_list_a )
=> ( ! [X2: list_a,Y: list_a] :
( ( member_list_a @ X2 @ A )
=> ( ( member_list_a @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_singleton_list_a @ A ) ) ) ).
% is_singletonI'
thf(fact_425_is__singletonI_H,axiom,
! [A: set_set_nat] :
( ( A != bot_bot_set_set_nat )
=> ( ! [X2: set_nat,Y: set_nat] :
( ( member_set_nat @ X2 @ A )
=> ( ( member_set_nat @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_singleton_set_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_426_is__singletonI_H,axiom,
! [A: set_set_a] :
( ( A != bot_bot_set_set_a )
=> ( ! [X2: set_a,Y: set_a] :
( ( member_set_a @ X2 @ A )
=> ( ( member_set_a @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_singleton_set_a @ A ) ) ) ).
% is_singletonI'
thf(fact_427_is__singletonI_H,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
=> ( ! [X2: a,Y: a] :
( ( member_a @ X2 @ A )
=> ( ( member_a @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_singleton_a @ A ) ) ) ).
% is_singletonI'
thf(fact_428_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y @ A )
=> ( X2 = Y ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_429_ordered__pairwise__balance_Oaxioms_I2_J,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,Lambda: nat] :
( ( incide4449361439798955450_set_a @ V_s @ B_s @ Lambda )
=> ( block_6207159848980890963_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) @ Lambda ) ) ).
% ordered_pairwise_balance.axioms(2)
thf(fact_430_ordered__pairwise__balance_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,Lambda: nat] :
( ( incide6880889959311561818ance_a @ V_s @ B_s @ Lambda )
=> ( block_5355636846524985331ance_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ Lambda ) ) ).
% ordered_pairwise_balance.axioms(2)
thf(fact_431_ordered__sym__bibd_Oaxioms_I2_J,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,K: nat,Lambda: nat] :
( ( incide3984264607369433128_set_a @ V_s @ B_s @ K @ Lambda )
=> ( symmetric_bibd_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) @ K @ Lambda ) ) ).
% ordered_sym_bibd.axioms(2)
thf(fact_432_ordered__sym__bibd_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat,Lambda: nat] :
( ( incide4194285476649307848bibd_a @ V_s @ B_s @ K @ Lambda )
=> ( symmetric_bibd_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ K @ Lambda ) ) ).
% ordered_sym_bibd.axioms(2)
thf(fact_433_ordered__incomplete__design_Oaxioms_I2_J,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,K: nat] :
( ( incide4280386766309144998_set_a @ V_s @ B_s @ K )
=> ( block_3501025933689710669_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) @ K ) ) ).
% ordered_incomplete_design.axioms(2)
thf(fact_434_ordered__incomplete__design_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat] :
( ( incide1377962018248667206sign_a @ V_s @ B_s @ K )
=> ( block_1438872132225661677sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ K ) ) ).
% ordered_incomplete_design.axioms(2)
thf(fact_435_ordered__sym__bibd__def,axiom,
( incide3984264607369433128_set_a
= ( ^ [V_s2: list_set_a,B_s2: list_set_set_a,K2: nat,Lambda2: nat] :
( ( incide532103555118123225_set_a @ V_s2 @ B_s2 @ K2 @ Lambda2 )
& ( symmetric_bibd_set_a @ ( set_set_a2 @ V_s2 ) @ ( mset_set_set_a @ B_s2 ) @ K2 @ Lambda2 ) ) ) ) ).
% ordered_sym_bibd_def
thf(fact_436_ordered__sym__bibd__def,axiom,
( incide4194285476649307848bibd_a
= ( ^ [V_s2: list_a,B_s2: list_set_a,K2: nat,Lambda2: nat] :
( ( incide4817766913905363833bibd_a @ V_s2 @ B_s2 @ K2 @ Lambda2 )
& ( symmetric_bibd_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) @ K2 @ Lambda2 ) ) ) ) ).
% ordered_sym_bibd_def
thf(fact_437_ordered__sym__bibd_Ointro,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,K: nat,Lambda: nat] :
( ( incide532103555118123225_set_a @ V_s @ B_s @ K @ Lambda )
=> ( ( symmetric_bibd_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) @ K @ Lambda )
=> ( incide3984264607369433128_set_a @ V_s @ B_s @ K @ Lambda ) ) ) ).
% ordered_sym_bibd.intro
thf(fact_438_ordered__sym__bibd_Ointro,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat,Lambda: nat] :
( ( incide4817766913905363833bibd_a @ V_s @ B_s @ K @ Lambda )
=> ( ( symmetric_bibd_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ K @ Lambda )
=> ( incide4194285476649307848bibd_a @ V_s @ B_s @ K @ Lambda ) ) ) ).
% ordered_sym_bibd.intro
thf(fact_439_ordered__incomplete__design__def,axiom,
( incide4280386766309144998_set_a
= ( ^ [V_s2: list_set_a,B_s2: list_set_set_a,K2: nat] :
( ( incide8999561475770975213_set_a @ V_s2 @ B_s2 @ K2 )
& ( block_3501025933689710669_set_a @ ( set_set_a2 @ V_s2 ) @ ( mset_set_set_a @ B_s2 ) @ K2 ) ) ) ) ).
% ordered_incomplete_design_def
thf(fact_440_ordered__incomplete__design__def,axiom,
( incide1377962018248667206sign_a
= ( ^ [V_s2: list_a,B_s2: list_set_a,K2: nat] :
( ( incide5219153079875704461sign_a @ V_s2 @ B_s2 @ K2 )
& ( block_1438872132225661677sign_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) @ K2 ) ) ) ) ).
% ordered_incomplete_design_def
thf(fact_441_ordered__incomplete__design_Ointro,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,K: nat] :
( ( incide8999561475770975213_set_a @ V_s @ B_s @ K )
=> ( ( block_3501025933689710669_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) @ K )
=> ( incide4280386766309144998_set_a @ V_s @ B_s @ K ) ) ) ).
% ordered_incomplete_design.intro
thf(fact_442_ordered__incomplete__design_Ointro,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat] :
( ( incide5219153079875704461sign_a @ V_s @ B_s @ K )
=> ( ( block_1438872132225661677sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ K )
=> ( incide1377962018248667206sign_a @ V_s @ B_s @ K ) ) ) ).
% ordered_incomplete_design.intro
thf(fact_443_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_444_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_445_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_446_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_447_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_448_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_449_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_450_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_451_b__non__zero,axiom,
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
!= zero_zero_nat ) ).
% b_non_zero
thf(fact_452_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_453_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_454_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_455_finite__block__sizes,axiom,
finite_finite_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ).
% finite_block_sizes
thf(fact_456_delete__block__subset,axiom,
! [B: set_a] : ( subseteq_mset_set_a @ ( design1146539425385464078lock_a @ ( mset_set_a @ b_s ) @ B ) @ ( mset_set_a @ b_s ) ) ).
% delete_block_subset
thf(fact_457_sys__block__sizes__uniform__single,axiom,
is_singleton_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) ).
% sys_block_sizes_uniform_single
thf(fact_458_v__non__zero,axiom,
ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% v_non_zero
thf(fact_459_b__positive,axiom,
ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% b_positive
thf(fact_460_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_461_sys__block__sizes__obtain__bl,axiom,
! [X3: nat] :
( ( member_nat @ X3 @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
& ( ( finite_card_a @ X2 )
= X3 ) ) ) ).
% sys_block_sizes_obtain_bl
thf(fact_462_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_463_incomplete,axiom,
ord_less_nat @ k @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% incomplete
thf(fact_464_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_465_multiple__block__in,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ) ).
% multiple_block_in
thf(fact_466_multiple__block__sizes__same,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) )
= ( design1769254222028858111izes_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ).
% multiple_block_sizes_same
thf(fact_467_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_468_multiple__blocks__sub,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( subseteq_mset_set_a @ ( mset_set_a @ b_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_blocks_sub
thf(fact_469_multiple__blocks__gt,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ).
% multiple_blocks_gt
thf(fact_470_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_471_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_472_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_473_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_474_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_475_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_476_dual__sys_Omultiple__block__in,axiom,
! [N: nat,B: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_nat @ B @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( member_set_nat @ B @ ( set_mset_set_nat @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ) ).
% dual_sys.multiple_block_in
thf(fact_477_multiple__orig__sub__system,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( sub_su7923802003039619913stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_orig_sub_system
thf(fact_478_multiple__orig__sub__des,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( sub_sub_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_orig_sub_des
thf(fact_479_dual__sys_Omultiple__block__sizes__same,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( design8152002643121538447es_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
= ( design8152002643121538447es_nat @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ).
% dual_sys.multiple_block_sizes_same
thf(fact_480_dual__sys_Omultiple__blocks__sub,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( subset6078030600694693471et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.multiple_blocks_sub
thf(fact_481_dual__sys_Omultiple__blocks__gt,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ ( size_s7462436076474991978et_nat @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ).
% dual_sys.multiple_blocks_gt
thf(fact_482_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_483_multiple__incomplete,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( block_1438872132225661677sign_a @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ k ) ) ).
% multiple_incomplete
thf(fact_484_card_Oempty,axiom,
( ( finite_card_set_a @ bot_bot_set_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_485_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_486_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_487_card_Oinfinite,axiom,
! [A: set_a] :
( ~ ( finite_finite_a @ A )
=> ( ( finite_card_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_488_card_Oinfinite,axiom,
! [A: set_set_nat] :
( ~ ( finite1152437895449049373et_nat @ A )
=> ( ( finite_card_set_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_489_card_Oinfinite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ A )
=> ( ( finite_card_nat @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_490_card_Oinfinite,axiom,
! [A: set_set_a] :
( ~ ( finite_finite_set_a @ A )
=> ( ( finite_card_set_a @ A )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_491_size__empty,axiom,
( ( size_s7462436076474991978et_nat @ zero_z3157962936165190495et_nat )
= zero_zero_nat ) ).
% size_empty
thf(fact_492_size__empty,axiom,
( ( size_s6566526139600085008_set_a @ zero_z5079479921072680283_set_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_493_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_494_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_495_repeat__mset__0,axiom,
! [M: multiset_set_nat] :
( ( repeat_mset_set_nat @ zero_zero_nat @ M )
= zero_z3157962936165190495et_nat ) ).
% repeat_mset_0
thf(fact_496_repeat__mset__0,axiom,
! [M: multiset_set_a] :
( ( repeat_mset_set_a @ zero_zero_nat @ M )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_0
thf(fact_497_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_498_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_499_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_500_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_501_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_502_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_503_incomplete__designI,axiom,
( ( ord_less_nat @ k @ ( finite_card_a @ ( set_a2 @ v_s ) ) )
=> ( block_1438872132225661677sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k ) ) ).
% incomplete_designI
thf(fact_504_card__ge__0__finite,axiom,
! [A: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A ) )
=> ( finite_finite_a @ A ) ) ).
% card_ge_0_finite
thf(fact_505_card__ge__0__finite,axiom,
! [A: set_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A ) )
=> ( finite1152437895449049373et_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_506_card__ge__0__finite,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
=> ( finite_finite_nat @ A ) ) ).
% card_ge_0_finite
thf(fact_507_card__ge__0__finite,axiom,
! [A: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A ) )
=> ( finite_finite_set_a @ A ) ) ).
% card_ge_0_finite
thf(fact_508_nonempty__has__size,axiom,
! [S: multiset_set_nat] :
( ( S != zero_z3157962936165190495et_nat )
= ( ord_less_nat @ zero_zero_nat @ ( size_s7462436076474991978et_nat @ S ) ) ) ).
% nonempty_has_size
thf(fact_509_nonempty__has__size,axiom,
! [S: multiset_set_a] :
( ( S != zero_z5079479921072680283_set_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ S ) ) ) ).
% nonempty_has_size
thf(fact_510_repeat__mset__cancel1,axiom,
! [A2: nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( ( repeat_mset_set_nat @ A2 @ A )
= ( repeat_mset_set_nat @ A2 @ B2 ) )
= ( ( A = B2 )
| ( A2 = zero_zero_nat ) ) ) ).
% repeat_mset_cancel1
thf(fact_511_repeat__mset__cancel1,axiom,
! [A2: nat,A: multiset_set_a,B2: multiset_set_a] :
( ( ( repeat_mset_set_a @ A2 @ A )
= ( repeat_mset_set_a @ A2 @ B2 ) )
= ( ( A = B2 )
| ( A2 = zero_zero_nat ) ) ) ).
% repeat_mset_cancel1
thf(fact_512_card__gt__0__iff,axiom,
! [A: set_set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A ) )
= ( ( A != bot_bot_set_set_nat )
& ( finite1152437895449049373et_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_513_card__gt__0__iff,axiom,
! [A: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A ) )
= ( ( A != bot_bot_set_set_a )
& ( finite_finite_set_a @ A ) ) ) ).
% card_gt_0_iff
thf(fact_514_card__gt__0__iff,axiom,
! [A: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A ) )
= ( ( A != bot_bot_set_a )
& ( finite_finite_a @ A ) ) ) ).
% card_gt_0_iff
thf(fact_515_card__gt__0__iff,axiom,
! [A: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A ) )
= ( ( A != bot_bot_set_nat )
& ( finite_finite_nat @ A ) ) ) ).
% card_gt_0_iff
thf(fact_516_repeat__mset__eq__empty__iff,axiom,
! [N: nat,A: multiset_set_nat] :
( ( ( repeat_mset_set_nat @ N @ A )
= zero_z3157962936165190495et_nat )
= ( ( N = zero_zero_nat )
| ( A = zero_z3157962936165190495et_nat ) ) ) ).
% repeat_mset_eq_empty_iff
thf(fact_517_repeat__mset__eq__empty__iff,axiom,
! [N: nat,A: multiset_set_a] :
( ( ( repeat_mset_set_a @ N @ A )
= zero_z5079479921072680283_set_a )
= ( ( N = zero_zero_nat )
| ( A = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_eq_empty_iff
thf(fact_518_card__eq__0__iff,axiom,
! [A: set_set_nat] :
( ( ( finite_card_set_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_set_nat )
| ~ ( finite1152437895449049373et_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_519_card__eq__0__iff,axiom,
! [A: set_set_a] :
( ( ( finite_card_set_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_520_card__eq__0__iff,axiom,
! [A: set_a] :
( ( ( finite_card_a @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_a )
| ~ ( finite_finite_a @ A ) ) ) ).
% card_eq_0_iff
thf(fact_521_card__eq__0__iff,axiom,
! [A: set_nat] :
( ( ( finite_card_nat @ A )
= zero_zero_nat )
= ( ( A = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A ) ) ) ).
% card_eq_0_iff
thf(fact_522_ordered__proper__design_Odual__sys__b__non__zero,axiom,
! [V_s: list_set_a,B_s: list_set_set_a] :
( ( incide2999377533768400724_set_a @ V_s @ B_s )
=> ( ( size_s7462436076474991978et_nat @ ( dual_d359914979145368543_set_a @ ( set_set_a2 @ V_s ) @ B_s ) )
!= zero_zero_nat ) ) ).
% ordered_proper_design.dual_sys_b_non_zero
thf(fact_523_ordered__proper__design_Odual__sys__b__non__zero,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ V_s ) @ B_s ) )
!= zero_zero_nat ) ) ).
% ordered_proper_design.dual_sys_b_non_zero
thf(fact_524_multiple__not__simple,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ( mset_set_a @ b_s )
!= zero_z5079479921072680283_set_a )
=> ~ ( design1338723777345758283stem_a @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) ) ) ) ).
% multiple_not_simple
thf(fact_525_block__design__multiple,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( block_block_design_a @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ k ) ) ).
% block_design_multiple
thf(fact_526_complement__incomplete,axiom,
block_1438872132225661677sign_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ k ) ).
% complement_incomplete
thf(fact_527_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_528_points__index__empty,axiom,
! [Ps: set_nat] :
( ( design6574611146354332593ex_nat @ zero_z3157962936165190495et_nat @ Ps )
= zero_zero_nat ) ).
% points_index_empty
thf(fact_529_points__index__empty,axiom,
! [Ps: set_a] :
( ( design254580327166089565ndex_a @ zero_z5079479921072680283_set_a @ Ps )
= zero_zero_nat ) ).
% points_index_empty
thf(fact_530_uniform__complement__block__size,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) )
=> ( ( finite_card_a @ Bl2 )
= ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ k ) ) ) ).
% uniform_complement_block_size
thf(fact_531_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_532_index__not__zero,axiom,
ord_less_eq_nat @ one_one_nat @ lambda ).
% index_not_zero
thf(fact_533_k__non__zero,axiom,
ord_less_eq_nat @ one_one_nat @ k ).
% k_non_zero
thf(fact_534_multiple__1__same,axiom,
( ( repeat_mset_set_a @ one_one_nat @ ( mset_set_a @ b_s ) )
= ( mset_set_a @ b_s ) ) ).
% multiple_1_same
thf(fact_535_block__comp__elem__alt__left,axiom,
! [X3: a,Bl2: set_a,Ps: set_a] :
( ( member_a @ X3 @ Bl2 )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) )
=> ~ ( member_a @ X3 @ Ps ) ) ) ).
% block_comp_elem_alt_left
thf(fact_536_block__comp__elem__alt__right,axiom,
! [Ps: set_a,Bl2: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ Ps )
=> ~ ( member_a @ X2 @ Bl2 ) )
=> ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ ( set_a2 @ v_s ) @ Bl2 ) ) ) ) ).
% block_comp_elem_alt_right
thf(fact_537_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 ) )
= ( ! [X: a] :
( ( member_a @ X @ Ps )
=> ~ ( member_a @ X @ Bl2 ) ) ) ) ) ).
% block_complement_elem_iff
thf(fact_538_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_539_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_540_t__design__min__v,axiom,
ord_less_nat @ one_one_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) ).
% t_design_min_v
thf(fact_541_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_542_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_543_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_544_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_545_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_546_dual__sys_Omultiple__1__same,axiom,
( ( repeat_mset_set_nat @ one_one_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) )
= ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ).
% dual_sys.multiple_1_same
thf(fact_547_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_548_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_549_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_550_index__zero__iff,axiom,
( ( lambda = zero_zero_nat )
= ( ! [X: set_a] :
( ( member_set_a @ X @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( finite_card_a @ X )
= one_one_nat ) ) ) ) ).
% index_zero_iff
thf(fact_551_block__design__axioms,axiom,
block_block_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k ).
% block_design_axioms
thf(fact_552_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_553_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_554_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_555_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_556_uniform__size__incomp,axiom,
! [Bl2: set_a] :
( ( ord_less_eq_nat @ k @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) )
=> ( ( 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 ) ) ) ) ) ) ).
% uniform_size_incomp
thf(fact_557_block__designI,axiom,
! [K3: nat] :
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( finite_card_a @ Bl )
= K3 ) )
=> ( block_block_design_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ K3 ) ) ).
% block_designI
thf(fact_558_uniform__complement,axiom,
( ( ord_less_eq_nat @ k @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) )
=> ( block_block_design_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ k ) ) ) ).
% uniform_complement
thf(fact_559_points__index__one__unique__block,axiom,
! [B2: multiset_set_nat,Ps: set_nat] :
( ( ( design6574611146354332593ex_nat @ B2 @ Ps )
= one_one_nat )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ ( set_mset_set_nat @ B2 ) )
& ( ord_less_eq_set_nat @ Ps @ X2 )
& ! [Y5: set_nat] :
( ( ( member_set_nat @ Y5 @ ( set_mset_set_nat @ B2 ) )
& ( ord_less_eq_set_nat @ Ps @ Y5 ) )
=> ( Y5 = X2 ) ) ) ) ).
% points_index_one_unique_block
thf(fact_560_points__index__one__unique__block,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= one_one_nat )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( set_mset_set_a @ B2 ) )
& ( ord_less_eq_set_a @ Ps @ X2 )
& ! [Y5: set_a] :
( ( ( member_set_a @ Y5 @ ( set_mset_set_a @ B2 ) )
& ( ord_less_eq_set_a @ Ps @ Y5 ) )
=> ( Y5 = X2 ) ) ) ) ).
% points_index_one_unique_block
thf(fact_561_points__index__one__unique,axiom,
! [B2: multiset_set_nat,Ps: set_nat,Bl2: set_nat,Bl3: set_nat] :
( ( ( design6574611146354332593ex_nat @ B2 @ Ps )
= one_one_nat )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ B2 ) )
=> ( ( ord_less_eq_set_nat @ Ps @ Bl2 )
=> ( ( member_set_nat @ Bl3 @ ( set_mset_set_nat @ B2 ) )
=> ( ( ord_less_eq_set_nat @ Ps @ Bl3 )
=> ( Bl2 = Bl3 ) ) ) ) ) ) ).
% points_index_one_unique
thf(fact_562_points__index__one__unique,axiom,
! [B2: multiset_set_a,Ps: set_a,Bl2: set_a,Bl3: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= one_one_nat )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ B2 ) )
=> ( ( ord_less_eq_set_a @ Ps @ Bl2 )
=> ( ( member_set_a @ Bl3 @ ( set_mset_set_a @ B2 ) )
=> ( ( ord_less_eq_set_a @ Ps @ Bl3 )
=> ( Bl2 = Bl3 ) ) ) ) ) ) ).
% points_index_one_unique
thf(fact_563_is__singleton__altdef,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( ( finite_card_a @ A4 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_564_is__singleton__altdef,axiom,
( is_singleton_set_a
= ( ^ [A4: set_set_a] :
( ( finite_card_set_a @ A4 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_565_is__singleton__altdef,axiom,
( is_singleton_nat
= ( ^ [A4: set_nat] :
( ( finite_card_nat @ A4 )
= one_one_nat ) ) ) ).
% is_singleton_altdef
thf(fact_566_ordered__block__design_Oaxioms_I2_J,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,K: nat] :
( ( incide8999561475770975213_set_a @ V_s @ B_s @ K )
=> ( block_4731189848718898310_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) @ K ) ) ).
% ordered_block_design.axioms(2)
thf(fact_567_ordered__block__design_Oaxioms_I2_J,axiom,
! [V_s: list_nat,B_s: list_set_nat,K: nat] :
( ( incide8589348598044109441gn_nat @ V_s @ B_s @ K )
=> ( block_625751327111516584gn_nat @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) @ K ) ) ).
% ordered_block_design.axioms(2)
thf(fact_568_ordered__block__design_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat] :
( ( incide5219153079875704461sign_a @ V_s @ B_s @ K )
=> ( block_block_design_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ K ) ) ).
% ordered_block_design.axioms(2)
thf(fact_569_incidence__system_Osys__block__sizes_Ocong,axiom,
design8152002643121538447es_nat = design8152002643121538447es_nat ).
% incidence_system.sys_block_sizes.cong
thf(fact_570_incidence__system_Osys__block__sizes_Ocong,axiom,
design1769254222028858111izes_a = design1769254222028858111izes_a ).
% incidence_system.sys_block_sizes.cong
thf(fact_571_incidence__system_Oreplication__numbers_Ocong,axiom,
design8835372594653258411bers_a = design8835372594653258411bers_a ).
% incidence_system.replication_numbers.cong
thf(fact_572_incidence__system_Oreplication__numbers_Ocong,axiom,
design3853898657598026467rs_nat = design3853898657598026467rs_nat ).
% incidence_system.replication_numbers.cong
thf(fact_573_incidence__system_Odesign__support_Ocong,axiom,
design4862117536649126062rt_nat = design4862117536649126062rt_nat ).
% incidence_system.design_support.cong
thf(fact_574_incidence__system_Odesign__support_Ocong,axiom,
design5397942185814921632port_a = design5397942185814921632port_a ).
% incidence_system.design_support.cong
thf(fact_575_incidence__system_Oincident_Ocong,axiom,
design3210447939978979927dent_a = design3210447939978979927dent_a ).
% incidence_system.incident.cong
thf(fact_576_incidence__system_Oincident_Ocong,axiom,
design8502206366797944887nt_nat = design8502206366797944887nt_nat ).
% incidence_system.incident.cong
thf(fact_577_ordered__block__design__def,axiom,
( incide8999561475770975213_set_a
= ( ^ [V_s2: list_set_a,B_s2: list_set_set_a,K2: nat] :
( ( incide2999377533768400724_set_a @ V_s2 @ B_s2 )
& ( block_4731189848718898310_set_a @ ( set_set_a2 @ V_s2 ) @ ( mset_set_set_a @ B_s2 ) @ K2 ) ) ) ) ).
% ordered_block_design_def
thf(fact_578_ordered__block__design__def,axiom,
( incide8589348598044109441gn_nat
= ( ^ [V_s2: list_nat,B_s2: list_set_nat,K2: nat] :
( ( incide1001368407746664282gn_nat @ V_s2 @ B_s2 )
& ( block_625751327111516584gn_nat @ ( set_nat2 @ V_s2 ) @ ( mset_set_nat @ B_s2 ) @ K2 ) ) ) ) ).
% ordered_block_design_def
thf(fact_579_ordered__block__design__def,axiom,
( incide5219153079875704461sign_a
= ( ^ [V_s2: list_a,B_s2: list_set_a,K2: nat] :
( ( incide3676903341588786676sign_a @ V_s2 @ B_s2 )
& ( block_block_design_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) @ K2 ) ) ) ) ).
% ordered_block_design_def
thf(fact_580_ordered__block__design_Ointro,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,K: nat] :
( ( incide2999377533768400724_set_a @ V_s @ B_s )
=> ( ( block_4731189848718898310_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) @ K )
=> ( incide8999561475770975213_set_a @ V_s @ B_s @ K ) ) ) ).
% ordered_block_design.intro
thf(fact_581_ordered__block__design_Ointro,axiom,
! [V_s: list_nat,B_s: list_set_nat,K: nat] :
( ( incide1001368407746664282gn_nat @ V_s @ B_s )
=> ( ( block_625751327111516584gn_nat @ ( set_nat2 @ V_s ) @ ( mset_set_nat @ B_s ) @ K )
=> ( incide8589348598044109441gn_nat @ V_s @ B_s @ K ) ) ) ).
% ordered_block_design.intro
thf(fact_582_ordered__block__design_Ointro,axiom,
! [V_s: list_a,B_s: list_set_a,K: nat] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( ( block_block_design_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ K )
=> ( incide5219153079875704461sign_a @ V_s @ B_s @ K ) ) ) ).
% ordered_block_design.intro
thf(fact_583_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_584_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_585_points__index__0__right__imp,axiom,
! [B2: multiset_set_nat,Ps: set_nat] :
( ! [B6: set_nat] :
( ( member_set_nat @ B6 @ ( set_mset_set_nat @ B2 ) )
=> ~ ( ord_less_eq_set_nat @ Ps @ B6 ) )
=> ( ( design6574611146354332593ex_nat @ B2 @ Ps )
= zero_zero_nat ) ) ).
% points_index_0_right_imp
thf(fact_586_points__index__0__right__imp,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ! [B6: set_a] :
( ( member_set_a @ B6 @ ( set_mset_set_a @ B2 ) )
=> ~ ( ord_less_eq_set_a @ Ps @ B6 ) )
=> ( ( design254580327166089565ndex_a @ B2 @ Ps )
= zero_zero_nat ) ) ).
% points_index_0_right_imp
thf(fact_587_points__index__0__left__imp,axiom,
! [B2: multiset_set_nat,Ps: set_nat,B: set_nat] :
( ( ( design6574611146354332593ex_nat @ B2 @ Ps )
= zero_zero_nat )
=> ( ( member_set_nat @ B @ ( set_mset_set_nat @ B2 ) )
=> ~ ( ord_less_eq_set_nat @ Ps @ B ) ) ) ).
% points_index_0_left_imp
thf(fact_588_points__index__0__left__imp,axiom,
! [B2: multiset_set_a,Ps: set_a,B: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= zero_zero_nat )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ B2 ) )
=> ~ ( ord_less_eq_set_a @ Ps @ B ) ) ) ).
% points_index_0_left_imp
thf(fact_589_points__index__0__iff,axiom,
! [B2: multiset_set_nat,Ps: set_nat] :
( ( ( design6574611146354332593ex_nat @ B2 @ Ps )
= zero_zero_nat )
= ( ! [B3: set_nat] :
( ( member_set_nat @ B3 @ ( set_mset_set_nat @ B2 ) )
=> ~ ( ord_less_eq_set_nat @ Ps @ B3 ) ) ) ) ).
% points_index_0_iff
thf(fact_590_points__index__0__iff,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= zero_zero_nat )
= ( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ B2 ) )
=> ~ ( ord_less_eq_set_a @ Ps @ B3 ) ) ) ) ).
% points_index_0_iff
thf(fact_591_simple__incidence__system_Osimple__block__size__eq__card,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design164292856788568387em_nat @ Point_set @ Block_collection )
=> ( ( size_s7462436076474991978et_nat @ Block_collection )
= ( finite_card_set_nat @ ( design4862117536649126062rt_nat @ Block_collection ) ) ) ) ).
% simple_incidence_system.simple_block_size_eq_card
thf(fact_592_simple__incidence__system_Osimple__block__size__eq__card,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1338723777345758283stem_a @ Point_set @ Block_collection )
=> ( ( size_s6566526139600085008_set_a @ Block_collection )
= ( finite_card_set_a @ ( design5397942185814921632port_a @ Block_collection ) ) ) ) ).
% simple_incidence_system.simple_block_size_eq_card
thf(fact_593_points__index__gt0__impl__existance,axiom,
! [B2: multiset_set_nat,Ps: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( design6574611146354332593ex_nat @ B2 @ Ps ) )
=> ? [Bl: set_nat] :
( ( member_set_nat @ Bl @ ( set_mset_set_nat @ B2 ) )
& ( ord_less_eq_set_nat @ Ps @ Bl ) ) ) ).
% points_index_gt0_impl_existance
thf(fact_594_points__index__gt0__impl__existance,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( design254580327166089565ndex_a @ B2 @ Ps ) )
=> ? [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ B2 ) )
& ( ord_less_eq_set_a @ Ps @ Bl ) ) ) ).
% points_index_gt0_impl_existance
thf(fact_595_n__inter__num__zero,axiom,
! [B1: set_a,B22: set_a] :
( ( member_set_a @ B1 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( member_set_a @ B22 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design735257067508376852mber_a @ B1 @ zero_zero_nat @ B22 )
= one_one_nat ) ) ) ).
% n_inter_num_zero
thf(fact_596_point__indices__elem__in,axiom,
! [Ps: set_a,T4: nat] :
( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ( ( finite_card_a @ Ps )
= T4 )
=> ( member_nat @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) @ ( design328527185268214962ices_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ T4 ) ) ) ) ).
% point_indices_elem_in
thf(fact_597_symmetric__bibdII,axiom,
( ( ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) )
= ( times_times_nat @ k @ ( minus_minus_nat @ k @ one_one_nat ) ) )
=> ( symmetric_bibd_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k @ lambda ) ) ).
% symmetric_bibdII
thf(fact_598_symmetric__not__admissable,axiom,
( ( ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) )
!= ( times_times_nat @ k @ ( minus_minus_nat @ k @ one_one_nat ) ) )
=> ~ ( symmetric_bibd_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k @ lambda ) ) ).
% symmetric_not_admissable
thf(fact_599_bibd__to__pbdI,axiom,
( ( lambda = one_one_nat )
=> ( block_k_PBD_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k ) ) ).
% bibd_to_pbdI
thf(fact_600_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_601_multiple__point__index,axiom,
! [N: nat,Ps: set_a] :
( ( design254580327166089565ndex_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ Ps )
= ( times_times_nat @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) @ N ) ) ).
% multiple_point_index
thf(fact_602_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_603_dual__sys_Omultiple__point__index,axiom,
! [N: nat,Ps: set_nat] :
( ( design6574611146354332593ex_nat @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ Ps )
= ( times_times_nat @ ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Ps ) @ N ) ) ).
% dual_sys.multiple_point_index
thf(fact_604_repeat__mset__right,axiom,
! [A2: nat,B: nat,A: multiset_set_nat] :
( ( repeat_mset_set_nat @ A2 @ ( repeat_mset_set_nat @ B @ A ) )
= ( repeat_mset_set_nat @ ( times_times_nat @ A2 @ B ) @ A ) ) ).
% repeat_mset_right
thf(fact_605_repeat__mset__right,axiom,
! [A2: nat,B: nat,A: multiset_set_a] :
( ( repeat_mset_set_a @ A2 @ ( repeat_mset_set_a @ B @ A ) )
= ( repeat_mset_set_a @ ( times_times_nat @ A2 @ B ) @ A ) ) ).
% repeat_mset_right
thf(fact_606_symmetric__condition__2,axiom,
( ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) )
= ( times_times_nat @ k @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ).
% symmetric_condition_2
thf(fact_607_multiple__blocks__num,axiom,
! [N: nat] :
( ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) )
= ( times_times_nat @ N @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ) ) ).
% multiple_blocks_num
thf(fact_608_dual__sys_Omultiple__blocks__num,axiom,
! [N: nat] :
( ( size_s7462436076474991978et_nat @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
= ( times_times_nat @ N @ ( size_s7462436076474991978et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.multiple_blocks_num
thf(fact_609_index__lt__const__rep,axiom,
! [Ps: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( set_a2 @ v_s ) )
=> ( ( Ps != bot_bot_set_a )
=> ( ord_less_eq_nat @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ) ).
% index_lt_const_rep
thf(fact_610_r__lt__eq__b,axiom,
ord_less_eq_nat @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% r_lt_eq_b
thf(fact_611_necess__cond__one__param__balance,axiom,
( ( ord_less_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ord_less_nat @ k @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ).
% necess_cond_one_param_balance
thf(fact_612_block__num__gt__rep,axiom,
ord_less_nat @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) ).
% block_num_gt_rep
thf(fact_613_symmetric__bibdIII,axiom,
( ( ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) )
= k )
=> ( symmetric_bibd_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k @ lambda ) ) ).
% symmetric_bibdIII
thf(fact_614_mult__le__cancel2,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_615_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_616_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_617_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_618_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_619_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_620_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_621_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_622_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_623_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_624_mult__cancel2,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ( times_times_nat @ M2 @ K3 )
= ( times_times_nat @ N @ K3 ) )
= ( ( M2 = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_625_mult__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M2 )
= ( times_times_nat @ K3 @ N ) )
= ( ( M2 = N )
| ( K3 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_626_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_627_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_628_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_629_nat__add__left__cancel__le,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K3 @ M2 ) @ ( plus_plus_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_630_necessary__condition__one,axiom,
( ( times_times_nat @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) )
= ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) ) ).
% necessary_condition_one
thf(fact_631_rep__value__sym,axiom,
( ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) )
= k ) ).
% rep_value_sym
thf(fact_632_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_633_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_634_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_635_mult__less__cancel2,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K3 ) @ ( times_times_nat @ N @ K3 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_636_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_637_index__lt__replication,axiom,
ord_less_nat @ lambda @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ).
% index_lt_replication
thf(fact_638_rep__not__zero,axiom,
( ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) )
!= zero_zero_nat ) ).
% rep_not_zero
thf(fact_639_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_640_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_641_Nat_Oadd__diff__assoc,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K3 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_642_Nat_Oadd__diff__assoc2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K3 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_643_Nat_Odiff__diff__right,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_644_r__gzero,axiom,
ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ).
% r_gzero
thf(fact_645_bibd__block__number,axiom,
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
= ( divide_divide_nat @ ( times_times_nat @ ( times_times_nat @ lambda @ ( finite_card_a @ ( set_a2 @ v_s ) ) ) @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( times_times_nat @ k @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ).
% bibd_block_number
thf(fact_646_necessary__condition__two,axiom,
( ( times_times_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) )
= ( times_times_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ k ) ) ).
% necessary_condition_two
thf(fact_647_symmetric__condition__1,axiom,
( ( ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) )
= ( times_times_nat @ k @ ( minus_minus_nat @ k @ one_one_nat ) ) )
=> ( ( ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) )
= ( finite_card_a @ ( set_a2 @ v_s ) ) )
& ( ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) )
= k ) ) ) ).
% symmetric_condition_1
thf(fact_648_ordered__constant__rep__axioms,axiom,
incide6922509864216205631_rep_a @ v_s @ b_s @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ).
% ordered_constant_rep_axioms
thf(fact_649_size__neq__size__imp__neq,axiom,
! [X3: multiset_set_nat,Y3: multiset_set_nat] :
( ( ( size_s7462436076474991978et_nat @ X3 )
!= ( size_s7462436076474991978et_nat @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_650_size__neq__size__imp__neq,axiom,
! [X3: multiset_set_a,Y3: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ X3 )
!= ( size_s6566526139600085008_set_a @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_651_size__neq__size__imp__neq,axiom,
! [X3: list_set_a,Y3: list_set_a] :
( ( ( size_size_list_set_a @ X3 )
!= ( size_size_list_set_a @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_652_size__neq__size__imp__neq,axiom,
! [X3: list_a,Y3: list_a] :
( ( ( size_size_list_a @ X3 )
!= ( size_size_list_a @ Y3 ) )
=> ( X3 != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_653_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K3: nat,B: nat] :
( ( P2 @ K3 )
=> ( ! [Y: nat] :
( ( P2 @ Y )
=> ( ord_less_eq_nat @ Y @ B ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y5: nat] :
( ( P2 @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_654_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_655_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_656_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_657_le__trans,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K3 )
=> ( ord_less_eq_nat @ I @ K3 ) ) ) ).
% le_trans
thf(fact_658_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_659_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_660_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_661_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_662_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_663_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_664_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_665_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N2 )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_666_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_667_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_668_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_669_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_670_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_671_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_672_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_673_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_674_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_675_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_676_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_677_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_678_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_679_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_680_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_681_eq__diff__iff,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ( minus_minus_nat @ M2 @ K3 )
= ( minus_minus_nat @ N @ K3 ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_682_le__diff__iff,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_683_Nat_Odiff__diff__eq,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_684_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_685_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_686_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_687_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_688_add__leE,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K3 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K3 @ N ) ) ) ).
% add_leE
thf(fact_689_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_690_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_691_add__leD1,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K3 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_692_add__leD2,axiom,
! [M2: nat,K3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K3 ) @ N )
=> ( ord_less_eq_nat @ K3 @ N ) ) ).
% add_leD2
thf(fact_693_le__Suc__ex,axiom,
! [K3: nat,L: nat] :
( ( ord_less_eq_nat @ K3 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K3 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_694_add__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_695_add__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ ( plus_plus_nat @ J @ K3 ) ) ) ).
% add_le_mono1
thf(fact_696_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_697_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_698_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K4: nat] :
( N3
= ( plus_plus_nat @ M4 @ K4 ) ) ) ) ).
% nat_le_iff_add
thf(fact_699_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_700_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_701_mult__le__mono,axiom,
! [I: nat,J: nat,K3: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K3 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_702_mult__le__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ).
% mult_le_mono1
thf(fact_703_mult__le__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ).
% mult_le_mono2
thf(fact_704_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K5: nat] :
( ( ord_less_eq_nat @ K5 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K5 )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K5 ) ) ) ) ).
% ex_least_nat_le
thf(fact_705_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_706_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K5: nat] :
( ( ord_less_nat @ zero_zero_nat @ K5 )
& ( ( plus_plus_nat @ I @ K5 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_707_mult__less__mono1,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ I @ K3 ) @ ( times_times_nat @ J @ K3 ) ) ) ) ).
% mult_less_mono1
thf(fact_708_mult__less__mono2,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_nat @ ( times_times_nat @ K3 @ I ) @ ( times_times_nat @ K3 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_709_less__diff__iff,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K3 @ M2 )
=> ( ( ord_less_eq_nat @ K3 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K3 ) @ ( minus_minus_nat @ N @ K3 ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_710_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_711_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M2: nat,K3: nat] :
( ! [M5: nat,N2: nat] :
( ( ord_less_nat @ M5 @ N2 )
=> ( ord_less_nat @ ( F2 @ M5 ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M2 ) @ K3 ) @ ( F2 @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_712_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_713_le__diff__conv,axiom,
! [J: nat,K3: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K3 ) ) ) ).
% le_diff_conv
thf(fact_714_Nat_Ole__diff__conv2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K3 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K3 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_715_Nat_Odiff__add__assoc,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K3 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K3 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_716_Nat_Odiff__add__assoc2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K3 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K3 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_717_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K3 )
= ( J
= ( plus_plus_nat @ K3 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_718_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_719_nat__diff__split__asm,axiom,
! [P2: nat > $o,A2: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A2 @ B ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D: nat] :
( ( A2
= ( plus_plus_nat @ B @ D ) )
& ~ ( P2 @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_720_nat__diff__split,axiom,
! [P2: nat > $o,A2: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A2 @ B ) )
= ( ( ( ord_less_nat @ A2 @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D: nat] :
( ( A2
= ( plus_plus_nat @ B @ D ) )
=> ( P2 @ D ) ) ) ) ).
% nat_diff_split
thf(fact_721_less__diff__conv2,axiom,
! [K3: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K3 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K3 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K3 ) ) ) ) ).
% less_diff_conv2
thf(fact_722_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M4: nat,N3: nat] : ( if_nat @ ( M4 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_723_complement__rep__number,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 ) ) ) ) )
=> ( block_6028206285060069402sign_a @ ( set_a2 @ v_s ) @ ( design8640656491286871389ocks_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) @ ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ ( mset_set_a @ b_s ) ) @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ) ) ) ).
% complement_rep_number
thf(fact_724_multiple__rep__number,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( block_6028206285060069402sign_a @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ ( times_times_nat @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) @ N ) ) ) ).
% multiple_rep_number
thf(fact_725_constant__rep__design__axioms,axiom,
block_6028206285060069402sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ).
% constant_rep_design_axioms
thf(fact_726_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_727_points__list__length,axiom,
( ( size_size_list_a @ v_s )
= ( finite_card_a @ ( set_a2 @ v_s ) ) ) ).
% points_list_length
thf(fact_728_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_729_psubsetI,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_730_psubsetI,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_set_a @ A @ B2 ) ) ) ).
% psubsetI
thf(fact_731_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_732_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_733_length__0__conv,axiom,
! [Xs: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_set_a ) ) ).
% length_0_conv
thf(fact_734_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_735_size__mset,axiom,
! [Xs: list_a] :
( ( size_size_multiset_a @ ( mset_a @ Xs ) )
= ( size_size_list_a @ Xs ) ) ).
% size_mset
thf(fact_736_size__mset,axiom,
! [Xs: list_set_nat] :
( ( size_s7462436076474991978et_nat @ ( mset_set_nat @ Xs ) )
= ( size_s3254054031482475050et_nat @ Xs ) ) ).
% size_mset
thf(fact_737_size__mset,axiom,
! [Xs: list_set_a] :
( ( size_s6566526139600085008_set_a @ ( mset_set_a @ Xs ) )
= ( size_size_list_set_a @ Xs ) ) ).
% size_mset
thf(fact_738_length__greater__0__conv,axiom,
! [Xs: list_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) )
= ( Xs != nil_set_a ) ) ).
% length_greater_0_conv
thf(fact_739_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_740_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_741_length__induct,axiom,
! [P2: list_set_a > $o,Xs: list_set_a] :
( ! [Xs2: list_set_a] :
( ! [Ys2: list_set_a] :
( ( ord_less_nat @ ( size_size_list_set_a @ Ys2 ) @ ( size_size_list_set_a @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_742_length__induct,axiom,
! [P2: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_743_mset__eq__length,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( mset_set_a @ Xs )
= ( mset_set_a @ Ys ) )
=> ( ( size_size_list_set_a @ Xs )
= ( size_size_list_set_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_744_mset__eq__length,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( mset_a @ Xs )
= ( mset_a @ Ys ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% mset_eq_length
thf(fact_745_not__psubset__empty,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_746_not__psubset__empty,axiom,
! [A: set_nat] :
~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_747_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_748_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_749_subset__psubset__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_750_subset__psubset__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_751_subset__not__subset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_752_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_753_psubset__subset__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_754_psubset__subset__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_755_psubset__imp__subset,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% psubset_imp_subset
thf(fact_756_psubset__imp__subset,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ A @ B2 ) ) ).
% psubset_imp_subset
thf(fact_757_psubset__eq,axiom,
( ord_less_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_758_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_759_psubsetE,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ~ ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ) ).
% psubsetE
thf(fact_760_psubsetE,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ) ).
% psubsetE
thf(fact_761_finite__psubset__induct,axiom,
! [A: set_set_nat,P2: set_set_nat > $o] :
( ( finite1152437895449049373et_nat @ A )
=> ( ! [A5: set_set_nat] :
( ( finite1152437895449049373et_nat @ A5 )
=> ( ! [B7: set_set_nat] :
( ( ord_less_set_set_nat @ B7 @ A5 )
=> ( P2 @ B7 ) )
=> ( P2 @ A5 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_762_finite__psubset__induct,axiom,
! [A: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ A )
=> ( ! [A5: set_set_a] :
( ( finite_finite_set_a @ A5 )
=> ( ! [B7: set_set_a] :
( ( ord_less_set_set_a @ B7 @ A5 )
=> ( P2 @ B7 ) )
=> ( P2 @ A5 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_763_finite__psubset__induct,axiom,
! [A: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A )
=> ( ! [A5: set_a] :
( ( finite_finite_a @ A5 )
=> ( ! [B7: set_a] :
( ( ord_less_set_a @ B7 @ A5 )
=> ( P2 @ B7 ) )
=> ( P2 @ A5 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_764_finite__psubset__induct,axiom,
! [A: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [A5: set_nat] :
( ( finite_finite_nat @ A5 )
=> ( ! [B7: set_nat] :
( ( ord_less_set_nat @ B7 @ A5 )
=> ( P2 @ B7 ) )
=> ( P2 @ A5 ) ) )
=> ( P2 @ A ) ) ) ).
% finite_psubset_induct
thf(fact_765_ordered__constant__rep_Oaxioms_I2_J,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,R: nat] :
( ( incide3293764431032925855_set_a @ V_s @ B_s @ R )
=> ( block_1497795055661977978_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) @ R ) ) ).
% ordered_constant_rep.axioms(2)
thf(fact_766_ordered__constant__rep_Oaxioms_I2_J,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat] :
( ( incide6922509864216205631_rep_a @ V_s @ B_s @ R )
=> ( block_6028206285060069402sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ R ) ) ).
% ordered_constant_rep.axioms(2)
thf(fact_767_list_Osize_I3_J,axiom,
( ( size_size_list_set_a @ nil_set_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_768_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_769_ordered__constant__rep_Ointro,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,R: nat] :
( ( incide2999377533768400724_set_a @ V_s @ B_s )
=> ( ( block_1497795055661977978_set_a @ ( set_set_a2 @ V_s ) @ ( mset_set_set_a @ B_s ) @ R )
=> ( incide3293764431032925855_set_a @ V_s @ B_s @ R ) ) ) ).
% ordered_constant_rep.intro
thf(fact_770_ordered__constant__rep_Ointro,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat] :
( ( incide3676903341588786676sign_a @ V_s @ B_s )
=> ( ( block_6028206285060069402sign_a @ ( set_a2 @ V_s ) @ ( mset_set_a @ B_s ) @ R )
=> ( incide6922509864216205631_rep_a @ V_s @ B_s @ R ) ) ) ).
% ordered_constant_rep.intro
thf(fact_771_ordered__constant__rep__def,axiom,
( incide3293764431032925855_set_a
= ( ^ [V_s2: list_set_a,B_s2: list_set_set_a,R2: nat] :
( ( incide2999377533768400724_set_a @ V_s2 @ B_s2 )
& ( block_1497795055661977978_set_a @ ( set_set_a2 @ V_s2 ) @ ( mset_set_set_a @ B_s2 ) @ R2 ) ) ) ) ).
% ordered_constant_rep_def
thf(fact_772_ordered__constant__rep__def,axiom,
( incide6922509864216205631_rep_a
= ( ^ [V_s2: list_a,B_s2: list_set_a,R2: nat] :
( ( incide3676903341588786676sign_a @ V_s2 @ B_s2 )
& ( block_6028206285060069402sign_a @ ( set_a2 @ V_s2 ) @ ( mset_set_a @ B_s2 ) @ R2 ) ) ) ) ).
% ordered_constant_rep_def
thf(fact_773_length__finite__permutations__of__set,axiom,
! [Xs: list_set_a,A: set_set_a] :
( ( member_list_set_a @ Xs @ ( multis2257881577744371681_set_a @ A ) )
=> ( ( size_size_list_set_a @ Xs )
= ( finite_card_set_a @ A ) ) ) ).
% length_finite_permutations_of_set
thf(fact_774_length__finite__permutations__of__set,axiom,
! [Xs: list_a,A: set_a] :
( ( member_list_a @ Xs @ ( multis2428024204330136193_set_a @ A ) )
=> ( ( size_size_list_a @ Xs )
= ( finite_card_a @ A ) ) ) ).
% length_finite_permutations_of_set
thf(fact_775_length__finite__permutations__of__set,axiom,
! [Xs: list_nat,A: set_nat] :
( ( member_list_nat @ Xs @ ( multis1655833086286526861et_nat @ A ) )
=> ( ( size_size_list_nat @ Xs )
= ( finite_card_nat @ A ) ) ) ).
% length_finite_permutations_of_set
thf(fact_776_length__finite__permutations__of__multiset,axiom,
! [Xs: list_a,A: multiset_a] :
( ( member_list_a @ Xs @ ( multis5886240593633752526iset_a @ A ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_multiset_a @ A ) ) ) ).
% length_finite_permutations_of_multiset
thf(fact_777_length__finite__permutations__of__multiset,axiom,
! [Xs: list_set_a,A: multiset_set_a] :
( ( member_list_set_a @ Xs @ ( multis5469701301851823918_set_a @ A ) )
=> ( ( size_size_list_set_a @ Xs )
= ( size_s6566526139600085008_set_a @ A ) ) ) ).
% length_finite_permutations_of_multiset
thf(fact_778_length__finite__permutations__of__multiset,axiom,
! [Xs: list_set_nat,A: multiset_set_nat] :
( ( member_list_set_nat @ Xs @ ( multis124346860217030838et_nat @ A ) )
=> ( ( size_s3254054031482475050et_nat @ Xs )
= ( size_s7462436076474991978et_nat @ A ) ) ) ).
% length_finite_permutations_of_multiset
thf(fact_779_finite__maxlen,axiom,
! [M: set_list_set_a] :
( ( finite1971793804006318733_set_a @ M )
=> ? [N2: nat] :
! [X4: list_set_a] :
( ( member_list_set_a @ X4 @ M )
=> ( ord_less_nat @ ( size_size_list_set_a @ X4 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_780_finite__maxlen,axiom,
! [M: set_list_a] :
( ( finite_finite_list_a @ M )
=> ? [N2: nat] :
! [X4: list_a] :
( ( member_list_a @ X4 @ M )
=> ( ord_less_nat @ ( size_size_list_a @ X4 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_781_ordered__simple__design_Osimp__blocks__length__card,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide371748008924627346sign_a @ V_s @ B_s )
=> ( ( size_size_list_set_a @ B_s )
= ( finite_card_set_a @ ( set_set_a2 @ B_s ) ) ) ) ).
% ordered_simple_design.simp_blocks_length_card
thf(fact_782_ordered__constant__rep_Oaxioms_I1_J,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat] :
( ( incide6922509864216205631_rep_a @ V_s @ B_s @ R )
=> ( incide3676903341588786676sign_a @ V_s @ B_s ) ) ).
% ordered_constant_rep.axioms(1)
thf(fact_783_length__pos__if__in__set,axiom,
! [X3: list_a,Xs: list_list_a] :
( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_784_length__pos__if__in__set,axiom,
! [X3: set_nat,Xs: list_set_nat] :
( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_785_length__pos__if__in__set,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_786_length__pos__if__in__set,axiom,
! [X3: set_a,Xs: list_set_a] :
( ( member_set_a @ X3 @ ( set_set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_set_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_787_length__pos__if__in__set,axiom,
! [X3: a,Xs: list_a] :
( ( member_a @ X3 @ ( set_a2 @ Xs ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_788_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_789_card__length,axiom,
! [Xs: list_set_a] : ( ord_less_eq_nat @ ( finite_card_set_a @ ( set_set_a2 @ Xs ) ) @ ( size_size_list_set_a @ Xs ) ) ).
% card_length
thf(fact_790_card__length,axiom,
! [Xs: list_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( set_a2 @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ).
% card_length
thf(fact_791_psubset__card__mono,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_set_set_nat @ A @ B2 )
=> ( ord_less_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_792_psubset__card__mono,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_set_set_a @ A @ B2 )
=> ( ord_less_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_793_psubset__card__mono,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_set_a @ A @ B2 )
=> ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_794_psubset__card__mono,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_795_ordered__incidence__system_Opoints__list__length,axiom,
! [V_s: list_set_a,B_s: list_set_set_a] :
( ( incide2166342132139297189_set_a @ V_s @ B_s )
=> ( ( size_size_list_set_a @ V_s )
= ( finite_card_set_a @ ( set_set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_list_length
thf(fact_796_ordered__incidence__system_Opoints__list__length,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( size_size_list_a @ V_s )
= ( finite_card_a @ ( set_a2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_list_length
thf(fact_797_ordered__incidence__system_Opoints__list__length,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( size_size_list_nat @ V_s )
= ( finite_card_nat @ ( set_nat2 @ V_s ) ) ) ) ).
% ordered_incidence_system.points_list_length
thf(fact_798_ordered__incidence__system_Oblocks__list__length,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( size_size_list_set_a @ B_s )
= ( size_s6566526139600085008_set_a @ ( mset_set_a @ B_s ) ) ) ) ).
% ordered_incidence_system.blocks_list_length
thf(fact_799_ordered__incidence__system_Oblocks__list__length,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( size_s3254054031482475050et_nat @ B_s )
= ( size_s7462436076474991978et_nat @ ( mset_set_nat @ B_s ) ) ) ) ).
% ordered_incidence_system.blocks_list_length
thf(fact_800_card__psubset,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_set_set_nat @ A @ B2 ) ) ) ) ).
% card_psubset
thf(fact_801_card__psubset,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_set_set_a @ A @ B2 ) ) ) ) ).
% card_psubset
thf(fact_802_card__psubset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A @ B2 ) ) ) ) ).
% card_psubset
thf(fact_803_card__psubset,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_set_a @ A @ B2 ) ) ) ) ).
% card_psubset
thf(fact_804_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_805_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_806_dual__is__block__design,axiom,
block_625751327111516584gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) ).
% dual_is_block_design
thf(fact_807_nat__mult__le__cancel__disj,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_808_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_809_Diff__cancel,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ A )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_810_Diff__cancel,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ A )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_811_empty__Diff,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_812_empty__Diff,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_813_Diff__empty,axiom,
! [A: set_a] :
( ( minus_minus_set_a @ A @ bot_bot_set_a )
= A ) ).
% Diff_empty
thf(fact_814_Diff__empty,axiom,
! [A: set_nat] :
( ( minus_minus_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Diff_empty
thf(fact_815_finite__Diff2,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
= ( finite1152437895449049373et_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_816_finite__Diff2,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
= ( finite_finite_set_a @ A ) ) ) ).
% finite_Diff2
thf(fact_817_finite__Diff2,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A @ B2 ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_Diff2
thf(fact_818_finite__Diff2,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_Diff2
thf(fact_819_finite__Diff,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_820_finite__Diff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_821_finite__Diff,axiom,
! [A: set_a,B2: set_a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( minus_minus_set_a @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_822_finite__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff
thf(fact_823_subset__mset_Oadd__diff__assoc,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( plus_p8712254050562127327et_nat @ C @ ( minus_7237264121398869807et_nat @ B @ A2 ) )
= ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ C @ B ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_824_subset__mset_Oadd__diff__assoc,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( plus_p2331992037799027419_set_a @ C @ ( minus_706656509937749387_set_a @ B @ A2 ) )
= ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ B ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_825_subset__mset_Oadd__diff__assoc2,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ B @ A2 ) @ C )
= ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ B @ C ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_826_subset__mset_Oadd__diff__assoc2,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ B @ A2 ) @ C )
= ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ B @ C ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_827_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat,C2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ B2 @ A )
=> ( ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ A @ B2 ) @ C2 )
= ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ B2 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_828_mset__subset__eq__multiset__union__diff__commute,axiom,
! [B2: multiset_set_a,A: multiset_set_a,C2: multiset_set_a] :
( ( subseteq_mset_set_a @ B2 @ A )
=> ( ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ A @ B2 ) @ C2 )
= ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A @ C2 ) @ B2 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_829_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_set_nat,A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ ( plus_p8712254050562127327et_nat @ C @ B ) )
= ( subset6078030600694693471et_nat @ A2 @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_830_subset__mset_Oadd__le__cancel__left,axiom,
! [C: multiset_set_a,A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A2 ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
= ( subseteq_mset_set_a @ A2 @ B ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_831_subset__mset_Oadd__le__cancel__right,axiom,
! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ C ) )
= ( subset6078030600694693471et_nat @ A2 @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_832_subset__mset_Oadd__le__cancel__right,axiom,
! [A2: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
= ( subseteq_mset_set_a @ A2 @ B ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_833_mset__subset__eq__mono__add__left__cancel,axiom,
! [C2: multiset_set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C2 @ A ) @ ( plus_p8712254050562127327et_nat @ C2 @ B2 ) )
= ( subset6078030600694693471et_nat @ A @ B2 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_834_mset__subset__eq__mono__add__left__cancel,axiom,
! [C2: multiset_set_a,A: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C2 @ A ) @ ( plus_p2331992037799027419_set_a @ C2 @ B2 ) )
= ( subseteq_mset_set_a @ A @ B2 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_835_mset__subset__eq__mono__add__right__cancel,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) )
= ( subset6078030600694693471et_nat @ A @ B2 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_836_mset__subset__eq__mono__add__right__cancel,axiom,
! [A: multiset_set_a,C2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C2 ) @ ( plus_p2331992037799027419_set_a @ B2 @ C2 ) )
= ( subseteq_mset_set_a @ A @ B2 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_837_union__eq__empty,axiom,
! [M: multiset_set_nat,N4: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ M @ N4 )
= zero_z3157962936165190495et_nat )
= ( ( M = zero_z3157962936165190495et_nat )
& ( N4 = zero_z3157962936165190495et_nat ) ) ) ).
% union_eq_empty
thf(fact_838_union__eq__empty,axiom,
! [M: multiset_set_a,N4: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ M @ N4 )
= zero_z5079479921072680283_set_a )
= ( ( M = zero_z5079479921072680283_set_a )
& ( N4 = zero_z5079479921072680283_set_a ) ) ) ).
% union_eq_empty
thf(fact_839_empty__eq__union,axiom,
! [M: multiset_set_nat,N4: multiset_set_nat] :
( ( zero_z3157962936165190495et_nat
= ( plus_p8712254050562127327et_nat @ M @ N4 ) )
= ( ( M = zero_z3157962936165190495et_nat )
& ( N4 = zero_z3157962936165190495et_nat ) ) ) ).
% empty_eq_union
thf(fact_840_empty__eq__union,axiom,
! [M: multiset_set_a,N4: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ M @ N4 ) )
= ( ( M = zero_z5079479921072680283_set_a )
& ( N4 = zero_z5079479921072680283_set_a ) ) ) ).
% empty_eq_union
thf(fact_841_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X3: multiset_set_nat,Y3: multiset_set_nat] :
( ( zero_z3157962936165190495et_nat
= ( plus_p8712254050562127327et_nat @ X3 @ Y3 ) )
= ( ( X3 = zero_z3157962936165190495et_nat )
& ( Y3 = zero_z3157962936165190495et_nat ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_842_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X3: multiset_set_a,Y3: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ X3 @ Y3 ) )
= ( ( X3 = zero_z5079479921072680283_set_a )
& ( Y3 = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_843_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X3: multiset_set_nat,Y3: multiset_set_nat] :
( ( ( plus_p8712254050562127327et_nat @ X3 @ Y3 )
= zero_z3157962936165190495et_nat )
= ( ( X3 = zero_z3157962936165190495et_nat )
& ( Y3 = zero_z3157962936165190495et_nat ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_844_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X3: multiset_set_a,Y3: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ X3 @ Y3 )
= zero_z5079479921072680283_set_a )
= ( ( X3 = zero_z5079479921072680283_set_a )
& ( Y3 = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_845_repeat__mset__distrib2,axiom,
! [N: nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( repeat_mset_set_nat @ N @ ( plus_p8712254050562127327et_nat @ A @ B2 ) )
= ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ N @ A ) @ ( repeat_mset_set_nat @ N @ B2 ) ) ) ).
% repeat_mset_distrib2
thf(fact_846_repeat__mset__distrib2,axiom,
! [N: nat,A: multiset_set_a,B2: multiset_set_a] :
( ( repeat_mset_set_a @ N @ ( plus_p2331992037799027419_set_a @ A @ B2 ) )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ N @ A ) @ ( repeat_mset_set_a @ N @ B2 ) ) ) ).
% repeat_mset_distrib2
thf(fact_847_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_848_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_849_dual__sys_Oalt__ordering__sysI,axiom,
! [Vs2: list_nat,Bs3: list_set_nat] :
( ( member_list_nat @ Vs2 @ ( multis1655833086286526861et_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
=> ( ( member_list_set_nat @ Bs3 @ ( multis124346860217030838et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
=> ( incide6998539924841383625em_nat @ Vs2 @ Bs3 ) ) ) ).
% dual_sys.alt_ordering_sysI
thf(fact_850_dual__sys_Oexists__ordering__sysI,axiom,
? [Vs: list_nat,Bs2: list_set_nat] :
( ( member_list_nat @ Vs @ ( multis1655833086286526861et_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) )
& ( member_list_set_nat @ Bs2 @ ( multis124346860217030838et_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) )
& ( incide6998539924841383625em_nat @ Vs @ Bs2 ) ) ).
% dual_sys.exists_ordering_sysI
thf(fact_851_Diff__eq__empty__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_852_Diff__eq__empty__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_853_subset__mset_Ole__add__same__cancel2,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ B @ A2 ) )
= ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ B ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_854_subset__mset_Ole__add__same__cancel2,axiom,
! [A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ B @ A2 ) )
= ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_855_subset__mset_Ole__add__same__cancel1,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ ( plus_p8712254050562127327et_nat @ A2 @ B ) )
= ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ B ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_856_subset__mset_Ole__add__same__cancel1,axiom,
! [A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ ( plus_p2331992037799027419_set_a @ A2 @ B ) )
= ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_857_subset__mset_Oadd__le__same__cancel2,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) @ B )
= ( subset6078030600694693471et_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_858_subset__mset_Oadd__le__same__cancel2,axiom,
! [A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B ) @ B )
= ( subseteq_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_859_subset__mset_Oadd__le__same__cancel1,axiom,
! [B: multiset_set_nat,A2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ B @ A2 ) @ B )
= ( subset6078030600694693471et_nat @ A2 @ zero_z3157962936165190495et_nat ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_860_subset__mset_Oadd__le__same__cancel1,axiom,
! [B: multiset_set_a,A2: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ B @ A2 ) @ B )
= ( subseteq_mset_set_a @ A2 @ zero_z5079479921072680283_set_a ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_861_size__union,axiom,
! [M: multiset_set_nat,N4: multiset_set_nat] :
( ( size_s7462436076474991978et_nat @ ( plus_p8712254050562127327et_nat @ M @ N4 ) )
= ( plus_plus_nat @ ( size_s7462436076474991978et_nat @ M ) @ ( size_s7462436076474991978et_nat @ N4 ) ) ) ).
% size_union
thf(fact_862_size__union,axiom,
! [M: multiset_set_a,N4: multiset_set_a] :
( ( size_s6566526139600085008_set_a @ ( plus_p2331992037799027419_set_a @ M @ N4 ) )
= ( plus_plus_nat @ ( size_s6566526139600085008_set_a @ M ) @ ( size_s6566526139600085008_set_a @ N4 ) ) ) ).
% size_union
thf(fact_863_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_864_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_865_dual__sys_Owf__invalid__point,axiom,
! [X3: nat,B: set_nat] :
( ~ ( member_nat @ X3 @ ( 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 @ X3 @ B ) ) ) ).
% dual_sys.wf_invalid_point
thf(fact_866_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_867_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_868_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_869_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_870_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_871_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_872_dual__sys_Orepeat__mset__block__point__rel,axiom,
! [B: set_nat,N: nat,X3: nat] :
( ( member_set_nat @ B @ ( set_mset_set_nat @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) )
=> ( ( member_nat @ X3 @ B )
=> ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) ) ) ) ).
% dual_sys.repeat_mset_block_point_rel
thf(fact_873_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_874_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_875_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_876_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_877_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_878_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_879_div__mult__mult1__if,axiom,
! [C: nat,A2: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_880_div__mult__mult2,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult2
thf(fact_881_div__mult__mult1,axiom,
! [C: nat,A2: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A2 @ B ) ) ) ).
% div_mult_mult1
thf(fact_882_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_883_nat__mult__less__cancel__disj,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_884_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_885_dual__sys_Omultiple__not__simple,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s )
!= zero_z3157962936165190495et_nat )
=> ~ ( design164292856788568387em_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ) ).
% dual_sys.multiple_not_simple
thf(fact_886_div__mult__self4,axiom,
! [B: nat,C: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A2 ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self4
thf(fact_887_div__mult__self3,axiom,
! [B: nat,C: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A2 ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self3
thf(fact_888_div__mult__self2,axiom,
! [B: nat,A2: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self2
thf(fact_889_div__mult__self1,axiom,
! [B: nat,A2: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B ) ) ) ) ).
% div_mult_self1
thf(fact_890_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_891_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_892_psubset__imp__ex__mem,axiom,
! [A: set_list_a,B2: set_list_a] :
( ( ord_less_set_list_a @ A @ B2 )
=> ? [B6: list_a] : ( member_list_a @ B6 @ ( minus_646659088055828811list_a @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_893_psubset__imp__ex__mem,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( ord_less_set_set_nat @ A @ B2 )
=> ? [B6: set_nat] : ( member_set_nat @ B6 @ ( minus_2163939370556025621et_nat @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_894_psubset__imp__ex__mem,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A @ B2 )
=> ? [B6: set_a] : ( member_set_a @ B6 @ ( minus_5736297505244876581_set_a @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_895_psubset__imp__ex__mem,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_896_psubset__imp__ex__mem,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ? [B6: nat] : ( member_nat @ B6 @ ( minus_minus_set_nat @ B2 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_897_psubsetD,axiom,
! [A: set_list_a,B2: set_list_a,C: list_a] :
( ( ord_less_set_list_a @ A @ B2 )
=> ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_898_psubsetD,axiom,
! [A: set_set_nat,B2: set_set_nat,C: set_nat] :
( ( ord_less_set_set_nat @ A @ B2 )
=> ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_899_psubsetD,axiom,
! [A: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A @ B2 )
=> ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_900_psubsetD,axiom,
! [A: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_901_psubsetD,axiom,
! [A: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_902_psubset__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_903_psubset__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_set_nat @ A @ B2 )
=> ( ( ord_less_set_nat @ B2 @ C2 )
=> ( ord_less_set_nat @ A @ C2 ) ) ) ).
% psubset_trans
thf(fact_904_neq__if__length__neq,axiom,
! [Xs: list_set_a,Ys: list_set_a] :
( ( ( size_size_list_set_a @ Xs )
!= ( size_size_list_set_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_905_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_906_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_set_a] :
( ( size_size_list_set_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_907_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_908_double__diff,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_909_double__diff,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A ) )
= A ) ) ) ).
% double_diff
thf(fact_910_Diff__subset,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_911_Diff__subset,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B2 ) @ A ) ).
% Diff_subset
thf(fact_912_Diff__mono,axiom,
! [A: set_nat,C2: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_913_Diff__mono,axiom,
! [A: set_a,C2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ B2 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_914_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_915_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_916_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_917_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_918_in__diffD,axiom,
! [A2: list_a,M: multiset_list_a,N4: multiset_list_a] :
( ( member_list_a @ A2 @ ( set_mset_list_a @ ( minus_7431248565939055793list_a @ M @ N4 ) ) )
=> ( member_list_a @ A2 @ ( set_mset_list_a @ M ) ) ) ).
% in_diffD
thf(fact_919_in__diffD,axiom,
! [A2: nat,M: multiset_nat,N4: multiset_nat] :
( ( member_nat @ A2 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ N4 ) ) )
=> ( member_nat @ A2 @ ( set_mset_nat @ M ) ) ) ).
% in_diffD
thf(fact_920_in__diffD,axiom,
! [A2: a,M: multiset_a,N4: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N4 ) ) )
=> ( member_a @ A2 @ ( set_mset_a @ M ) ) ) ).
% in_diffD
thf(fact_921_in__diffD,axiom,
! [A2: set_nat,M: multiset_set_nat,N4: multiset_set_nat] :
( ( member_set_nat @ A2 @ ( set_mset_set_nat @ ( minus_7237264121398869807et_nat @ M @ N4 ) ) )
=> ( member_set_nat @ A2 @ ( set_mset_set_nat @ M ) ) ) ).
% in_diffD
thf(fact_922_in__diffD,axiom,
! [A2: set_a,M: multiset_set_a,N4: multiset_set_a] :
( ( member_set_a @ A2 @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N4 ) ) )
=> ( member_set_a @ A2 @ ( set_mset_set_a @ M ) ) ) ).
% in_diffD
thf(fact_923_diff__subset__eq__self,axiom,
! [M: multiset_set_nat,N4: multiset_set_nat] : ( subset6078030600694693471et_nat @ ( minus_7237264121398869807et_nat @ M @ N4 ) @ M ) ).
% diff_subset_eq_self
thf(fact_924_diff__subset__eq__self,axiom,
! [M: multiset_set_a,N4: multiset_set_a] : ( subseteq_mset_set_a @ ( minus_706656509937749387_set_a @ M @ N4 ) @ M ) ).
% diff_subset_eq_self
thf(fact_925_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_926_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_927_Multiset_Odiff__cancel,axiom,
! [A: multiset_set_nat] :
( ( minus_7237264121398869807et_nat @ A @ A )
= zero_z3157962936165190495et_nat ) ).
% Multiset.diff_cancel
thf(fact_928_Multiset_Odiff__cancel,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ A )
= zero_z5079479921072680283_set_a ) ).
% Multiset.diff_cancel
thf(fact_929_union__iff,axiom,
! [A2: list_a,A: multiset_list_a,B2: multiset_list_a] :
( ( member_list_a @ A2 @ ( set_mset_list_a @ ( plus_p690419498615200257list_a @ A @ B2 ) ) )
= ( ( member_list_a @ A2 @ ( set_mset_list_a @ A ) )
| ( member_list_a @ A2 @ ( set_mset_list_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_930_union__iff,axiom,
! [A2: nat,A: multiset_nat,B2: multiset_nat] :
( ( member_nat @ A2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ B2 ) ) )
= ( ( member_nat @ A2 @ ( set_mset_nat @ A ) )
| ( member_nat @ A2 @ ( set_mset_nat @ B2 ) ) ) ) ).
% union_iff
thf(fact_931_union__iff,axiom,
! [A2: a,A: multiset_a,B2: multiset_a] :
( ( member_a @ A2 @ ( set_mset_a @ ( plus_plus_multiset_a @ A @ B2 ) ) )
= ( ( member_a @ A2 @ ( set_mset_a @ A ) )
| ( member_a @ A2 @ ( set_mset_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_932_union__iff,axiom,
! [A2: set_nat,A: multiset_set_nat,B2: multiset_set_nat] :
( ( member_set_nat @ A2 @ ( set_mset_set_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) ) )
= ( ( member_set_nat @ A2 @ ( set_mset_set_nat @ A ) )
| ( member_set_nat @ A2 @ ( set_mset_set_nat @ B2 ) ) ) ) ).
% union_iff
thf(fact_933_union__iff,axiom,
! [A2: set_a,A: multiset_set_a,B2: multiset_set_a] :
( ( member_set_a @ A2 @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ B2 ) ) )
= ( ( member_set_a @ A2 @ ( set_mset_set_a @ A ) )
| ( member_set_a @ A2 @ ( set_mset_set_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_934_mset__subset__eq__exists__conv,axiom,
( subset6078030600694693471et_nat
= ( ^ [A4: multiset_set_nat,B4: multiset_set_nat] :
? [C3: multiset_set_nat] :
( B4
= ( plus_p8712254050562127327et_nat @ A4 @ C3 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_935_mset__subset__eq__exists__conv,axiom,
( subseteq_mset_set_a
= ( ^ [A4: multiset_set_a,B4: multiset_set_a] :
? [C3: multiset_set_a] :
( B4
= ( plus_p2331992037799027419_set_a @ A4 @ C3 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_936_mset__subset__eq__add__right,axiom,
! [B2: multiset_set_nat,A: multiset_set_nat] : ( subset6078030600694693471et_nat @ B2 @ ( plus_p8712254050562127327et_nat @ A @ B2 ) ) ).
% mset_subset_eq_add_right
thf(fact_937_mset__subset__eq__add__right,axiom,
! [B2: multiset_set_a,A: multiset_set_a] : ( subseteq_mset_set_a @ B2 @ ( plus_p2331992037799027419_set_a @ A @ B2 ) ) ).
% mset_subset_eq_add_right
thf(fact_938_mset__subset__eq__mono__add,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat,C2: multiset_set_nat,D2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A @ B2 )
=> ( ( subset6078030600694693471et_nat @ C2 @ D2 )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A @ C2 ) @ ( plus_p8712254050562127327et_nat @ B2 @ D2 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_939_mset__subset__eq__mono__add,axiom,
! [A: multiset_set_a,B2: multiset_set_a,C2: multiset_set_a,D2: multiset_set_a] :
( ( subseteq_mset_set_a @ A @ B2 )
=> ( ( subseteq_mset_set_a @ C2 @ D2 )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A @ C2 ) @ ( plus_p2331992037799027419_set_a @ B2 @ D2 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_940_mset__subset__eq__add__left,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] : ( subset6078030600694693471et_nat @ A @ ( plus_p8712254050562127327et_nat @ A @ B2 ) ) ).
% mset_subset_eq_add_left
thf(fact_941_mset__subset__eq__add__left,axiom,
! [A: multiset_set_a,B2: multiset_set_a] : ( subseteq_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B2 ) ) ).
% mset_subset_eq_add_left
thf(fact_942_subset__mset_Oadd__le__imp__le__right,axiom,
! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ C ) )
=> ( subset6078030600694693471et_nat @ A2 @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_943_subset__mset_Oadd__le__imp__le__right,axiom,
! [A2: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) )
=> ( subseteq_mset_set_a @ A2 @ B ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_944_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_set_nat,A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ ( plus_p8712254050562127327et_nat @ C @ B ) )
=> ( subset6078030600694693471et_nat @ A2 @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_945_subset__mset_Oadd__le__imp__le__left,axiom,
! [C: multiset_set_a,A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A2 ) @ ( plus_p2331992037799027419_set_a @ C @ B ) )
=> ( subseteq_mset_set_a @ A2 @ B ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_946_subset__mset_Oadd__right__mono,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_947_subset__mset_Oadd__right__mono,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ ( plus_p2331992037799027419_set_a @ B @ C ) ) ) ).
% subset_mset.add_right_mono
thf(fact_948_subset__mset_Oadd__left__mono,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ ( plus_p8712254050562127327et_nat @ C @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_949_subset__mset_Oadd__left__mono,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A2 ) @ ( plus_p2331992037799027419_set_a @ C @ B ) ) ) ).
% subset_mset.add_left_mono
thf(fact_950_subset__mset_Ole__iff__add,axiom,
( subset6078030600694693471et_nat
= ( ^ [A3: multiset_set_nat,B3: multiset_set_nat] :
? [C4: multiset_set_nat] :
( B3
= ( plus_p8712254050562127327et_nat @ A3 @ C4 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_951_subset__mset_Ole__iff__add,axiom,
( subseteq_mset_set_a
= ( ^ [A3: multiset_set_a,B3: multiset_set_a] :
? [C4: multiset_set_a] :
( B3
= ( plus_p2331992037799027419_set_a @ A3 @ C4 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_952_subset__mset_Oless__eqE,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ~ ! [C5: multiset_set_nat] :
( B
!= ( plus_p8712254050562127327et_nat @ A2 @ C5 ) ) ) ).
% subset_mset.less_eqE
thf(fact_953_subset__mset_Oless__eqE,axiom,
! [A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ~ ! [C5: multiset_set_a] :
( B
!= ( plus_p2331992037799027419_set_a @ A2 @ C5 ) ) ) ).
% subset_mset.less_eqE
thf(fact_954_subset__mset_Oadd__mono,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat,D3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( subset6078030600694693471et_nat @ C @ D3 )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ ( plus_p8712254050562127327et_nat @ B @ D3 ) ) ) ) ).
% subset_mset.add_mono
thf(fact_955_subset__mset_Oadd__mono,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a,D3: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( subseteq_mset_set_a @ C @ D3 )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ ( plus_p2331992037799027419_set_a @ B @ D3 ) ) ) ) ).
% subset_mset.add_mono
thf(fact_956_empty__neutral_I2_J,axiom,
! [X3: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ X3 @ zero_z3157962936165190495et_nat )
= X3 ) ).
% empty_neutral(2)
thf(fact_957_empty__neutral_I2_J,axiom,
! [X3: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ X3 @ zero_z5079479921072680283_set_a )
= X3 ) ).
% empty_neutral(2)
thf(fact_958_empty__neutral_I1_J,axiom,
! [X3: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ zero_z3157962936165190495et_nat @ X3 )
= X3 ) ).
% empty_neutral(1)
thf(fact_959_empty__neutral_I1_J,axiom,
! [X3: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ X3 )
= X3 ) ).
% empty_neutral(1)
thf(fact_960_subset__mset_Odiff__add,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ B @ A2 ) @ A2 )
= B ) ) ).
% subset_mset.diff_add
thf(fact_961_subset__mset_Odiff__add,axiom,
! [A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ B @ A2 ) @ A2 )
= B ) ) ).
% subset_mset.diff_add
thf(fact_962_subset__mset_Ole__add__diff,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( subset6078030600694693471et_nat @ C @ ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ B @ C ) @ A2 ) ) ) ).
% subset_mset.le_add_diff
thf(fact_963_subset__mset_Ole__add__diff,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( subseteq_mset_set_a @ C @ ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ B @ C ) @ A2 ) ) ) ).
% subset_mset.le_add_diff
thf(fact_964_subset__mset_Ole__diff__conv2,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( subset6078030600694693471et_nat @ C @ ( minus_7237264121398869807et_nat @ B @ A2 ) )
= ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_965_subset__mset_Ole__diff__conv2,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( subseteq_mset_set_a @ C @ ( minus_706656509937749387_set_a @ B @ A2 ) )
= ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ C @ A2 ) @ B ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_966_subset__mset_Odiff__add__assoc,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ C @ B ) @ A2 )
= ( plus_p8712254050562127327et_nat @ C @ ( minus_7237264121398869807et_nat @ B @ A2 ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_967_subset__mset_Odiff__add__assoc,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ B ) @ A2 )
= ( plus_p2331992037799027419_set_a @ C @ ( minus_706656509937749387_set_a @ B @ A2 ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_968_subset__mset_Odiff__add__assoc2,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ B @ C ) @ A2 )
= ( plus_p8712254050562127327et_nat @ ( minus_7237264121398869807et_nat @ B @ A2 ) @ C ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_969_subset__mset_Odiff__add__assoc2,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ B @ C ) @ A2 )
= ( plus_p2331992037799027419_set_a @ ( minus_706656509937749387_set_a @ B @ A2 ) @ C ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_970_subset__mset_Odiff__diff__right,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( minus_7237264121398869807et_nat @ C @ ( minus_7237264121398869807et_nat @ B @ A2 ) )
= ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ C @ A2 ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_971_subset__mset_Odiff__diff__right,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( minus_706656509937749387_set_a @ C @ ( minus_706656509937749387_set_a @ B @ A2 ) )
= ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ C @ A2 ) @ B ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_972_subset__mset_Oadd__diff__inverse,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( plus_p8712254050562127327et_nat @ A2 @ ( minus_7237264121398869807et_nat @ B @ A2 ) )
= B ) ) ).
% subset_mset.add_diff_inverse
thf(fact_973_subset__mset_Oadd__diff__inverse,axiom,
! [A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( plus_p2331992037799027419_set_a @ A2 @ ( minus_706656509937749387_set_a @ B @ A2 ) )
= B ) ) ).
% subset_mset.add_diff_inverse
thf(fact_974_subset__mset_Ole__imp__diff__is__add,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( ( ( minus_7237264121398869807et_nat @ B @ A2 )
= C )
= ( B
= ( plus_p8712254050562127327et_nat @ C @ A2 ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_975_subset__mset_Ole__imp__diff__is__add,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( subseteq_mset_set_a @ A2 @ B )
=> ( ( ( minus_706656509937749387_set_a @ B @ A2 )
= C )
= ( B
= ( plus_p2331992037799027419_set_a @ C @ A2 ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_976_subset__eq__diff__conv,axiom,
! [A: multiset_set_nat,C2: multiset_set_nat,B2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ ( minus_7237264121398869807et_nat @ A @ C2 ) @ B2 )
= ( subset6078030600694693471et_nat @ A @ ( plus_p8712254050562127327et_nat @ B2 @ C2 ) ) ) ).
% subset_eq_diff_conv
thf(fact_977_subset__eq__diff__conv,axiom,
! [A: multiset_set_a,C2: multiset_set_a,B2: multiset_set_a] :
( ( subseteq_mset_set_a @ ( minus_706656509937749387_set_a @ A @ C2 ) @ B2 )
= ( subseteq_mset_set_a @ A @ ( plus_p2331992037799027419_set_a @ B2 @ C2 ) ) ) ).
% subset_eq_diff_conv
thf(fact_978_multiset__diff__union__assoc,axiom,
! [C2: multiset_set_nat,B2: multiset_set_nat,A: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ C2 @ B2 )
=> ( ( minus_7237264121398869807et_nat @ ( plus_p8712254050562127327et_nat @ A @ B2 ) @ C2 )
= ( plus_p8712254050562127327et_nat @ A @ ( minus_7237264121398869807et_nat @ B2 @ C2 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_979_multiset__diff__union__assoc,axiom,
! [C2: multiset_set_a,B2: multiset_set_a,A: multiset_set_a] :
( ( subseteq_mset_set_a @ C2 @ B2 )
=> ( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ A @ B2 ) @ C2 )
= ( plus_p2331992037799027419_set_a @ A @ ( minus_706656509937749387_set_a @ B2 @ C2 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_980_all__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N )
=> ( P2 @ M4 ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
=> ( P2 @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_981_ex__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M4: nat] :
( ( ord_less_nat @ M4 @ N )
& ( P2 @ M4 ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
& ( P2 @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_982_Diff__eq__empty__iff__mset,axiom,
! [A: multiset_set_nat,B2: multiset_set_nat] :
( ( ( minus_7237264121398869807et_nat @ A @ B2 )
= zero_z3157962936165190495et_nat )
= ( subset6078030600694693471et_nat @ A @ B2 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_983_Diff__eq__empty__iff__mset,axiom,
! [A: multiset_set_a,B2: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ A @ B2 )
= zero_z5079479921072680283_set_a )
= ( subseteq_mset_set_a @ A @ B2 ) ) ).
% Diff_eq_empty_iff_mset
thf(fact_984_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X3: multiset_set_nat,Y3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ X3 @ zero_z3157962936165190495et_nat )
=> ( ( subset6078030600694693471et_nat @ Y3 @ zero_z3157962936165190495et_nat )
=> ( ( ( plus_p8712254050562127327et_nat @ X3 @ Y3 )
= zero_z3157962936165190495et_nat )
= ( ( X3 = zero_z3157962936165190495et_nat )
& ( Y3 = zero_z3157962936165190495et_nat ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_985_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [X3: multiset_set_a,Y3: multiset_set_a] :
( ( subseteq_mset_set_a @ X3 @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ Y3 @ zero_z5079479921072680283_set_a )
=> ( ( ( plus_p2331992037799027419_set_a @ X3 @ Y3 )
= zero_z5079479921072680283_set_a )
= ( ( X3 = zero_z5079479921072680283_set_a )
& ( Y3 = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_986_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X3: multiset_set_nat,Y3: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ X3 )
=> ( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ Y3 )
=> ( ( ( plus_p8712254050562127327et_nat @ X3 @ Y3 )
= zero_z3157962936165190495et_nat )
= ( ( X3 = zero_z3157962936165190495et_nat )
& ( Y3 = zero_z3157962936165190495et_nat ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_987_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [X3: multiset_set_a,Y3: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ X3 )
=> ( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ Y3 )
=> ( ( ( plus_p2331992037799027419_set_a @ X3 @ Y3 )
= zero_z5079479921072680283_set_a )
= ( ( X3 = zero_z5079479921072680283_set_a )
& ( Y3 = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_988_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ zero_z3157962936165190495et_nat )
=> ( ( subset6078030600694693471et_nat @ B @ zero_z3157962936165190495et_nat )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) @ zero_z3157962936165190495et_nat ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_989_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_990_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ A2 )
=> ( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ B )
=> ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ B ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_991_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A2 )
=> ( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_992_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_set_nat,B: multiset_set_nat,A2: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ C )
=> ( ( subset6078030600694693471et_nat @ B @ A2 )
=> ( subset6078030600694693471et_nat @ B @ ( plus_p8712254050562127327et_nat @ A2 @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_993_subset__mset_Oadd__increasing2,axiom,
! [C: multiset_set_a,B: multiset_set_a,A2: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ C )
=> ( ( subseteq_mset_set_a @ B @ A2 )
=> ( subseteq_mset_set_a @ B @ ( plus_p2331992037799027419_set_a @ A2 @ C ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_994_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_set_nat,A2: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ C @ zero_z3157962936165190495et_nat )
=> ( ( subset6078030600694693471et_nat @ A2 @ B )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_995_subset__mset_Oadd__decreasing2,axiom,
! [C: multiset_set_a,A2: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ C @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ A2 @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_996_subset__mset_Oadd__increasing,axiom,
! [A2: multiset_set_nat,B: multiset_set_nat,C: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ zero_z3157962936165190495et_nat @ A2 )
=> ( ( subset6078030600694693471et_nat @ B @ C )
=> ( subset6078030600694693471et_nat @ B @ ( plus_p8712254050562127327et_nat @ A2 @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_997_subset__mset_Oadd__increasing,axiom,
! [A2: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( subseteq_mset_set_a @ zero_z5079479921072680283_set_a @ A2 )
=> ( ( subseteq_mset_set_a @ B @ C )
=> ( subseteq_mset_set_a @ B @ ( plus_p2331992037799027419_set_a @ A2 @ C ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_998_subset__mset_Oadd__decreasing,axiom,
! [A2: multiset_set_nat,C: multiset_set_nat,B: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ A2 @ zero_z3157962936165190495et_nat )
=> ( ( subset6078030600694693471et_nat @ C @ B )
=> ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ A2 @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_999_subset__mset_Oadd__decreasing,axiom,
! [A2: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( subseteq_mset_set_a @ A2 @ zero_z5079479921072680283_set_a )
=> ( ( subseteq_mset_set_a @ C @ B )
=> ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ C ) @ B ) ) ) ).
% subset_mset.add_decreasing
thf(fact_1000_left__diff__repeat__mset__distrib_H,axiom,
! [I: nat,J: nat,U: multiset_set_nat] :
( ( repeat_mset_set_nat @ ( minus_minus_nat @ I @ J ) @ U )
= ( minus_7237264121398869807et_nat @ ( repeat_mset_set_nat @ I @ U ) @ ( repeat_mset_set_nat @ J @ U ) ) ) ).
% left_diff_repeat_mset_distrib'
thf(fact_1001_left__diff__repeat__mset__distrib_H,axiom,
! [I: nat,J: nat,U: multiset_set_a] :
( ( repeat_mset_set_a @ ( minus_minus_nat @ I @ J ) @ U )
= ( minus_706656509937749387_set_a @ ( repeat_mset_set_a @ I @ U ) @ ( repeat_mset_set_a @ J @ U ) ) ) ).
% left_diff_repeat_mset_distrib'
thf(fact_1002_point__index__diff,axiom,
! [B23: multiset_set_nat] :
( design6574611146354332593ex_nat
= ( ^ [B12: multiset_set_nat,Ps2: set_nat] : ( minus_minus_nat @ ( design6574611146354332593ex_nat @ ( plus_p8712254050562127327et_nat @ B12 @ B23 ) @ Ps2 ) @ ( design6574611146354332593ex_nat @ B23 @ Ps2 ) ) ) ) ).
% point_index_diff
thf(fact_1003_point__index__diff,axiom,
! [B23: multiset_set_a] :
( design254580327166089565ndex_a
= ( ^ [B12: multiset_set_a,Ps2: set_a] : ( minus_minus_nat @ ( design254580327166089565ndex_a @ ( plus_p2331992037799027419_set_a @ B12 @ B23 ) @ Ps2 ) @ ( design254580327166089565ndex_a @ B23 @ Ps2 ) ) ) ) ).
% point_index_diff
thf(fact_1004_point__index__distrib,axiom,
! [B13: multiset_set_nat,B23: multiset_set_nat,Ps: set_nat] :
( ( design6574611146354332593ex_nat @ ( plus_p8712254050562127327et_nat @ B13 @ B23 ) @ Ps )
= ( plus_plus_nat @ ( design6574611146354332593ex_nat @ B13 @ Ps ) @ ( design6574611146354332593ex_nat @ B23 @ Ps ) ) ) ).
% point_index_distrib
thf(fact_1005_point__index__distrib,axiom,
! [B13: multiset_set_a,B23: multiset_set_a,Ps: set_a] :
( ( design254580327166089565ndex_a @ ( plus_p2331992037799027419_set_a @ B13 @ B23 ) @ Ps )
= ( plus_plus_nat @ ( design254580327166089565ndex_a @ B13 @ Ps ) @ ( design254580327166089565ndex_a @ B23 @ Ps ) ) ) ).
% point_index_distrib
thf(fact_1006_repeat__mset__distrib,axiom,
! [M2: nat,N: nat,A: multiset_set_nat] :
( ( repeat_mset_set_nat @ ( plus_plus_nat @ M2 @ N ) @ A )
= ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ M2 @ A ) @ ( repeat_mset_set_nat @ N @ A ) ) ) ).
% repeat_mset_distrib
thf(fact_1007_repeat__mset__distrib,axiom,
! [M2: nat,N: nat,A: multiset_set_a] :
( ( repeat_mset_set_a @ ( plus_plus_nat @ M2 @ N ) @ A )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ M2 @ A ) @ ( repeat_mset_set_a @ N @ A ) ) ) ).
% repeat_mset_distrib
thf(fact_1008_left__add__mult__distrib__mset,axiom,
! [I: nat,U: multiset_set_nat,J: nat,K3: multiset_set_nat] :
( ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ I @ U ) @ ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ J @ U ) @ K3 ) )
= ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K3 ) ) ).
% left_add_mult_distrib_mset
thf(fact_1009_left__add__mult__distrib__mset,axiom,
! [I: nat,U: multiset_set_a,J: nat,K3: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I @ U ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J @ U ) @ K3 ) )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( plus_plus_nat @ I @ J ) @ U ) @ K3 ) ) ).
% left_add_mult_distrib_mset
thf(fact_1010_card__less__sym__Diff,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1011_card__less__sym__Diff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1012_card__less__sym__Diff,axiom,
! [A: set_a,B2: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1013_card__less__sym__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_1014_card__le__sym__Diff,axiom,
! [A: set_set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1015_card__le__sym__Diff,axiom,
! [A: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1016_card__le__sym__Diff,axiom,
! [A: set_a,B2: set_a] :
( ( finite_finite_a @ A )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1017_card__le__sym__Diff,axiom,
! [A: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_1018_diff__size__le__size__Diff,axiom,
! [M: multiset_set_nat,M6: multiset_set_nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ M ) @ ( size_s7462436076474991978et_nat @ M6 ) ) @ ( size_s7462436076474991978et_nat @ ( minus_7237264121398869807et_nat @ M @ M6 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_1019_diff__size__le__size__Diff,axiom,
! [M: multiset_set_a,M6: multiset_set_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M ) @ ( size_s6566526139600085008_set_a @ M6 ) ) @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M @ M6 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_1020_size__Diff__submset,axiom,
! [M: multiset_set_nat,M6: multiset_set_nat] :
( ( subset6078030600694693471et_nat @ M @ M6 )
=> ( ( size_s7462436076474991978et_nat @ ( minus_7237264121398869807et_nat @ M6 @ M ) )
= ( minus_minus_nat @ ( size_s7462436076474991978et_nat @ M6 ) @ ( size_s7462436076474991978et_nat @ M ) ) ) ) ).
% size_Diff_submset
thf(fact_1021_size__Diff__submset,axiom,
! [M: multiset_set_a,M6: multiset_set_a] :
( ( subseteq_mset_set_a @ M @ M6 )
=> ( ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M6 @ M ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M6 ) @ ( size_s6566526139600085008_set_a @ M ) ) ) ) ).
% size_Diff_submset
thf(fact_1022_card__Diff__subset,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ( ord_le6893508408891458716et_nat @ B2 @ A )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1023_card__Diff__subset,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1024_card__Diff__subset,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1025_card__Diff__subset,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_1026_diff__card__le__card__Diff,axiom,
! [B2: set_set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A ) @ ( finite_card_set_nat @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1027_diff__card__le__card__Diff,axiom,
! [B2: set_set_a,A: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_a @ A ) @ ( finite_card_set_a @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1028_diff__card__le__card__Diff,axiom,
! [B2: set_a,A: set_a] :
( ( finite_finite_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A ) @ ( finite_card_a @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1029_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_1030_mset__subseteq__add__iff1,axiom,
! [J: nat,I: nat,U: multiset_set_nat,M2: multiset_set_nat,N: multiset_set_nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ I @ U ) @ M2 ) @ ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ J @ U ) @ N ) )
= ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% mset_subseteq_add_iff1
thf(fact_1031_mset__subseteq__add__iff1,axiom,
! [J: nat,I: nat,U: multiset_set_a,M2: multiset_set_a,N: multiset_set_a] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I @ U ) @ M2 ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J @ U ) @ N ) )
= ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% mset_subseteq_add_iff1
thf(fact_1032_mset__subseteq__add__iff2,axiom,
! [I: nat,J: nat,U: multiset_set_nat,M2: multiset_set_nat,N: multiset_set_nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( subset6078030600694693471et_nat @ ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ I @ U ) @ M2 ) @ ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ J @ U ) @ N ) )
= ( subset6078030600694693471et_nat @ M2 @ ( plus_p8712254050562127327et_nat @ ( repeat_mset_set_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% mset_subseteq_add_iff2
thf(fact_1033_mset__subseteq__add__iff2,axiom,
! [I: nat,J: nat,U: multiset_set_a,M2: multiset_set_a,N: multiset_set_a] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( subseteq_mset_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I @ U ) @ M2 ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J @ U ) @ N ) )
= ( subseteq_mset_set_a @ M2 @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% mset_subseteq_add_iff2
thf(fact_1034_dual__blocks__wf,axiom,
! [B: set_nat,V: set_a,Bs3: list_set_a] :
( ( member_set_nat @ B @ ( set_mset_set_nat @ ( dual_dual_blocks_a @ V @ Bs3 ) ) )
=> ( ord_less_eq_set_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ Bs3 ) ) ) ) ).
% dual_blocks_wf
thf(fact_1035_ordered__constant__rep_Odual__is__block__design,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,R: nat] :
( ( incide3293764431032925855_set_a @ V_s @ B_s @ R )
=> ( block_625751327111516584gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7045802815848406192_set_a @ B_s ) ) @ ( dual_d359914979145368543_set_a @ ( set_set_a2 @ V_s ) @ B_s ) @ R ) ) ).
% ordered_constant_rep.dual_is_block_design
thf(fact_1036_ordered__constant__rep_Odual__is__block__design,axiom,
! [V_s: list_a,B_s: list_set_a,R: nat] :
( ( incide6922509864216205631_rep_a @ V_s @ B_s @ R )
=> ( block_625751327111516584gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ B_s ) ) @ ( dual_dual_blocks_a @ ( set_a2 @ V_s ) @ B_s ) @ R ) ) ).
% ordered_constant_rep.dual_is_block_design
thf(fact_1037_ordered__incidence__system_Odual__blocks__v,axiom,
! [V_s: list_a,B_s: list_set_a] :
( ( incide1624170830610365509stem_a @ V_s @ B_s )
=> ( ( 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 ) ) ) ) ).
% ordered_incidence_system.dual_blocks_v
thf(fact_1038_ordered__incidence__system_Odual__blocks__v,axiom,
! [V_s: list_nat,B_s: list_set_nat] :
( ( incide6998539924841383625em_nat @ V_s @ B_s )
=> ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ B_s ) ) )
= ( size_s7462436076474991978et_nat @ ( mset_set_nat @ B_s ) ) ) ) ).
% ordered_incidence_system.dual_blocks_v
thf(fact_1039_nat__mult__eq__cancel__disj,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ( times_times_nat @ K3 @ M2 )
= ( times_times_nat @ K3 @ N ) )
= ( ( K3 = zero_zero_nat )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1040_div__le__mono,axiom,
! [M2: nat,N: nat,K3: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K3 ) @ ( divide_divide_nat @ N @ K3 ) ) ) ).
% div_le_mono
thf(fact_1041_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1042_nat__mult__eq__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ( times_times_nat @ K3 @ M2 )
= ( times_times_nat @ K3 @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1043_nat__mult__less__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1044_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1045_nat__mult__div__cancel__disj,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ( K3 = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= zero_zero_nat ) )
& ( ( K3 != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1046_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1047_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1048_div__add__self1,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A2 ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_1049_div__add__self2,axiom,
! [B: nat,A2: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_1050_nat__mult__le__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1051_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1052_div__le__mono2,axiom,
! [M2: nat,N: nat,K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K3 @ N ) @ ( divide_divide_nat @ K3 @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1053_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1054_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1055_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1056_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1057_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M2
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1058_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1059_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1060_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1061_div__less__iff__less__mult,axiom,
! [Q2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q2 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1062_nat__mult__div__cancel1,axiom,
! [K3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
=> ( ( divide_divide_nat @ ( times_times_nat @ K3 @ M2 ) @ ( times_times_nat @ K3 @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1063_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1064_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1065_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q2 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q2 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1066_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1067_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1068_split__div,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P2 @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I4: nat,J3: nat] :
( ( ( ord_less_nat @ J3 @ N )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) ) )
=> ( P2 @ I4 ) ) ) ) ) ).
% split_div
thf(fact_1069_dual__sys_Opoint__indices__elem__in,axiom,
! [Ps: set_nat,T4: nat] :
( ( ord_less_eq_set_nat @ Ps @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) )
=> ( ( ( finite_card_nat @ Ps )
= T4 )
=> ( 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 ) @ T4 ) ) ) ) ).
% dual_sys.point_indices_elem_in
thf(fact_1070_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_1071_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_1072_Diff__idemp,axiom,
! [A: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B2 ) @ B2 )
= ( minus_minus_set_a @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_1073_Diff__idemp,axiom,
! [A: set_nat,B2: set_nat] :
( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ B2 ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ).
% Diff_idemp
thf(fact_1074_Diff__iff,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
= ( ( member_list_a @ C @ A )
& ~ ( member_list_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1075_Diff__iff,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
= ( ( member_set_nat @ C @ A )
& ~ ( member_set_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1076_Diff__iff,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
= ( ( member_set_a @ C @ A )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1077_Diff__iff,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
= ( ( member_a @ C @ A )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1078_Diff__iff,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_1079_DiffI,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ A )
=> ( ~ ( member_list_a @ C @ B2 )
=> ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_1080_DiffI,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( ~ ( member_set_nat @ C @ B2 )
=> ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_1081_DiffI,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_1082_DiffI,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ A )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_1083_DiffI,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) ) ) ) ).
% DiffI
thf(fact_1084_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_1085_diff__diff__add__mset,axiom,
! [M: multiset_set_a,N4: multiset_set_a,P2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ N4 ) @ P2 )
= ( minus_706656509937749387_set_a @ M @ ( plus_p2331992037799027419_set_a @ N4 @ P2 ) ) ) ).
% diff_diff_add_mset
thf(fact_1086_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_1087_Multiset_Odiff__add,axiom,
! [M: multiset_set_a,N4: multiset_set_a,Q: multiset_set_a] :
( ( minus_706656509937749387_set_a @ M @ ( plus_p2331992037799027419_set_a @ N4 @ Q ) )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ N4 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_1088_diff__union__cancelL,axiom,
! [N4: multiset_set_a,M: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ N4 @ M ) @ N4 )
= M ) ).
% diff_union_cancelL
thf(fact_1089_diff__union__cancelR,axiom,
! [M: multiset_set_a,N4: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( plus_p2331992037799027419_set_a @ M @ N4 ) @ N4 )
= M ) ).
% diff_union_cancelR
thf(fact_1090_DiffD2,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ~ ( member_list_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1091_DiffD2,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
=> ~ ( member_set_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1092_DiffD2,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
=> ~ ( member_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1093_DiffD2,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_1094_DiffD2,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_1095_DiffD1,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ( member_list_a @ C @ A ) ) ).
% DiffD1
thf(fact_1096_DiffD1,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
=> ( member_set_nat @ C @ A ) ) ).
% DiffD1
thf(fact_1097_DiffD1,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
=> ( member_set_a @ C @ A ) ) ).
% DiffD1
thf(fact_1098_DiffD1,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
=> ( member_a @ C @ A ) ) ).
% DiffD1
thf(fact_1099_DiffD1,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_1100_DiffE,axiom,
! [C: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ C @ ( minus_646659088055828811list_a @ A @ B2 ) )
=> ~ ( ( member_list_a @ C @ A )
=> ( member_list_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1101_DiffE,axiom,
! [C: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A @ B2 ) )
=> ~ ( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1102_DiffE,axiom,
! [C: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A @ B2 ) )
=> ~ ( ( member_set_a @ C @ A )
=> ( member_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1103_DiffE,axiom,
! [C: a,A: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A @ B2 ) )
=> ~ ( ( member_a @ C @ A )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1104_DiffE,axiom,
! [C: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B2 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_1105_Multiset_Odiff__right__commute,axiom,
! [M: multiset_set_a,N4: multiset_set_a,Q: multiset_set_a] :
( ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ N4 ) @ Q )
= ( minus_706656509937749387_set_a @ ( minus_706656509937749387_set_a @ M @ Q ) @ N4 ) ) ).
% Multiset.diff_right_commute
thf(fact_1106_ordered__pairwise__balance_Odual__is__const__intersect__des,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,Lambda: nat] :
( ( incide4449361439798955450_set_a @ V_s @ B_s @ Lambda )
=> ( ( ord_less_nat @ zero_zero_nat @ Lambda )
=> ( design137120128173859224gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7045802815848406192_set_a @ B_s ) ) @ ( dual_d359914979145368543_set_a @ ( set_set_a2 @ V_s ) @ B_s ) @ Lambda ) ) ) ).
% ordered_pairwise_balance.dual_is_const_intersect_des
thf(fact_1107_ordered__pairwise__balance_Odual__is__const__intersect__des,axiom,
! [V_s: list_a,B_s: list_set_a,Lambda: nat] :
( ( incide6880889959311561818ance_a @ V_s @ B_s @ Lambda )
=> ( ( 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 ) ) ) ).
% ordered_pairwise_balance.dual_is_const_intersect_des
thf(fact_1108_ordered__pairwise__balance_Odual__is__simp__const__inter__des,axiom,
! [V_s: list_set_a,B_s: list_set_set_a,Lambda: nat] :
( ( incide4449361439798955450_set_a @ V_s @ B_s @ Lambda )
=> ( ( ord_less_nat @ zero_zero_nat @ Lambda )
=> ( ! [Bl: set_set_a] :
( ( member_set_set_a @ Bl @ ( set_mset_set_set_a @ ( mset_set_set_a @ B_s ) ) )
=> ( ( ord_less_nat @ ( finite_card_set_a @ Bl ) @ ( finite_card_set_a @ ( set_set_a2 @ V_s ) ) )
& ( member_set_set_a @ Bl @ ( set_mset_set_set_a @ ( mset_set_set_a @ B_s ) ) ) ) )
=> ( design8545500683235687882gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s7045802815848406192_set_a @ B_s ) ) @ ( dual_d359914979145368543_set_a @ ( set_set_a2 @ V_s ) @ B_s ) @ Lambda ) ) ) ) ).
% ordered_pairwise_balance.dual_is_simp_const_inter_des
thf(fact_1109_ordered__pairwise__balance_Odual__is__simp__const__inter__des,axiom,
! [V_s: list_nat,B_s: list_set_nat,Lambda: nat] :
( ( incide3388802471754236788ce_nat @ V_s @ B_s @ Lambda )
=> ( ( ord_less_nat @ zero_zero_nat @ Lambda )
=> ( ! [Bl: set_nat] :
( ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( mset_set_nat @ B_s ) ) )
=> ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ ( set_nat2 @ V_s ) ) )
& ( member_set_nat @ Bl @ ( set_mset_set_nat @ ( mset_set_nat @ B_s ) ) ) ) )
=> ( design8545500683235687882gn_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ B_s ) ) @ ( dual_dual_blocks_nat @ ( set_nat2 @ V_s ) @ B_s ) @ Lambda ) ) ) ) ).
% ordered_pairwise_balance.dual_is_simp_const_inter_des
thf(fact_1110_ordered__pairwise__balance_Odual__is__simp__const__inter__des,axiom,
! [V_s: list_a,B_s: list_set_a,Lambda: nat] :
( ( incide6880889959311561818ance_a @ V_s @ B_s @ Lambda )
=> ( ( 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 ) ) ) ) ).
% ordered_pairwise_balance.dual_is_simp_const_inter_des
thf(fact_1111_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_1112_dual__sys_Omultiple__orig__sub__system,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( 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 ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( size_size_list_set_a @ b_s ) ) @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) ) ) ).
% dual_sys.multiple_orig_sub_system
thf(fact_1113_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_1114_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_1115_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_1116_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_1117_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_1118_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_1119_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_1120_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_1121_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_1122_dual__sys_Oblock__comp__elem__alt__left,axiom,
! [X3: nat,Bl2: set_nat,Ps: set_nat] :
( ( member_nat @ X3 @ 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 @ X3 @ Ps ) ) ) ).
% dual_sys.block_comp_elem_alt_left
thf(fact_1123_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 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ Ps )
=> ~ ( member_nat @ X2 @ 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_1124_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 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ Ps )
=> ~ ( member_nat @ X @ Bl2 ) ) ) ) ) ).
% dual_sys.block_complement_elem_iff
thf(fact_1125_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_1126_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_1127_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_1128_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_1129_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_1130_multiple__bibd,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( bibd_a @ ( set_a2 @ v_s ) @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ k @ ( times_times_nat @ lambda @ N ) ) ) ).
% multiple_bibd
thf(fact_1131_insert__absorb2,axiom,
! [X3: nat,A: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A ) )
= ( insert_nat @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_1132_insert__absorb2,axiom,
! [X3: a,A: set_a] :
( ( insert_a @ X3 @ ( insert_a @ X3 @ A ) )
= ( insert_a @ X3 @ A ) ) ).
% insert_absorb2
thf(fact_1133_insert__iff,axiom,
! [A2: list_a,B: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ ( insert_list_a @ B @ A ) )
= ( ( A2 = B )
| ( member_list_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_1134_insert__iff,axiom,
! [A2: set_nat,B: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_set_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_1135_insert__iff,axiom,
! [A2: set_a,B: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B @ A ) )
= ( ( A2 = B )
| ( member_set_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_1136_insert__iff,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
= ( ( A2 = B )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_1137_insert__iff,axiom,
! [A2: a,B: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A ) )
= ( ( A2 = B )
| ( member_a @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_1138_insertCI,axiom,
! [A2: list_a,B2: set_list_a,B: list_a] :
( ( ~ ( member_list_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_list_a @ A2 @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1139_insertCI,axiom,
! [A2: set_nat,B2: set_set_nat,B: set_nat] :
( ( ~ ( member_set_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1140_insertCI,axiom,
! [A2: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_set_a @ A2 @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1141_insertCI,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1142_insertCI,axiom,
! [A2: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A2 @ B2 )
=> ( A2 = B ) )
=> ( member_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_1143_positive__ints,axiom,
! [X3: nat] :
( ( member_nat @ X3 @ ( insert_nat @ k @ bot_bot_set_nat ) )
=> ( ord_less_nat @ zero_zero_nat @ X3 ) ) ).
% positive_ints
thf(fact_1144_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_1145_singletonI,axiom,
! [A2: list_a] : ( member_list_a @ A2 @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) ) ).
% singletonI
thf(fact_1146_singletonI,axiom,
! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% singletonI
thf(fact_1147_singletonI,axiom,
! [A2: set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_1148_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_1149_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_1150_insert__subset,axiom,
! [X3: list_a,A: set_list_a,B2: set_list_a] :
( ( ord_le8861187494160871172list_a @ ( insert_list_a @ X3 @ A ) @ B2 )
= ( ( member_list_a @ X3 @ B2 )
& ( ord_le8861187494160871172list_a @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_1151_insert__subset,axiom,
! [X3: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ A ) @ B2 )
= ( ( member_set_nat @ X3 @ B2 )
& ( ord_le6893508408891458716et_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_1152_insert__subset,axiom,
! [X3: set_a,A: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X3 @ A ) @ B2 )
= ( ( member_set_a @ X3 @ B2 )
& ( ord_le3724670747650509150_set_a @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_1153_insert__subset,axiom,
! [X3: nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A ) @ B2 )
= ( ( member_nat @ X3 @ B2 )
& ( ord_less_eq_set_nat @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_1154_insert__subset,axiom,
! [X3: a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A ) @ B2 )
= ( ( member_a @ X3 @ B2 )
& ( ord_less_eq_set_a @ A @ B2 ) ) ) ).
% insert_subset
thf(fact_1155_finite__insert,axiom,
! [A2: a,A: set_a] :
( ( finite_finite_a @ ( insert_a @ A2 @ A ) )
= ( finite_finite_a @ A ) ) ).
% finite_insert
thf(fact_1156_finite__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( insert_set_nat @ A2 @ A ) )
= ( finite1152437895449049373et_nat @ A ) ) ).
% finite_insert
thf(fact_1157_finite__insert,axiom,
! [A2: nat,A: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A2 @ A ) )
= ( finite_finite_nat @ A ) ) ).
% finite_insert
thf(fact_1158_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_1159_Diff__insert0,axiom,
! [X3: list_a,A: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X3 @ A )
=> ( ( minus_646659088055828811list_a @ A @ ( insert_list_a @ X3 @ B2 ) )
= ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1160_Diff__insert0,axiom,
! [X3: set_nat,A: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X3 @ A )
=> ( ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ X3 @ B2 ) )
= ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1161_Diff__insert0,axiom,
! [X3: set_a,A: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ X3 @ B2 ) )
= ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1162_Diff__insert0,axiom,
! [X3: a,A: set_a,B2: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ( minus_minus_set_a @ A @ ( insert_a @ X3 @ B2 ) )
= ( minus_minus_set_a @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1163_Diff__insert0,axiom,
! [X3: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X3 @ B2 ) )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% Diff_insert0
thf(fact_1164_insert__Diff1,axiom,
! [X3: list_a,B2: set_list_a,A: set_list_a] :
( ( member_list_a @ X3 @ B2 )
=> ( ( minus_646659088055828811list_a @ ( insert_list_a @ X3 @ A ) @ B2 )
= ( minus_646659088055828811list_a @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1165_insert__Diff1,axiom,
! [X3: set_nat,B2: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ X3 @ B2 )
=> ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A ) @ B2 )
= ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1166_insert__Diff1,axiom,
! [X3: set_a,B2: set_set_a,A: set_set_a] :
( ( member_set_a @ X3 @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X3 @ A ) @ B2 )
= ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1167_insert__Diff1,axiom,
! [X3: a,B2: set_a,A: set_a] :
( ( member_a @ X3 @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A ) @ B2 )
= ( minus_minus_set_a @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1168_insert__Diff1,axiom,
! [X3: nat,B2: set_nat,A: set_nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A ) @ B2 )
= ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% insert_Diff1
thf(fact_1169_block__sizes,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( member_nat @ ( finite_card_a @ Bl2 ) @ ( insert_nat @ k @ bot_bot_set_nat ) ) ) ).
% block_sizes
thf(fact_1170_sys__block__size__subset,axiom,
ord_less_eq_set_nat @ ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) ) @ ( insert_nat @ k @ bot_bot_set_nat ) ).
% sys_block_size_subset
thf(fact_1171_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_1172_bibd__axioms,axiom,
bibd_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ k @ lambda ).
% bibd_axioms
thf(fact_1173_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_1174_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_1175_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_1176_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_1177_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_1178_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_1179_finite__Diff__insert,axiom,
! [A: set_set_nat,A2: set_nat,B2: set_set_nat] :
( ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ B2 ) ) )
= ( finite1152437895449049373et_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1180_finite__Diff__insert,axiom,
! [A: set_set_a,A2: set_a,B2: set_set_a] :
( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ B2 ) ) )
= ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1181_finite__Diff__insert,axiom,
! [A: set_a,A2: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1182_finite__Diff__insert,axiom,
! [A: set_nat,A2: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_1183_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_1184_is__singletonI,axiom,
! [X3: a] : ( is_singleton_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_1185_is__singletonI,axiom,
! [X3: nat] : ( is_singleton_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).
% is_singletonI
thf(fact_1186_card__Diff__insert,axiom,
! [A2: list_a,A: set_list_a,B2: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ( ~ ( member_list_a @ A2 @ B2 )
=> ( ( finite_card_list_a @ ( minus_646659088055828811list_a @ A @ ( insert_list_a @ A2 @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_list_a @ ( minus_646659088055828811list_a @ A @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1187_card__Diff__insert,axiom,
! [A2: set_nat,A: set_set_nat,B2: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ~ ( member_set_nat @ A2 @ B2 )
=> ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A @ ( insert_set_nat @ A2 @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1188_card__Diff__insert,axiom,
! [A2: set_a,A: set_set_a,B2: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ A2 @ B2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A @ ( insert_set_a @ A2 @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1189_card__Diff__insert,axiom,
! [A2: a,A: set_a,B2: set_a] :
( ( member_a @ A2 @ A )
=> ( ~ ( member_a @ A2 @ B2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A @ ( insert_a @ A2 @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1190_card__Diff__insert,axiom,
! [A2: nat,A: set_nat,B2: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ A2 @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_1191_sys__block__sizes__uniform,axiom,
( ( design1769254222028858111izes_a @ ( mset_set_a @ b_s ) )
= ( insert_nat @ k @ bot_bot_set_nat ) ) ).
% sys_block_sizes_uniform
thf(fact_1192_rep__numbers__constant,axiom,
( ( design8835372594653258411bers_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) )
= ( insert_nat @ ( divide_divide_nat @ ( times_times_nat @ lambda @ ( minus_minus_nat @ ( finite_card_a @ ( set_a2 @ v_s ) ) @ one_one_nat ) ) @ ( minus_minus_nat @ k @ one_one_nat ) ) @ bot_bot_set_nat ) ) ).
% rep_numbers_constant
thf(fact_1193__092_060Lambda_062__PBD__axioms,axiom,
block_Lambda_PBD_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ lambda @ ( insert_nat @ k @ bot_bot_set_nat ) ).
% \<Lambda>_PBD_axioms
thf(fact_1194_mk__disjoint__insert,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ? [B5: set_list_a] :
( ( A
= ( insert_list_a @ A2 @ B5 ) )
& ~ ( member_list_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_1195_mk__disjoint__insert,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ? [B5: set_set_nat] :
( ( A
= ( insert_set_nat @ A2 @ B5 ) )
& ~ ( member_set_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_1196_mk__disjoint__insert,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ? [B5: set_set_a] :
( ( A
= ( insert_set_a @ A2 @ B5 ) )
& ~ ( member_set_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_1197_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B5: set_nat] :
( ( A
= ( insert_nat @ A2 @ B5 ) )
& ~ ( member_nat @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_1198_mk__disjoint__insert,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ? [B5: set_a] :
( ( A
= ( insert_a @ A2 @ B5 ) )
& ~ ( member_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_1199_insert__commute,axiom,
! [X3: nat,Y3: nat,A: set_nat] :
( ( insert_nat @ X3 @ ( insert_nat @ Y3 @ A ) )
= ( insert_nat @ Y3 @ ( insert_nat @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_1200_insert__commute,axiom,
! [X3: a,Y3: a,A: set_a] :
( ( insert_a @ X3 @ ( insert_a @ Y3 @ A ) )
= ( insert_a @ Y3 @ ( insert_a @ X3 @ A ) ) ) ).
% insert_commute
thf(fact_1201_insert__eq__iff,axiom,
! [A2: list_a,A: set_list_a,B: list_a,B2: set_list_a] :
( ~ ( member_list_a @ A2 @ A )
=> ( ~ ( member_list_a @ B @ B2 )
=> ( ( ( insert_list_a @ A2 @ A )
= ( insert_list_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_list_a] :
( ( A
= ( insert_list_a @ B @ C3 ) )
& ~ ( member_list_a @ B @ C3 )
& ( B2
= ( insert_list_a @ A2 @ C3 ) )
& ~ ( member_list_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1202_insert__eq__iff,axiom,
! [A2: set_nat,A: set_set_nat,B: set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ A2 @ A )
=> ( ~ ( member_set_nat @ B @ B2 )
=> ( ( ( insert_set_nat @ A2 @ A )
= ( insert_set_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_set_nat] :
( ( A
= ( insert_set_nat @ B @ C3 ) )
& ~ ( member_set_nat @ B @ C3 )
& ( B2
= ( insert_set_nat @ A2 @ C3 ) )
& ~ ( member_set_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1203_insert__eq__iff,axiom,
! [A2: set_a,A: set_set_a,B: set_a,B2: set_set_a] :
( ~ ( member_set_a @ A2 @ A )
=> ( ~ ( member_set_a @ B @ B2 )
=> ( ( ( insert_set_a @ A2 @ A )
= ( insert_set_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_set_a] :
( ( A
= ( insert_set_a @ B @ C3 ) )
& ~ ( member_set_a @ B @ C3 )
& ( B2
= ( insert_set_a @ A2 @ C3 ) )
& ~ ( member_set_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1204_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B @ C3 ) )
& ~ ( member_nat @ B @ C3 )
& ( B2
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1205_insert__eq__iff,axiom,
! [A2: a,A: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A2 @ A )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A2 @ A )
= ( insert_a @ B @ B2 ) )
= ( ( ( A2 = B )
=> ( A = B2 ) )
& ( ( A2 != B )
=> ? [C3: set_a] :
( ( A
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B2
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_1206_insert__absorb,axiom,
! [A2: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ A )
=> ( ( insert_list_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1207_insert__absorb,axiom,
! [A2: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ A )
=> ( ( insert_set_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1208_insert__absorb,axiom,
! [A2: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ A )
=> ( ( insert_set_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1209_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1210_insert__absorb,axiom,
! [A2: a,A: set_a] :
( ( member_a @ A2 @ A )
=> ( ( insert_a @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_1211_insert__ident,axiom,
! [X3: list_a,A: set_list_a,B2: set_list_a] :
( ~ ( member_list_a @ X3 @ A )
=> ( ~ ( member_list_a @ X3 @ B2 )
=> ( ( ( insert_list_a @ X3 @ A )
= ( insert_list_a @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_1212_insert__ident,axiom,
! [X3: set_nat,A: set_set_nat,B2: set_set_nat] :
( ~ ( member_set_nat @ X3 @ A )
=> ( ~ ( member_set_nat @ X3 @ B2 )
=> ( ( ( insert_set_nat @ X3 @ A )
= ( insert_set_nat @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_1213_insert__ident,axiom,
! [X3: set_a,A: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X3 @ A )
=> ( ~ ( member_set_a @ X3 @ B2 )
=> ( ( ( insert_set_a @ X3 @ A )
= ( insert_set_a @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_1214_insert__ident,axiom,
! [X3: nat,A: set_nat,B2: set_nat] :
( ~ ( member_nat @ X3 @ A )
=> ( ~ ( member_nat @ X3 @ B2 )
=> ( ( ( insert_nat @ X3 @ A )
= ( insert_nat @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_1215_insert__ident,axiom,
! [X3: a,A: set_a,B2: set_a] :
( ~ ( member_a @ X3 @ A )
=> ( ~ ( member_a @ X3 @ B2 )
=> ( ( ( insert_a @ X3 @ A )
= ( insert_a @ X3 @ B2 ) )
= ( A = B2 ) ) ) ) ).
% insert_ident
thf(fact_1216_Set_Oset__insert,axiom,
! [X3: list_a,A: set_list_a] :
( ( member_list_a @ X3 @ A )
=> ~ ! [B5: set_list_a] :
( ( A
= ( insert_list_a @ X3 @ B5 ) )
=> ( member_list_a @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_1217_Set_Oset__insert,axiom,
! [X3: set_nat,A: set_set_nat] :
( ( member_set_nat @ X3 @ A )
=> ~ ! [B5: set_set_nat] :
( ( A
= ( insert_set_nat @ X3 @ B5 ) )
=> ( member_set_nat @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_1218_Set_Oset__insert,axiom,
! [X3: set_a,A: set_set_a] :
( ( member_set_a @ X3 @ A )
=> ~ ! [B5: set_set_a] :
( ( A
= ( insert_set_a @ X3 @ B5 ) )
=> ( member_set_a @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_1219_Set_Oset__insert,axiom,
! [X3: nat,A: set_nat] :
( ( member_nat @ X3 @ A )
=> ~ ! [B5: set_nat] :
( ( A
= ( insert_nat @ X3 @ B5 ) )
=> ( member_nat @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_1220_Set_Oset__insert,axiom,
! [X3: a,A: set_a] :
( ( member_a @ X3 @ A )
=> ~ ! [B5: set_a] :
( ( A
= ( insert_a @ X3 @ B5 ) )
=> ( member_a @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_1221_insertI2,axiom,
! [A2: list_a,B2: set_list_a,B: list_a] :
( ( member_list_a @ A2 @ B2 )
=> ( member_list_a @ A2 @ ( insert_list_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1222_insertI2,axiom,
! [A2: set_nat,B2: set_set_nat,B: set_nat] :
( ( member_set_nat @ A2 @ B2 )
=> ( member_set_nat @ A2 @ ( insert_set_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1223_insertI2,axiom,
! [A2: set_a,B2: set_set_a,B: set_a] :
( ( member_set_a @ A2 @ B2 )
=> ( member_set_a @ A2 @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1224_insertI2,axiom,
! [A2: nat,B2: set_nat,B: nat] :
( ( member_nat @ A2 @ B2 )
=> ( member_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1225_insertI2,axiom,
! [A2: a,B2: set_a,B: a] :
( ( member_a @ A2 @ B2 )
=> ( member_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_1226_insertI1,axiom,
! [A2: list_a,B2: set_list_a] : ( member_list_a @ A2 @ ( insert_list_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_1227_insertI1,axiom,
! [A2: set_nat,B2: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_1228_insertI1,axiom,
! [A2: set_a,B2: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_1229_insertI1,axiom,
! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).
% insertI1
thf(fact_1230_insertI1,axiom,
! [A2: a,B2: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B2 ) ) ).
% insertI1
thf(fact_1231_insertE,axiom,
! [A2: list_a,B: list_a,A: set_list_a] :
( ( member_list_a @ A2 @ ( insert_list_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member_list_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_1232_insertE,axiom,
! [A2: set_nat,B: set_nat,A: set_set_nat] :
( ( member_set_nat @ A2 @ ( insert_set_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_1233_insertE,axiom,
! [A2: set_a,B: set_a,A: set_set_a] :
( ( member_set_a @ A2 @ ( insert_set_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member_set_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_1234_insertE,axiom,
! [A2: nat,B: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B @ A ) )
=> ( ( A2 != B )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_1235_insertE,axiom,
! [A2: a,B: a,A: set_a] :
( ( member_a @ A2 @ ( insert_a @ B @ A ) )
=> ( ( A2 != B )
=> ( member_a @ A2 @ A ) ) ) ).
% insertE
thf(fact_1236_finite_OinsertI,axiom,
! [A: set_a,A2: a] :
( ( finite_finite_a @ A )
=> ( finite_finite_a @ ( insert_a @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1237_finite_OinsertI,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( finite1152437895449049373et_nat @ A )
=> ( finite1152437895449049373et_nat @ ( insert_set_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1238_finite_OinsertI,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( finite_finite_nat @ ( insert_nat @ A2 @ A ) ) ) ).
% finite.insertI
thf(fact_1239_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_1240_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_1241_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_1242_insert__not__empty,axiom,
! [A2: a,A: set_a] :
( ( insert_a @ A2 @ A )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_1243_insert__not__empty,axiom,
! [A2: nat,A: set_nat] :
( ( insert_nat @ A2 @ A )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_1244_doubleton__eq__iff,axiom,
! [A2: a,B: a,C: a,D3: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D3 @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B = D3 ) )
| ( ( A2 = D3 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1245_doubleton__eq__iff,axiom,
! [A2: nat,B: nat,C: nat,D3: nat] :
( ( ( insert_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D3 @ bot_bot_set_nat ) ) )
= ( ( ( A2 = C )
& ( B = D3 ) )
| ( ( A2 = D3 )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_1246_singleton__iff,axiom,
! [B: list_a,A2: list_a] :
( ( member_list_a @ B @ ( insert_list_a @ A2 @ bot_bot_set_list_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_1247_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_1248_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_1249_singleton__iff,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_1250_singleton__iff,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B = A2 ) ) ).
% singleton_iff
thf(fact_1251_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_1252_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_1253_singletonD,axiom,
! [B: a,A2: a] :
( ( member_a @ B @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_1254_singletonD,axiom,
! [B: nat,A2: nat] :
( ( member_nat @ B @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B = A2 ) ) ).
% singletonD
thf(fact_1255_dual__sys_Ostr__del__block__del__point,axiom,
! [X3: nat] :
( ~ ( member_set_nat @ ( insert_nat @ X3 @ 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 @ X3 @ bot_bot_set_nat ) )
= ( design4832208198062110345ks_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ X3 ) ) ) ).
% dual_sys.str_del_block_del_point
thf(fact_1256_K__block__design__axioms,axiom,
block_4943671260865530057sign_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) @ ( insert_nat @ k @ bot_bot_set_nat ) ).
% K_block_design_axioms
thf(fact_1257_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_1258_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_1259_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_1260_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_1261_str__del__block__del__point,axiom,
! [X3: a] :
( ~ ( member_set_a @ ( insert_a @ X3 @ bot_bot_set_a ) @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( design4241783006516448631lock_a @ ( mset_set_a @ b_s ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= ( design6411949732824333445ocks_a @ ( mset_set_a @ b_s ) @ X3 ) ) ) ).
% str_del_block_del_point
thf(fact_1262_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_1263_dual__sys_Oblock__original__count__le,axiom,
! [N: nat,B: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( count_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ B ) @ ( count_set_nat @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ B ) ) ) ).
% dual_sys.block_original_count_le
thf(fact_1264_dual__sys_Omultiple__block__multiplicity,axiom,
! [N: nat,Bl2: set_nat] :
( ( count_set_nat @ ( repeat_mset_set_nat @ N @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) ) @ Bl2 )
= ( times_times_nat @ ( count_set_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ Bl2 ) @ N ) ) ).
% dual_sys.multiple_block_multiplicity
thf(fact_1265_remove__complete__blocks__set__pbd,axiom,
! [X3: nat,A: multiset_set_a] :
( ( ord_less_nat @ X3 @ lambda )
=> ( ( ( size_s6566526139600085008_set_a @ A )
= X3 )
=> ( ( subset_mset_set_a @ A @ ( mset_set_a @ b_s ) )
=> ( ! [A6: set_a] :
( ( member_set_a @ A6 @ ( set_mset_set_a @ A ) )
=> ( A6
= ( 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 @ X3 ) ) ) ) ) ) ).
% remove_complete_blocks_set_pbd
thf(fact_1266_simple__alt__def__all,axiom,
! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( count_set_a @ ( mset_set_a @ b_s ) @ X4 )
= one_one_nat ) ) ).
% simple_alt_def_all
thf(fact_1267_points__index__count__min,axiom,
! [N: nat,Bl2: set_a,Ps: set_a] :
( ( ord_less_eq_nat @ N @ ( count_set_a @ ( mset_set_a @ b_s ) @ Bl2 ) )
=> ( ( ord_less_eq_set_a @ Ps @ Bl2 )
=> ( ord_less_eq_nat @ N @ ( design254580327166089565ndex_a @ ( mset_set_a @ b_s ) @ Ps ) ) ) ) ).
% points_index_count_min
thf(fact_1268_block__original__count__le,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ B ) @ ( count_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ B ) ) ) ).
% block_original_count_le
thf(fact_1269_count__complete__lt__balance,axiom,
ord_less_eq_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ ( set_a2 @ v_s ) ) @ lambda ).
% count_complete_lt_balance
thf(fact_1270_const__inter__multiplicity__one,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ lambda @ ( finite_card_a @ Bl2 ) )
=> ( ( count_set_a @ ( mset_set_a @ b_s ) @ Bl2 )
= one_one_nat ) ) ) ).
% const_inter_multiplicity_one
thf(fact_1271_simple__not__multiplicity,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( ord_less_nat @ one_one_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ B ) )
=> ~ ( design1338723777345758283stem_a @ ( set_a2 @ v_s ) @ ( mset_set_a @ b_s ) ) ) ) ).
% simple_not_multiplicity
thf(fact_1272_sym__block__mult__one,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( mset_set_a @ b_s ) ) )
=> ( ( count_set_a @ ( mset_set_a @ b_s ) @ Bl2 )
= one_one_nat ) ) ).
% sym_block_mult_one
thf(fact_1273_multiple__block__multiplicity,axiom,
! [N: nat,Bl2: set_a] :
( ( count_set_a @ ( repeat_mset_set_a @ N @ ( mset_set_a @ b_s ) ) @ Bl2 )
= ( times_times_nat @ ( count_set_a @ ( mset_set_a @ b_s ) @ Bl2 ) @ N ) ) ).
% multiple_block_multiplicity
% 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,
! [X3: nat,Y3: nat] :
( ( if_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y3: nat] :
( ( if_nat @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( design6574611146354332593ex_nat @ ( dual_dual_blocks_a @ ( set_a2 @ v_s ) @ b_s ) @ ps )
= lambda ) ).
%------------------------------------------------------------------------------