TPTP Problem File: SLH0771^1.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/0015_Design_Extras/prob_00597_020933__27922216_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1409 ( 510 unt; 137 typ; 0 def)
% Number of atoms : 3403 (1100 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 10384 ( 463 ~; 47 |; 210 &;8148 @)
% ( 0 <=>;1516 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 271 ( 271 >; 0 *; 0 +; 0 <<)
% Number of symbols : 126 ( 125 usr; 15 con; 0-3 aty)
% Number of variables : 2960 ( 99 ^;2791 !; 70 ?;2960 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-18 15:45:37.692
%------------------------------------------------------------------------------
% Could-be-implicit typings (12)
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__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
multiset_set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
multiset_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $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__Set__Oset_It__Nat__Onat_J,type,
set_nat: $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 (125)
thf(sy_c_Design__Basics_Odesign_001t__Nat__Onat,type,
design_design_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Odesign_001t__Set__Oset_Itf__a_J,type,
design_design_set_a: set_set_a > multiset_set_set_a > $o ).
thf(sy_c_Design__Basics_Odesign_001tf__a,type,
design_design_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Ofinite__incidence__system_001t__Nat__Onat,type,
design5426232790142929158em_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Ofinite__incidence__system_001t__Set__Oset_Itf__a_J,type,
design1749870844763721896_set_a: set_set_a > multiset_set_set_a > $o ).
thf(sy_c_Design__Basics_Ofinite__incidence__system_001tf__a,type,
design9187838744727572296stem_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_001t__Nat__Onat,type,
design3753904077504641269em_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Oincidence__system_001t__Set__Oset_Itf__a_J,type,
design9013482484999600761_set_a: set_set_a > multiset_set_set_a > $o ).
thf(sy_c_Design__Basics_Oincidence__system_001tf__a,type,
design1863209521793301785stem_a: set_a > multiset_set_a > $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_001t__Set__Oset_Itf__a_J,type,
design4243878040612417342_set_a: set_set_a > set_set_a > set_set_a ).
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_001tf__a,type,
design5397942185814921632port_a: multiset_set_a > set_set_a ).
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_001tf__a,type,
design328527185268214962ices_a: set_a > multiset_set_a > 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_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_Opoint__replication__number_001t__Nat__Onat,type,
design3571518413069006949er_nat: multiset_set_nat > nat > nat ).
thf(sy_c_Design__Basics_Opoint__replication__number_001t__Set__Oset_Itf__a_J,type,
design5008467512594872073_set_a: multiset_set_set_a > set_a > nat ).
thf(sy_c_Design__Basics_Opoint__replication__number_001tf__a,type,
design6637022207325878697mber_a: multiset_set_a > a > nat ).
thf(sy_c_Design__Basics_Opoints__index_001t__Nat__Onat,type,
design6574611146354332593ex_nat: multiset_set_nat > set_nat > nat ).
thf(sy_c_Design__Basics_Opoints__index_001t__Set__Oset_Itf__a_J,type,
design88022138586678973_set_a: multiset_set_set_a > set_set_a > nat ).
thf(sy_c_Design__Basics_Opoints__index_001tf__a,type,
design254580327166089565ndex_a: multiset_set_a > set_a > nat ).
thf(sy_c_Design__Basics_Oproper__design_001t__Nat__Onat,type,
design435815215503836206gn_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Oproper__design_001tf__a,type,
design7287791228148780576sign_a: set_a > multiset_set_a > $o ).
thf(sy_c_Design__Basics_Osimple__design_001t__Nat__Onat,type,
design7861764274488435984gn_nat: set_nat > multiset_set_nat > $o ).
thf(sy_c_Design__Basics_Osimple__design_001tf__a,type,
design3982635895484621246sign_a: set_a > multiset_set_a > $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_001t__Set__Oset_Itf__a_J,type,
design2124024510932123798_set_a: set_set_a > multiset_set_set_a > 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__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_001t__Set__Oset_Itf__a_J,type,
design7860908649167014820_set_a: multiset_set_set_a > set_set_a > multiset_set_set_a ).
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_001tf__a,type,
design2964366272795260673oint_a: set_a > a > set_a ).
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_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_001t__Set__Oset_Itf__a_J,type,
design7586725432863044395_set_a: set_set_a > set_a > set_set_a ).
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_001tf__a,type,
design6411949732824333445ocks_a: multiset_set_a > a > multiset_set_a ).
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_001tf__a,type,
design5657747894866638574ocks_a: multiset_set_a > a > multiset_set_a ).
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_Itf__a_J,type,
finite_card_set_a: set_set_a > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J,type,
finite_finite_set_a: set_set_a > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
minus_8522176038001411705et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
minus_706656509937749387_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
minus_3765977307040488491iset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
plus_p6334493942879108393et_nat: multiset_nat > multiset_nat > multiset_nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
plus_p2331992037799027419_set_a: multiset_set_a > multiset_set_a > multiset_set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
zero_z7348594199698428585et_nat: multiset_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
zero_z3157962936165190495et_nat: multiset_set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
zero_z6396401802697562811_set_a: multiset_set_set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
zero_z5079479921072680283_set_a: multiset_set_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
zero_zero_multiset_a: multiset_a ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
sup_sup_set_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Multiset_Omultiset_Ocount_001t__Nat__Onat,type,
count_nat: multiset_nat > nat > nat ).
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_Omultiset_Ocount_001tf__a,type,
count_a: multiset_a > a > nat ).
thf(sy_c_Multiset_Orepeat__mset_001t__Nat__Onat,type,
repeat_mset_nat: nat > multiset_nat > multiset_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_It__Set__Oset_Itf__a_J_J,type,
repeat3222187171979612824_set_a: nat > multiset_set_set_a > multiset_set_set_a ).
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_Orepeat__mset_001tf__a,type,
repeat_mset_a: nat > multiset_a > multiset_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_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
size_s5917832649809541300et_nat: multiset_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_It__Nat__Onat_J_J,type,
size_s7462436076474991978et_nat: multiset_set_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
size_s6566526139600085008_set_a: multiset_set_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_Itf__a_J,type,
size_size_multiset_a: multiset_a > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
bot_bo4176661893541381648_set_a: multiset_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_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__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le5777773500796000884et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le5765082015083327056_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_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__Multiset__Omultiset_It__Nat__Onat_J,type,
ord_le6602235886369790592et_nat: multiset_nat > multiset_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Multiset__Omultiset_It__Set__Oset_Itf__a_J_J,type,
ord_le7905258569527593284_set_a: multiset_set_a > multiset_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_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_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_060m_062,type,
m: nat ).
thf(sy_v_block__collection,type,
block_collection: multiset_set_a ).
thf(sy_v_point__set,type,
point_set: set_a ).
% Relevant facts (1271)
thf(fact_0_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 @ block_collection @ P @ Bs )
= block_collection ) ) ).
% add_point_existing_blocks
thf(fact_1_v__eq0__imp__b__eq__0,axiom,
( ( ( finite_card_a @ point_set )
= zero_zero_nat )
=> ( ( size_s6566526139600085008_set_a @ block_collection )
= zero_zero_nat ) ) ).
% v_eq0_imp_b_eq_0
thf(fact_2_del__invalid__point__blocks,axiom,
! [P: a] :
( ~ ( member_a @ P @ point_set )
=> ( ( design6411949732824333445ocks_a @ block_collection @ P )
= block_collection ) ) ).
% del_invalid_point_blocks
thf(fact_3_delete__invalid__pt__strong__eq,axiom,
! [P: a] :
( ~ ( member_a @ P @ point_set )
=> ( block_collection
= ( design5657747894866638574ocks_a @ block_collection @ P ) ) ) ).
% delete_invalid_pt_strong_eq
thf(fact_4_del__point__block__count,axiom,
! [P: a] :
( ( size_s6566526139600085008_set_a @ ( design6411949732824333445ocks_a @ block_collection @ P ) )
= ( size_s6566526139600085008_set_a @ block_collection ) ) ).
% del_point_block_count
thf(fact_5_const__intersect__design__axioms,axiom,
design9190424834980853558sign_a @ point_set @ block_collection @ m ).
% const_intersect_design_axioms
thf(fact_6_block__size__lt__v,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) ) ) ).
% block_size_lt_v
thf(fact_7_complete__block__size__eq__points,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( ( finite_card_a @ Bl2 )
= ( finite_card_a @ point_set ) )
=> ( Bl2 = point_set ) ) ) ).
% complete_block_size_eq_points
thf(fact_8_b__non__zero,axiom,
( ( size_s6566526139600085008_set_a @ block_collection )
!= zero_zero_nat ) ).
% b_non_zero
thf(fact_9_proper__design__axioms,axiom,
design7287791228148780576sign_a @ point_set @ block_collection ).
% proper_design_axioms
thf(fact_10_del__invalid__point,axiom,
! [P: a] :
( ~ ( member_a @ P @ point_set )
=> ( ( design108908007054065099oint_a @ point_set @ P )
= point_set ) ) ).
% del_invalid_point
thf(fact_11_finite__incidence__system__axioms,axiom,
design9187838744727572296stem_a @ point_set @ block_collection ).
% finite_incidence_system_axioms
thf(fact_12_assms,axiom,
m = zero_zero_nat ).
% assms
thf(fact_13_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 @ block_collection @ P ) ) ) ) ).
% delete_point_strong_block_not_in
thf(fact_14_delete__point__strong__block__in__orig,axiom,
! [Bl2: set_a,P: a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) )
=> ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) ) ).
% delete_point_strong_block_in_orig
thf(fact_15_delete__point__strong__block__in__iff,axiom,
! [Bl2: set_a,P: a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) )
= ( ~ ( member_a @ P @ Bl2 ) ) ) ) ).
% delete_point_strong_block_in_iff
thf(fact_16_delete__point__strong__block__in,axiom,
! [P: a,Bl2: set_a] :
( ~ ( member_a @ P @ Bl2 )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) ) ) ) ).
% delete_point_strong_block_in
thf(fact_17_delete__point__p__not__in__bl__blocks,axiom,
! [P: a] :
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ~ ( member_a @ P @ Bl ) )
=> ( ( design6411949732824333445ocks_a @ block_collection @ P )
= block_collection ) ) ).
% delete_point_p_not_in_bl_blocks
thf(fact_18_wf__invalid__point,axiom,
! [X: a,B: set_a] :
( ~ ( member_a @ X @ point_set )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ~ ( member_a @ X @ B ) ) ) ).
% wf_invalid_point
thf(fact_19_strong__del__point__finite,axiom,
! [P: a] : ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) ).
% strong_del_point_finite
thf(fact_20_delete__point__finite,axiom,
! [P: a] : ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design6411949732824333445ocks_a @ block_collection @ P ) ) ).
% delete_point_finite
thf(fact_21_incidence__alt__def,axiom,
! [P: a,B: set_a] :
( ( member_a @ P @ point_set )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( ( design3210447939978979927dent_a @ block_collection @ P @ B )
= ( member_a @ P @ B ) ) ) ) ).
% incidence_alt_def
thf(fact_22_proper__designI,axiom,
( ( ( size_s6566526139600085008_set_a @ block_collection )
!= zero_zero_nat )
=> ( design7287791228148780576sign_a @ point_set @ block_collection ) ) ).
% proper_designI
thf(fact_23_design__support__def,axiom,
( ( design5397942185814921632port_a @ block_collection )
= ( set_mset_set_a @ block_collection ) ) ).
% design_support_def
thf(fact_24_const__intersect__design_Oaxioms_I1_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,M: nat] :
( ( design9190424834980853558sign_a @ Point_set @ Block_collection @ M )
=> ( design7287791228148780576sign_a @ Point_set @ Block_collection ) ) ).
% const_intersect_design.axioms(1)
thf(fact_25_finite__incidence__system_Ocomplete__block__size__eq__points,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ( finite_card_nat @ Bl2 )
= ( finite_card_nat @ Point_set ) )
=> ( Bl2 = Point_set ) ) ) ) ).
% finite_incidence_system.complete_block_size_eq_points
thf(fact_26_finite__incidence__system_Ocomplete__block__size__eq__points,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ( finite_card_a @ Bl2 )
= ( finite_card_a @ Point_set ) )
=> ( Bl2 = Point_set ) ) ) ) ).
% finite_incidence_system.complete_block_size_eq_points
thf(fact_27_add__point__to__blocks__finite,axiom,
! [P: a,Bs: set_set_a] : ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ ( design2935547469388721088ocks_a @ block_collection @ P @ Bs ) ) ).
% add_point_to_blocks_finite
thf(fact_28_add__point__finite,axiom,
! [P: a] : ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ block_collection ) ).
% add_point_finite
thf(fact_29_add__delete__point__inv,axiom,
! [P: a] :
( ~ ( member_a @ P @ point_set )
=> ( ( design108908007054065099oint_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ P )
= point_set ) ) ).
% add_delete_point_inv
thf(fact_30_sys__block__sizes__obtain__bl,axiom,
! [X: nat] :
( ( member_nat @ X @ ( design1769254222028858111izes_a @ block_collection ) )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ ( set_mset_set_a @ block_collection ) )
& ( ( finite_card_a @ X2 )
= X ) ) ) ).
% sys_block_sizes_obtain_bl
thf(fact_31_sys__block__sizes__in,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( member_nat @ ( finite_card_a @ Bl2 ) @ ( design1769254222028858111izes_a @ block_collection ) ) ) ).
% sys_block_sizes_in
thf(fact_32_complement__finite,axiom,
design9187838744727572296stem_a @ point_set @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ).
% complement_finite
thf(fact_33_delete__block__fin__incidence__sys,axiom,
! [B: set_a] : ( design9187838744727572296stem_a @ point_set @ ( design1146539425385464078lock_a @ block_collection @ B ) ) ).
% delete_block_fin_incidence_sys
thf(fact_34_b__non__zero__imp__v__non__zero,axiom,
( ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ point_set ) ) ) ).
% b_non_zero_imp_v_non_zero
thf(fact_35_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_36_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_37_block__complement__inv,axiom,
! [Bl2: set_a,Bl22: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( ( design6447616907850319326ment_a @ point_set @ Bl2 )
= Bl22 )
=> ( ( design6447616907850319326ment_a @ point_set @ Bl22 )
= Bl2 ) ) ) ).
% block_complement_inv
thf(fact_38_del__block__b_I2_J,axiom,
! [Bl2: set_a] :
( ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ block_collection @ Bl2 ) )
= ( size_s6566526139600085008_set_a @ block_collection ) ) ) ).
% del_block_b(2)
thf(fact_39_delete__invalid__block__eq,axiom,
! [B: set_a] :
( ~ ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( ( design1146539425385464078lock_a @ block_collection @ B )
= block_collection ) ) ).
% delete_invalid_block_eq
thf(fact_40_v__non__zero,axiom,
ord_less_nat @ zero_zero_nat @ ( finite_card_a @ point_set ) ).
% v_non_zero
thf(fact_41_b__positive,axiom,
ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ block_collection ) ).
% b_positive
thf(fact_42_block__size__gt__0,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Bl2 ) ) ) ).
% block_size_gt_0
thf(fact_43_incomplete__alt__in,axiom,
! [Bl2: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) )
=> ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) ) ).
% incomplete_alt_in
thf(fact_44_incomplete__alt__size,axiom,
! [Bl2: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) )
=> ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) ) ) ).
% incomplete_alt_size
thf(fact_45_obtain__comp__block__orig,axiom,
! [Bl1: set_a] :
( ( member_set_a @ Bl1 @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ) )
=> ~ ! [Bl23: set_a] :
( ( member_set_a @ Bl23 @ ( set_mset_set_a @ block_collection ) )
=> ( Bl1
!= ( design6447616907850319326ment_a @ point_set @ Bl23 ) ) ) ) ).
% obtain_comp_block_orig
thf(fact_46_mem__Collect__eq,axiom,
! [A: set_a,P2: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: a,P2: a > $o] :
( ( member_a @ A @ ( collect_a @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: nat,P2: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P2 ) )
= ( P2 @ A ) ) ).
% mem_Collect_eq
thf(fact_49_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_53_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_54_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_55_block__comp__incomplete,axiom,
! [Bl2: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( design6447616907850319326ment_a @ point_set @ Bl2 ) ) ) ) ).
% block_comp_incomplete
thf(fact_56_complement__proper__design,axiom,
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) ) )
=> ( design7287791228148780576sign_a @ point_set @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ) ) ).
% complement_proper_design
thf(fact_57_add__existing__point,axiom,
! [P: a] :
( ( member_a @ P @ point_set )
=> ( ( design2964366272795260673oint_a @ point_set @ P )
= point_set ) ) ).
% add_existing_point
thf(fact_58_complement__same__b,axiom,
( ( size_s6566526139600085008_set_a @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) )
= ( size_s6566526139600085008_set_a @ block_collection ) ) ).
% complement_same_b
thf(fact_59_incomplete__alt__imp,axiom,
! [Bl2: set_a] :
( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) ) ) ) ).
% incomplete_alt_imp
thf(fact_60_simple__const__inter__block__size,axiom,
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ord_less_nat @ m @ ( finite_card_a @ Bl ) ) )
=> ( design3982635895484621246sign_a @ point_set @ block_collection ) ) ).
% simple_const_inter_block_size
thf(fact_61_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_62_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_63_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P2 @ M2 ) )
=> ( P2 @ N2 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_64_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_65_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_66_less__not__refl2,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ N @ M3 )
=> ( M3 != N ) ) ).
% less_not_refl2
thf(fact_67_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_68_nat__neq__iff,axiom,
! [M3: nat,N: nat] :
( ( M3 != N )
= ( ( ord_less_nat @ M3 @ N )
| ( ord_less_nat @ N @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_69_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P2 @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P2 @ M2 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_70_gr__implies__not0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_71_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_72_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_73_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_74_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_75_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_76_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_77_less__imp__le__nat,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_imp_le_nat
thf(fact_78_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_79_less__or__eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( ( ord_less_nat @ M3 @ N )
| ( M3 = N ) )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_80_le__neq__implies__less,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( M3 != N )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_81_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_82_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ zero_zero_nat )
=> ? [K: nat] :
( ( ord_less_eq_nat @ K @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ~ ( P2 @ I3 ) )
& ( P2 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_83_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_84_le__trans,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K2 )
=> ( ord_less_eq_nat @ I @ K2 ) ) ) ).
% le_trans
thf(fact_85_eq__imp__le,axiom,
! [M3: nat,N: nat] :
( ( M3 = N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% eq_imp_le
thf(fact_86_le__antisym,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( ord_less_eq_nat @ N @ M3 )
=> ( M3 = N ) ) ) ).
% le_antisym
thf(fact_87_nat__le__linear,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
| ( ord_less_eq_nat @ N @ M3 ) ) ).
% nat_le_linear
thf(fact_88_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B: nat] :
( ( P2 @ K2 )
=> ( ! [Y2: nat] :
( ( P2 @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P2 @ X2 )
& ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_89_size__neq__size__imp__neq,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ X )
!= ( size_s6566526139600085008_set_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_90_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_91_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_92_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_93_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_94_del__block__proper,axiom,
! [Bl2: set_a] :
( ( ord_less_nat @ one_one_nat @ ( size_s6566526139600085008_set_a @ block_collection ) )
=> ( design7287791228148780576sign_a @ point_set @ ( design1146539425385464078lock_a @ block_collection @ Bl2 ) ) ) ).
% del_block_proper
thf(fact_95_block__comp__incomplete__nempty,axiom,
! [Bl2: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) )
=> ( ( design6447616907850319326ment_a @ point_set @ Bl2 )
!= bot_bot_set_a ) ) ).
% block_comp_incomplete_nempty
thf(fact_96_finite__incidence__system_Oblock__comp__incomplete,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design2875492832550762736nt_nat @ Point_set @ Bl2 ) ) ) ) ) ).
% finite_incidence_system.block_comp_incomplete
thf(fact_97_finite__incidence__system_Oblock__comp__incomplete,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( design6447616907850319326ment_a @ Point_set @ Bl2 ) ) ) ) ) ).
% finite_incidence_system.block_comp_incomplete
thf(fact_98_complement__design,axiom,
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) ) )
=> ( design_design_a @ point_set @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ) ) ).
% complement_design
thf(fact_99_proper__design_Ocomplement__proper__design,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design435815215503836206gn_nat @ Point_set @ Block_collection )
=> ( ! [Bl: set_nat] :
( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) ) ) )
=> ( design435815215503836206gn_nat @ Point_set @ ( design5569578106646884273ks_nat @ Point_set @ Block_collection ) ) ) ) ).
% proper_design.complement_proper_design
thf(fact_100_proper__design_Ocomplement__proper__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) ) )
=> ( design7287791228148780576sign_a @ Point_set @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) ) ) ).
% proper_design.complement_proper_design
thf(fact_101_incomplete__block__proper__subset,axiom,
! [Bl2: set_a] :
( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) ) )
=> ( ord_less_set_a @ Bl2 @ point_set ) ) ).
% incomplete_block_proper_subset
thf(fact_102_mset__size__ne0__set__card,axiom,
! [A2: multiset_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_multiset_a @ A2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( set_mset_a @ A2 ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_103_mset__size__ne0__set__card,axiom,
! [A2: multiset_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s5917832649809541300et_nat @ A2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( set_mset_nat @ A2 ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_104_mset__size__ne0__set__card,axiom,
! [A2: multiset_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ A2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ ( set_mset_set_a @ A2 ) ) ) ) ).
% mset_size_ne0_set_card
thf(fact_105_del__add__block__inv,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ block_collection @ Bl2 ) @ Bl2 )
= block_collection ) ) ).
% del_add_block_inv
thf(fact_106_del__invalid__add__block__eq,axiom,
! [Bl2: set_a] :
( ~ ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( design4001997691126659652lock_a @ ( design1146539425385464078lock_a @ block_collection @ Bl2 ) @ Bl2 )
= ( design4001997691126659652lock_a @ block_collection @ Bl2 ) ) ) ).
% del_invalid_add_block_eq
thf(fact_107_block__sizes__non__empty,axiom,
( ( block_collection != zero_z5079479921072680283_set_a )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( design1769254222028858111izes_a @ block_collection ) ) ) ) ).
% block_sizes_non_empty
thf(fact_108_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_109_design__points__nempty,axiom,
point_set != bot_bot_set_a ).
% design_points_nempty
thf(fact_110_design__blocks__nempty,axiom,
block_collection != zero_z5079479921072680283_set_a ).
% design_blocks_nempty
thf(fact_111_blocks__nempty__alt,axiom,
! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ block_collection ) )
=> ( X4 != bot_bot_set_a ) ) ).
% blocks_nempty_alt
thf(fact_112_blocks__nempty,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( Bl2 != bot_bot_set_a ) ) ).
% blocks_nempty
thf(fact_113_block__set__nempty__imp__block__ex,axiom,
( ( block_collection != zero_z5079479921072680283_set_a )
=> ? [Bl: set_a] : ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) ) ) ).
% block_set_nempty_imp_block_ex
thf(fact_114_wf__design,axiom,
design_design_a @ point_set @ block_collection ).
% wf_design
thf(fact_115_block__set__nempty__imp__points,axiom,
( ( block_collection != zero_z5079479921072680283_set_a )
=> ( point_set != bot_bot_set_a ) ) ).
% block_set_nempty_imp_points
thf(fact_116_delete__block__design,axiom,
! [Bl2: set_a] : ( design_design_a @ point_set @ ( design1146539425385464078lock_a @ block_collection @ Bl2 ) ) ).
% delete_block_design
thf(fact_117_add__point__design,axiom,
! [P: a] : ( design_design_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ block_collection ) ).
% add_point_design
thf(fact_118_strong__del__point__design,axiom,
! [P: a] : ( design_design_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) ).
% strong_del_point_design
thf(fact_119_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_120_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_121_proper__designII,axiom,
! [V: set_a,B2: multiset_set_a] :
( ( design_design_a @ V @ B2 )
=> ( ( B2 != zero_z5079479921072680283_set_a )
=> ( design7287791228148780576sign_a @ V @ B2 ) ) ) ).
% proper_designII
thf(fact_122_designI,axiom,
( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ block_collection ) )
=> ( B3 != bot_bot_set_a ) )
=> ( ( block_collection != zero_z5079479921072680283_set_a )
=> ( ( point_set != bot_bot_set_a )
=> ( design_design_a @ point_set @ block_collection ) ) ) ) ).
% designI
thf(fact_123_n__inter__num__zero,axiom,
! [B1: set_a,B22: set_a] :
( ( member_set_a @ B1 @ ( set_mset_set_a @ block_collection ) )
=> ( ( member_set_a @ B22 @ ( set_mset_set_a @ block_collection ) )
=> ( ( design735257067508376852mber_a @ B1 @ zero_zero_nat @ B22 )
= one_one_nat ) ) ) ).
% n_inter_num_zero
thf(fact_124_design_Oblocks__nempty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( Bl2 != bot_bot_set_a ) ) ) ).
% design.blocks_nempty
thf(fact_125_design_Oblocks__nempty,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( Bl2 != bot_bot_set_nat ) ) ) ).
% design.blocks_nempty
thf(fact_126_design_Oblocks__nempty__alt,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ Block_collection ) )
=> ( X4 != bot_bot_set_a ) ) ) ).
% design.blocks_nempty_alt
thf(fact_127_design_Oblocks__nempty__alt,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ! [X4: set_nat] :
( ( member_set_nat @ X4 @ ( set_mset_set_nat @ Block_collection ) )
=> ( X4 != bot_bot_set_nat ) ) ) ).
% design.blocks_nempty_alt
thf(fact_128_mset__nempty__set__nempty,axiom,
! [A2: multiset_set_a] :
( ( A2 != zero_z5079479921072680283_set_a )
= ( ( set_mset_set_a @ A2 )
!= bot_bot_set_set_a ) ) ).
% mset_nempty_set_nempty
thf(fact_129_mset__nempty__set__nempty,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
= ( ( set_mset_a @ A2 )
!= bot_bot_set_a ) ) ).
% mset_nempty_set_nempty
thf(fact_130_mset__nempty__set__nempty,axiom,
! [A2: multiset_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
= ( ( set_mset_nat @ A2 )
!= bot_bot_set_nat ) ) ).
% mset_nempty_set_nempty
thf(fact_131_simple__design_Oaxioms_I1_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design3982635895484621246sign_a @ Point_set @ Block_collection )
=> ( design_design_a @ Point_set @ Block_collection ) ) ).
% simple_design.axioms(1)
thf(fact_132_design_Owf__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ Point_set @ Block_collection ) ) ).
% design.wf_design
thf(fact_133_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_134_design_Oblock__set__nempty__imp__points,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( Block_collection != zero_z5079479921072680283_set_a )
=> ( Point_set != bot_bot_set_a ) ) ) ).
% design.block_set_nempty_imp_points
thf(fact_135_design_Oblock__set__nempty__imp__points,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( Block_collection != zero_z3157962936165190495et_nat )
=> ( Point_set != bot_bot_set_nat ) ) ) ).
% design.block_set_nempty_imp_points
thf(fact_136_finite__incidence__system_OdesignI,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ! [B3: set_nat] :
( ( member_set_nat @ B3 @ ( set_mset_set_nat @ Block_collection ) )
=> ( B3 != bot_bot_set_nat ) )
=> ( ( Block_collection != zero_z3157962936165190495et_nat )
=> ( ( Point_set != bot_bot_set_nat )
=> ( design_design_nat @ Point_set @ Block_collection ) ) ) ) ) ).
% finite_incidence_system.designI
thf(fact_137_finite__incidence__system_OdesignI,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ Block_collection ) )
=> ( B3 != bot_bot_set_a ) )
=> ( ( Block_collection != zero_z5079479921072680283_set_a )
=> ( ( Point_set != bot_bot_set_a )
=> ( design_design_a @ Point_set @ Block_collection ) ) ) ) ) ).
% finite_incidence_system.designI
thf(fact_138_design_Oaxioms_I1_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ Point_set @ Block_collection ) ) ).
% design.axioms(1)
thf(fact_139_proper__design_Oaxioms_I1_J,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( design_design_a @ Point_set @ Block_collection ) ) ).
% proper_design.axioms(1)
thf(fact_140_proper__design_Odesign__points__nempty,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design435815215503836206gn_nat @ Point_set @ Block_collection )
=> ( Point_set != bot_bot_set_nat ) ) ).
% proper_design.design_points_nempty
thf(fact_141_proper__design_Odesign__points__nempty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( Point_set != bot_bot_set_a ) ) ).
% proper_design.design_points_nempty
thf(fact_142_proper__design_Odesign__blocks__nempty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( Block_collection != zero_z5079479921072680283_set_a ) ) ).
% proper_design.design_blocks_nempty
thf(fact_143_design_Oblock__size__lt__v,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ Point_set ) ) ) ) ).
% design.block_size_lt_v
thf(fact_144_design_Oblock__size__lt__v,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ Point_set ) ) ) ) ).
% design.block_size_lt_v
thf(fact_145_design_Ov__eq0__imp__b__eq__0,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( ( finite_card_nat @ Point_set )
= zero_zero_nat )
=> ( ( size_s7462436076474991978et_nat @ Block_collection )
= zero_zero_nat ) ) ) ).
% design.v_eq0_imp_b_eq_0
thf(fact_146_design_Ov__eq0__imp__b__eq__0,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( ( finite_card_a @ Point_set )
= zero_zero_nat )
=> ( ( size_s6566526139600085008_set_a @ Block_collection )
= zero_zero_nat ) ) ) ).
% design.v_eq0_imp_b_eq_0
thf(fact_147_design_Oproper__designI,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( ( size_s6566526139600085008_set_a @ Block_collection )
!= zero_zero_nat )
=> ( design7287791228148780576sign_a @ Point_set @ Block_collection ) ) ) ).
% design.proper_designI
thf(fact_148_design_Oblock__size__gt__0,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ Bl2 ) ) ) ) ).
% design.block_size_gt_0
thf(fact_149_design_Oblock__size__gt__0,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Bl2 ) ) ) ) ).
% design.block_size_gt_0
thf(fact_150_design_Ob__non__zero__imp__v__non__zero,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s7462436076474991978et_nat @ Block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ Point_set ) ) ) ) ).
% design.b_non_zero_imp_v_non_zero
thf(fact_151_design_Ob__non__zero__imp__v__non__zero,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ Block_collection ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Point_set ) ) ) ) ).
% design.b_non_zero_imp_v_non_zero
thf(fact_152_finite__incidence__system_Oincomplete__block__proper__subset,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) ) )
=> ( ord_less_set_nat @ Bl2 @ Point_set ) ) ) ).
% finite_incidence_system.incomplete_block_proper_subset
thf(fact_153_finite__incidence__system_Oincomplete__block__proper__subset,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) ) )
=> ( ord_less_set_a @ Bl2 @ Point_set ) ) ) ).
% finite_incidence_system.incomplete_block_proper_subset
thf(fact_154_design_Ocomplement__design,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ! [Bl: set_nat] :
( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_nat @ Bl ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) ) ) )
=> ( design_design_nat @ Point_set @ ( design5569578106646884273ks_nat @ Point_set @ Block_collection ) ) ) ) ).
% design.complement_design
thf(fact_155_design_Ocomplement__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ord_less_nat @ ( finite_card_a @ Bl ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) ) )
=> ( design_design_a @ Point_set @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) ) ) ).
% design.complement_design
thf(fact_156_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_157_zero__reorient,axiom,
! [X: multiset_set_a] :
( ( zero_z5079479921072680283_set_a = X )
= ( X = zero_z5079479921072680283_set_a ) ) ).
% zero_reorient
thf(fact_158_proper__design_Ois__proper,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( design7287791228148780576sign_a @ Point_set @ Block_collection ) ) ).
% proper_design.is_proper
thf(fact_159_incidence__system_Oblock__complement_Ocong,axiom,
design6447616907850319326ment_a = design6447616907850319326ment_a ).
% incidence_system.block_complement.cong
thf(fact_160_incidence__system_Ocomplement__blocks_Ocong,axiom,
design8640656491286871389ocks_a = design8640656491286871389ocks_a ).
% incidence_system.complement_blocks.cong
thf(fact_161_finite__incidence__system_Oblock__comp__incomplete__nempty,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ Point_set ) )
& ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) ) )
=> ( ( design2875492832550762736nt_nat @ Point_set @ Bl2 )
!= bot_bot_set_nat ) ) ) ).
% finite_incidence_system.block_comp_incomplete_nempty
thf(fact_162_finite__incidence__system_Oblock__comp__incomplete__nempty,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ( ord_less_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ Point_set ) )
& ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) ) )
=> ( ( design6447616907850319326ment_a @ Point_set @ Bl2 )
!= bot_bot_set_a ) ) ) ).
% finite_incidence_system.block_comp_incomplete_nempty
thf(fact_163_incidence__system_Osys__block__sizes_Ocong,axiom,
design1769254222028858111izes_a = design1769254222028858111izes_a ).
% incidence_system.sys_block_sizes.cong
thf(fact_164_incidence__system_Odesign__support_Ocong,axiom,
design5397942185814921632port_a = design5397942185814921632port_a ).
% incidence_system.design_support.cong
thf(fact_165_proper__design_Odel__block__proper,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ one_one_nat @ ( size_s6566526139600085008_set_a @ Block_collection ) )
=> ( design7287791228148780576sign_a @ Point_set @ ( design1146539425385464078lock_a @ Block_collection @ Bl2 ) ) ) ) ).
% proper_design.del_block_proper
thf(fact_166_incidence__system_Oincident_Ocong,axiom,
design3210447939978979927dent_a = design3210447939978979927dent_a ).
% incidence_system.incident.cong
thf(fact_167_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_168_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_169_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_170_gr__implies__not__zero,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_171_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_172_finite__incidence__system_Ocomplement__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ Point_set @ ( design8640656491286871389ocks_a @ Point_set @ Block_collection ) ) ) ).
% finite_incidence_system.complement_finite
thf(fact_173_const__intersect__design_Osimple__const__inter__block__size,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,M: nat] :
( ( design137120128173859224gn_nat @ Point_set @ Block_collection @ M )
=> ( ! [Bl: set_nat] :
( ( member_set_nat @ Bl @ ( set_mset_set_nat @ Block_collection ) )
=> ( ord_less_nat @ M @ ( finite_card_nat @ Bl ) ) )
=> ( design7861764274488435984gn_nat @ Point_set @ Block_collection ) ) ) ).
% const_intersect_design.simple_const_inter_block_size
thf(fact_174_const__intersect__design_Osimple__const__inter__block__size,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,M: nat] :
( ( design9190424834980853558sign_a @ Point_set @ Block_collection @ M )
=> ( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_nat @ M @ ( finite_card_a @ Bl ) ) )
=> ( design3982635895484621246sign_a @ Point_set @ Block_collection ) ) ) ).
% const_intersect_design.simple_const_inter_block_size
thf(fact_175_elem__exists__non__empty__set,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
=> ~ ! [X2: set_a] :
~ ( member_set_a @ X2 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_176_elem__exists__non__empty__set,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ~ ! [X2: a] :
~ ( member_a @ X2 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_177_elem__exists__non__empty__set,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ~ ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) ) ).
% elem_exists_non_empty_set
thf(fact_178_proper__design_Ob__non__zero,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ( size_s6566526139600085008_set_a @ Block_collection )
!= zero_zero_nat ) ) ).
% proper_design.b_non_zero
thf(fact_179_proper__design_Ov__non__zero,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design435815215503836206gn_nat @ Point_set @ Block_collection )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ Point_set ) ) ) ).
% proper_design.v_non_zero
thf(fact_180_proper__design_Ov__non__zero,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ Point_set ) ) ) ).
% proper_design.v_non_zero
thf(fact_181_finite__incidence__system_Oblock__size__lt__order,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ Bl2 ) @ ( finite_card_nat @ Point_set ) ) ) ) ).
% finite_incidence_system.block_size_lt_order
thf(fact_182_finite__incidence__system_Oblock__size__lt__order,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ Bl2 ) @ ( finite_card_a @ Point_set ) ) ) ) ).
% finite_incidence_system.block_size_lt_order
thf(fact_183_proper__design_Ob__positive,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ Block_collection ) ) ) ).
% proper_design.b_positive
thf(fact_184_size__empty,axiom,
( ( size_s6566526139600085008_set_a @ zero_z5079479921072680283_set_a )
= zero_zero_nat ) ).
% size_empty
thf(fact_185_size__eq__0__iff__empty,axiom,
! [M5: multiset_set_a] :
( ( ( size_s6566526139600085008_set_a @ M5 )
= zero_zero_nat )
= ( M5 = zero_z5079479921072680283_set_a ) ) ).
% size_eq_0_iff_empty
thf(fact_186_set__mset__empty,axiom,
( ( set_mset_set_a @ zero_z5079479921072680283_set_a )
= bot_bot_set_set_a ) ).
% set_mset_empty
thf(fact_187_set__mset__empty,axiom,
( ( set_mset_a @ zero_zero_multiset_a )
= bot_bot_set_a ) ).
% set_mset_empty
thf(fact_188_set__mset__empty,axiom,
( ( set_mset_nat @ zero_z7348594199698428585et_nat )
= bot_bot_set_nat ) ).
% set_mset_empty
thf(fact_189_set__mset__eq__empty__iff,axiom,
! [M5: multiset_set_a] :
( ( ( set_mset_set_a @ M5 )
= bot_bot_set_set_a )
= ( M5 = zero_z5079479921072680283_set_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_190_set__mset__eq__empty__iff,axiom,
! [M5: multiset_a] :
( ( ( set_mset_a @ M5 )
= bot_bot_set_a )
= ( M5 = zero_zero_multiset_a ) ) ).
% set_mset_eq_empty_iff
thf(fact_191_set__mset__eq__empty__iff,axiom,
! [M5: multiset_nat] :
( ( ( set_mset_nat @ M5 )
= bot_bot_set_nat )
= ( M5 = zero_z7348594199698428585et_nat ) ) ).
% set_mset_eq_empty_iff
thf(fact_192_card_Oempty,axiom,
( ( finite_card_a @ bot_bot_set_a )
= zero_zero_nat ) ).
% card.empty
thf(fact_193_card_Oempty,axiom,
( ( finite_card_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% card.empty
thf(fact_194_add__block__design__cond,axiom,
! [Bl2: set_a] :
( ( ord_less_eq_set_a @ Bl2 @ point_set )
=> ( ( Bl2 != bot_bot_set_a )
=> ( design_design_a @ point_set @ ( design4001997691126659652lock_a @ block_collection @ Bl2 ) ) ) ) ).
% add_block_design_cond
thf(fact_195_empty__inter__implies__rep__one,axiom,
! [X: a] :
( ( m = zero_zero_nat )
=> ( ( member_a @ X @ point_set )
=> ( ord_less_eq_nat @ ( design6637022207325878697mber_a @ block_collection @ X ) @ one_one_nat ) ) ) ).
% empty_inter_implies_rep_one
thf(fact_196_const__inter__multiplicity__one,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( ord_less_nat @ m @ ( finite_card_a @ Bl2 ) )
=> ( ( count_set_a @ block_collection @ Bl2 )
= one_one_nat ) ) ) ).
% const_inter_multiplicity_one
thf(fact_197_add__block__fin__cond,axiom,
! [B: set_a] :
( ( ord_less_eq_set_a @ B @ point_set )
=> ( design9187838744727572296stem_a @ point_set @ ( design4001997691126659652lock_a @ block_collection @ B ) ) ) ).
% add_block_fin_cond
thf(fact_198_add__del__block__inv,axiom,
! [Bl2: set_a] :
( ( ord_less_eq_set_a @ Bl2 @ point_set )
=> ( ( design1146539425385464078lock_a @ ( design4001997691126659652lock_a @ block_collection @ Bl2 ) @ Bl2 )
= block_collection ) ) ).
% add_del_block_inv
thf(fact_199_add__block__rep__number__not__in,axiom,
! [X: a,B: set_a] :
( ~ ( member_a @ X @ B )
=> ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ block_collection @ B ) @ X )
= ( design6637022207325878697mber_a @ block_collection @ X ) ) ) ).
% add_block_rep_number_not_in
thf(fact_200_block__comp__elem__alt__left,axiom,
! [X: a,Bl2: set_a,Ps: set_a] :
( ( member_a @ X @ Bl2 )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ point_set @ Bl2 ) )
=> ~ ( member_a @ X @ Ps ) ) ) ).
% block_comp_elem_alt_left
thf(fact_201_block__comp__elem__alt__right,axiom,
! [Ps: set_a,Bl2: set_a] :
( ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ! [X2: a] :
( ( member_a @ X2 @ Ps )
=> ~ ( member_a @ X2 @ Bl2 ) )
=> ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ point_set @ Bl2 ) ) ) ) ).
% block_comp_elem_alt_right
thf(fact_202_block__complement__elem__iff,axiom,
! [Ps: set_a,Bl2: set_a] :
( ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ point_set @ Bl2 ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ Ps )
=> ~ ( member_a @ X3 @ Bl2 ) ) ) ) ) ).
% block_complement_elem_iff
thf(fact_203_block__complement__subset__points,axiom,
! [Ps: set_a,Bl2: set_a] :
( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ point_set @ Bl2 ) )
=> ( ord_less_eq_set_a @ Ps @ point_set ) ) ).
% block_complement_subset_points
thf(fact_204_block__sizes__non__empty__set,axiom,
( ( block_collection != zero_z5079479921072680283_set_a )
=> ( ( design1769254222028858111izes_a @ block_collection )
!= bot_bot_set_nat ) ) ).
% block_sizes_non_empty_set
thf(fact_205_delete__point__blocks__sub,axiom,
! [B: set_a,P: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ block_collection @ P ) ) )
=> ~ ! [Bl: set_a] :
~ ( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
& ( ord_less_eq_set_a @ B @ Bl ) ) ) ).
% delete_point_blocks_sub
thf(fact_206_wellformed,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( ord_less_eq_set_a @ B @ point_set ) ) ).
% wellformed
thf(fact_207_rep__number__non__zero__system__point,axiom,
! [X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ block_collection @ X ) )
=> ( member_a @ X @ point_set ) ) ).
% rep_number_non_zero_system_point
thf(fact_208_complete__block__all__subsets,axiom,
! [Bl2: set_a,Ps: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( ( finite_card_a @ Bl2 )
= ( finite_card_a @ point_set ) )
=> ( ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ord_less_eq_set_a @ Ps @ Bl2 ) ) ) ) ).
% complete_block_all_subsets
thf(fact_209_complement__blocks__wf,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ) )
=> ( ord_less_eq_set_a @ Bl2 @ point_set ) ) ).
% complement_blocks_wf
thf(fact_210_point__in__block__rep__min__iff,axiom,
! [X: a] :
( ( member_a @ X @ point_set )
=> ? [Bl: set_a] :
( ( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
& ( member_a @ X @ Bl ) )
= ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ block_collection @ X ) ) ) ) ).
% point_in_block_rep_min_iff
thf(fact_211_count__empty,axiom,
! [A: set_a] :
( ( count_set_a @ zero_z5079479921072680283_set_a @ A )
= zero_zero_nat ) ).
% count_empty
thf(fact_212_count__greater__zero__iff,axiom,
! [M5: multiset_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_a @ M5 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ).
% count_greater_zero_iff
thf(fact_213_count__greater__zero__iff,axiom,
! [M5: multiset_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( count_nat @ M5 @ X ) )
= ( member_nat @ X @ ( set_mset_nat @ M5 ) ) ) ).
% count_greater_zero_iff
thf(fact_214_count__greater__zero__iff,axiom,
! [M5: multiset_set_a,X: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( count_set_a @ M5 @ X ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M5 ) ) ) ).
% count_greater_zero_iff
thf(fact_215_count__greater__eq__one__iff,axiom,
! [M5: multiset_a,X: a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_a @ M5 @ X ) )
= ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_216_count__greater__eq__one__iff,axiom,
! [M5: multiset_nat,X: nat] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_nat @ M5 @ X ) )
= ( member_nat @ X @ ( set_mset_nat @ M5 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_217_count__greater__eq__one__iff,axiom,
! [M5: multiset_set_a,X: set_a] :
( ( ord_less_eq_nat @ one_one_nat @ ( count_set_a @ M5 @ X ) )
= ( member_set_a @ X @ ( set_mset_set_a @ M5 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_218_point__rep__non__existance,axiom,
! [X: a] :
( ~ ( member_a @ X @ point_set )
=> ( ( design6637022207325878697mber_a @ block_collection @ X )
= zero_zero_nat ) ) ).
% point_rep_non_existance
thf(fact_219_multiset__eq__iff,axiom,
( ( ^ [Y4: multiset_set_a,Z: multiset_set_a] : ( Y4 = Z ) )
= ( ^ [M6: multiset_set_a,N4: multiset_set_a] :
! [A3: set_a] :
( ( count_set_a @ M6 @ A3 )
= ( count_set_a @ N4 @ A3 ) ) ) ) ).
% multiset_eq_iff
thf(fact_220_multiset__eqI,axiom,
! [A2: multiset_set_a,B2: multiset_set_a] :
( ! [X2: set_a] :
( ( count_set_a @ A2 @ X2 )
= ( count_set_a @ B2 @ X2 ) )
=> ( A2 = B2 ) ) ).
% multiset_eqI
thf(fact_221_count__inject,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ( count_set_a @ X )
= ( count_set_a @ Y ) )
= ( X = Y ) ) ).
% count_inject
thf(fact_222_count__eq__zero__iff,axiom,
! [M5: multiset_a,X: a] :
( ( ( count_a @ M5 @ X )
= zero_zero_nat )
= ( ~ ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_223_count__eq__zero__iff,axiom,
! [M5: multiset_nat,X: nat] :
( ( ( count_nat @ M5 @ X )
= zero_zero_nat )
= ( ~ ( member_nat @ X @ ( set_mset_nat @ M5 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_224_count__eq__zero__iff,axiom,
! [M5: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M5 @ X )
= zero_zero_nat )
= ( ~ ( member_set_a @ X @ ( set_mset_set_a @ M5 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_225_count__inI,axiom,
! [M5: multiset_a,X: a] :
( ( ( count_a @ M5 @ X )
!= zero_zero_nat )
=> ( member_a @ X @ ( set_mset_a @ M5 ) ) ) ).
% count_inI
thf(fact_226_count__inI,axiom,
! [M5: multiset_nat,X: nat] :
( ( ( count_nat @ M5 @ X )
!= zero_zero_nat )
=> ( member_nat @ X @ ( set_mset_nat @ M5 ) ) ) ).
% count_inI
thf(fact_227_count__inI,axiom,
! [M5: multiset_set_a,X: set_a] :
( ( ( count_set_a @ M5 @ X )
!= zero_zero_nat )
=> ( member_set_a @ X @ ( set_mset_set_a @ M5 ) ) ) ).
% count_inI
thf(fact_228_zero__multiset_Orep__eq,axiom,
( ( count_set_a @ zero_z5079479921072680283_set_a )
= ( ^ [A3: set_a] : zero_zero_nat ) ) ).
% zero_multiset.rep_eq
thf(fact_229_set__count__size__min,axiom,
! [N: nat,A2: multiset_set_a,A: set_a] :
( ( ord_less_eq_nat @ N @ ( count_set_a @ A2 @ A ) )
=> ( ord_less_eq_nat @ N @ ( size_s6566526139600085008_set_a @ A2 ) ) ) ).
% set_count_size_min
thf(fact_230_max__point__rep,axiom,
! [B2: multiset_set_a,X: a] : ( ord_less_eq_nat @ ( design6637022207325878697mber_a @ B2 @ X ) @ ( size_s6566526139600085008_set_a @ B2 ) ) ).
% max_point_rep
thf(fact_231_rep__number__g0__exists,axiom,
! [B2: multiset_set_set_a,X: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( design5008467512594872073_set_a @ B2 @ X ) )
=> ~ ! [B3: set_set_a] :
( ( member_set_set_a @ B3 @ ( set_mset_set_set_a @ B2 ) )
=> ~ ( member_set_a @ X @ B3 ) ) ) ).
% rep_number_g0_exists
thf(fact_232_rep__number__g0__exists,axiom,
! [B2: multiset_set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( design3571518413069006949er_nat @ B2 @ X ) )
=> ~ ! [B3: set_nat] :
( ( member_set_nat @ B3 @ ( set_mset_set_nat @ B2 ) )
=> ~ ( member_nat @ X @ B3 ) ) ) ).
% rep_number_g0_exists
thf(fact_233_rep__number__g0__exists,axiom,
! [B2: multiset_set_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ ( design6637022207325878697mber_a @ B2 @ X ) )
=> ~ ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ B2 ) )
=> ~ ( member_a @ X @ B3 ) ) ) ).
% rep_number_g0_exists
thf(fact_234_finite__incidence__system_Ocomplete__block__all__subsets,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat,Ps: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ( finite_card_nat @ Bl2 )
= ( finite_card_nat @ Point_set ) )
=> ( ( ord_less_eq_set_nat @ Ps @ Point_set )
=> ( ord_less_eq_set_nat @ Ps @ Bl2 ) ) ) ) ) ).
% finite_incidence_system.complete_block_all_subsets
thf(fact_235_finite__incidence__system_Ocomplete__block__all__subsets,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a,Ps: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ( finite_card_a @ Bl2 )
= ( finite_card_a @ Point_set ) )
=> ( ( ord_less_eq_set_a @ Ps @ Point_set )
=> ( ord_less_eq_set_a @ Ps @ Bl2 ) ) ) ) ) ).
% finite_incidence_system.complete_block_all_subsets
thf(fact_236_const__intersect__design_Oempty__inter__implies__rep__one,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,M: nat,X: set_a] :
( ( design2124024510932123798_set_a @ Point_set @ Block_collection @ M )
=> ( ( M = zero_zero_nat )
=> ( ( member_set_a @ X @ Point_set )
=> ( ord_less_eq_nat @ ( design5008467512594872073_set_a @ Block_collection @ X ) @ one_one_nat ) ) ) ) ).
% const_intersect_design.empty_inter_implies_rep_one
thf(fact_237_const__intersect__design_Oempty__inter__implies__rep__one,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,M: nat,X: nat] :
( ( design137120128173859224gn_nat @ Point_set @ Block_collection @ M )
=> ( ( M = zero_zero_nat )
=> ( ( member_nat @ X @ Point_set )
=> ( ord_less_eq_nat @ ( design3571518413069006949er_nat @ Block_collection @ X ) @ one_one_nat ) ) ) ) ).
% const_intersect_design.empty_inter_implies_rep_one
thf(fact_238_const__intersect__design_Oempty__inter__implies__rep__one,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,M: nat,X: a] :
( ( design9190424834980853558sign_a @ Point_set @ Block_collection @ M )
=> ( ( M = zero_zero_nat )
=> ( ( member_a @ X @ Point_set )
=> ( ord_less_eq_nat @ ( design6637022207325878697mber_a @ Block_collection @ X ) @ one_one_nat ) ) ) ) ).
% const_intersect_design.empty_inter_implies_rep_one
thf(fact_239_const__intersect__design_Oconst__inter__multiplicity__one,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,M: nat,Bl2: set_nat] :
( ( design137120128173859224gn_nat @ Point_set @ Block_collection @ M )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( ord_less_nat @ M @ ( finite_card_nat @ Bl2 ) )
=> ( ( count_set_nat @ Block_collection @ Bl2 )
= one_one_nat ) ) ) ) ).
% const_intersect_design.const_inter_multiplicity_one
thf(fact_240_const__intersect__design_Oconst__inter__multiplicity__one,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,M: nat,Bl2: set_a] :
( ( design9190424834980853558sign_a @ Point_set @ Block_collection @ M )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( ord_less_nat @ M @ ( finite_card_a @ Bl2 ) )
=> ( ( count_set_a @ Block_collection @ Bl2 )
= one_one_nat ) ) ) ) ).
% const_intersect_design.const_inter_multiplicity_one
thf(fact_241_multiset__nonemptyE,axiom,
! [A2: multiset_a] :
( ( A2 != zero_zero_multiset_a )
=> ~ ! [X2: a] :
~ ( member_a @ X2 @ ( set_mset_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_242_multiset__nonemptyE,axiom,
! [A2: multiset_nat] :
( ( A2 != zero_z7348594199698428585et_nat )
=> ~ ! [X2: nat] :
~ ( member_nat @ X2 @ ( set_mset_nat @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_243_multiset__nonemptyE,axiom,
! [A2: multiset_set_a] :
( ( A2 != zero_z5079479921072680283_set_a )
=> ~ ! [X2: set_a] :
~ ( member_set_a @ X2 @ ( set_mset_set_a @ A2 ) ) ) ).
% multiset_nonemptyE
thf(fact_244_design_On__inter__num__zero,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B1: set_a,B22: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B1 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( member_set_a @ B22 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( design735257067508376852mber_a @ B1 @ zero_zero_nat @ B22 )
= one_one_nat ) ) ) ) ).
% design.n_inter_num_zero
thf(fact_245_nonempty__has__size,axiom,
! [S2: multiset_set_a] :
( ( S2 != zero_z5079479921072680283_set_a )
= ( ord_less_nat @ zero_zero_nat @ ( size_s6566526139600085008_set_a @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_246_simple__not__multiplicity,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( ( ord_less_nat @ one_one_nat @ ( count_set_a @ block_collection @ B ) )
=> ~ ( design1338723777345758283stem_a @ point_set @ block_collection ) ) ) ).
% simple_not_multiplicity
thf(fact_247_replication__numbers__non__empty,axiom,
( ( point_set != bot_bot_set_a )
=> ( ( design8835372594653258411bers_a @ point_set @ block_collection )
!= bot_bot_set_nat ) ) ).
% replication_numbers_non_empty
thf(fact_248_obtain__point__with__rep,axiom,
! [R: nat] :
( ( member_nat @ R @ ( design8835372594653258411bers_a @ point_set @ block_collection ) )
=> ? [X2: a] :
( ( member_a @ X2 @ point_set )
& ( ( design6637022207325878697mber_a @ block_collection @ X2 )
= R ) ) ) ).
% obtain_point_with_rep
thf(fact_249_point__rep__number__in__set,axiom,
! [X: a] :
( ( member_a @ X @ point_set )
=> ( member_nat @ ( design6637022207325878697mber_a @ block_collection @ X ) @ ( design8835372594653258411bers_a @ point_set @ block_collection ) ) ) ).
% point_rep_number_in_set
thf(fact_250_wf__design__iff,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( design_design_a @ point_set @ block_collection )
= ( ( ord_less_eq_set_a @ Bl2 @ point_set )
& ( finite_finite_a @ point_set )
& ( Bl2 != bot_bot_set_a ) ) ) ) ).
% wf_design_iff
thf(fact_251_complement__rep__number,axiom,
! [X: a,R: nat] :
( ( member_a @ X @ point_set )
=> ( ( ( design6637022207325878697mber_a @ block_collection @ X )
= R )
=> ( ( design6637022207325878697mber_a @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) @ X )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ block_collection ) @ R ) ) ) ) ).
% complement_rep_number
thf(fact_252_psubsetI,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_253_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_254_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_255_finite__sets,axiom,
finite_finite_a @ point_set ).
% finite_sets
thf(fact_256_finite__blocks,axiom,
! [B: set_a] :
( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( finite_finite_a @ B ) ) ).
% finite_blocks
thf(fact_257_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_258_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_259_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_260_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X3: set_a] :
~ ( member_set_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_261_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_262_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_263_Collect__empty__eq,axiom,
! [P2: a > $o] :
( ( ( collect_a @ P2 )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_264_Collect__empty__eq,axiom,
! [P2: nat > $o] :
( ( ( collect_nat @ P2 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_265_empty__Collect__eq,axiom,
! [P2: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P2 ) )
= ( ! [X3: a] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_266_empty__Collect__eq,axiom,
! [P2: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P2 ) )
= ( ! [X3: nat] :
~ ( P2 @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_267_subsetI,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( member_set_a @ X2 @ B2 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_268_subsetI,axiom,
! [A2: set_nat,B2: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( member_nat @ X2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_269_subsetI,axiom,
! [A2: set_a,B2: set_a] :
( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( member_a @ X2 @ B2 ) )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% subsetI
thf(fact_270_subset__antisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_271_del__point__order,axiom,
! [P: a] :
( ( member_a @ P @ point_set )
=> ( ( finite_card_a @ ( design108908007054065099oint_a @ point_set @ P ) )
= ( minus_minus_nat @ ( finite_card_a @ point_set ) @ one_one_nat ) ) ) ).
% del_point_order
thf(fact_272_block__complement__size,axiom,
! [B: set_a] :
( ( ord_less_eq_set_a @ B @ point_set )
=> ( ( finite_card_a @ ( design6447616907850319326ment_a @ point_set @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ point_set ) @ ( finite_card_a @ B ) ) ) ) ).
% block_complement_size
thf(fact_273_del__block__b_I1_J,axiom,
! [Bl2: set_a] :
( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( size_s6566526139600085008_set_a @ ( design1146539425385464078lock_a @ block_collection @ Bl2 ) )
= ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ block_collection ) @ one_one_nat ) ) ) ).
% del_block_b(1)
thf(fact_274_zero__diff,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ A )
= zero_z5079479921072680283_set_a ) ).
% zero_diff
thf(fact_275_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_276_diff__zero,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% diff_zero
thf(fact_277_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_278_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ A )
= zero_z5079479921072680283_set_a ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_279_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_280_subset__empty,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_281_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_282_finite__set__mset,axiom,
! [M5: multiset_a] : ( finite_finite_a @ ( set_mset_a @ M5 ) ) ).
% finite_set_mset
thf(fact_283_finite__set__mset,axiom,
! [M5: multiset_nat] : ( finite_finite_nat @ ( set_mset_nat @ M5 ) ) ).
% finite_set_mset
thf(fact_284_finite__set__mset,axiom,
! [M5: multiset_set_a] : ( finite_finite_set_a @ ( set_mset_set_a @ M5 ) ) ).
% finite_set_mset
thf(fact_285_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_286_diff__self__eq__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ M3 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_287_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_288_card_Oinfinite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ A2 )
=> ( ( finite_card_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_289_card_Oinfinite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ A2 )
=> ( ( finite_card_nat @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_290_card_Oinfinite,axiom,
! [A2: set_set_a] :
( ~ ( finite_finite_set_a @ A2 )
=> ( ( finite_card_set_a @ A2 )
= zero_zero_nat ) ) ).
% card.infinite
thf(fact_291_zero__less__diff,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M3 ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% zero_less_diff
thf(fact_292_diff__is__0__eq,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% diff_is_0_eq
thf(fact_293_diff__is__0__eq_H,axiom,
! [M3: nat,N: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_294_card__0__eq,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_set_a ) ) ) ).
% card_0_eq
thf(fact_295_card__0__eq,axiom,
! [A2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_a ) ) ) ).
% card_0_eq
thf(fact_296_card__0__eq,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( A2 = bot_bot_set_nat ) ) ) ).
% card_0_eq
thf(fact_297_finite__sysI,axiom,
( ( finite_finite_a @ point_set )
=> ( design9187838744727572296stem_a @ point_set @ block_collection ) ) ).
% finite_sysI
thf(fact_298_diff__commute,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_299_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_300_incidence__system_Oreplication__numbers_Ocong,axiom,
design8835372594653258411bers_a = design8835372594653258411bers_a ).
% incidence_system.replication_numbers.cong
thf(fact_301_finite__has__minimal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ X2 @ A )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_302_finite__has__minimal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ X2 @ A )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_303_finite__has__maximal2,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ A @ A2 )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( ord_less_eq_nat @ A @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_304_finite__has__maximal2,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ A @ A2 )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ( ord_less_eq_set_a @ A @ X2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_305_finite_OemptyI,axiom,
finite_finite_set_a @ bot_bot_set_set_a ).
% finite.emptyI
thf(fact_306_finite_OemptyI,axiom,
finite_finite_a @ bot_bot_set_a ).
% finite.emptyI
thf(fact_307_finite_OemptyI,axiom,
finite_finite_nat @ bot_bot_set_nat ).
% finite.emptyI
thf(fact_308_infinite__imp__nonempty,axiom,
! [S2: set_set_a] :
( ~ ( finite_finite_set_a @ S2 )
=> ( S2 != bot_bot_set_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_309_infinite__imp__nonempty,axiom,
! [S2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ( S2 != bot_bot_set_a ) ) ).
% infinite_imp_nonempty
thf(fact_310_infinite__imp__nonempty,axiom,
! [S2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ( S2 != bot_bot_set_nat ) ) ).
% infinite_imp_nonempty
thf(fact_311_finite__subset,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( finite_finite_nat @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% finite_subset
thf(fact_312_finite__subset,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% finite_subset
thf(fact_313_finite__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( finite_finite_a @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% finite_subset
thf(fact_314_infinite__super,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ord_less_eq_set_nat @ S2 @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_super
thf(fact_315_infinite__super,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_super
thf(fact_316_infinite__super,axiom,
! [S2: set_a,T2: set_a] :
( ( ord_less_eq_set_a @ S2 @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_super
thf(fact_317_rev__finite__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( finite_finite_nat @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_318_rev__finite__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( finite_finite_set_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_319_rev__finite__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( finite_finite_a @ A2 ) ) ) ).
% rev_finite_subset
thf(fact_320_minus__nat_Odiff__0,axiom,
! [M3: nat] :
( ( minus_minus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% minus_nat.diff_0
thf(fact_321_diffs0__imp__equal,axiom,
! [M3: nat,N: nat] :
( ( ( minus_minus_nat @ M3 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M3 )
= zero_zero_nat )
=> ( M3 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_322_diff__less__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_nat @ M3 @ N )
=> ( ( ord_less_nat @ M3 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ) ).
% diff_less_mono2
thf(fact_323_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less_nat @ J @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_324_diff__le__mono2,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M3 ) ) ) ).
% diff_le_mono2
thf(fact_325_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_326_diff__le__self,axiom,
! [M3: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ).
% diff_le_self
thf(fact_327_diff__le__mono,axiom,
! [M3: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M3 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_328_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M3 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_329_le__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_330_eq__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ( minus_minus_nat @ M3 @ K2 )
= ( minus_minus_nat @ N @ K2 ) )
= ( M3 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_331_finite__psubset__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ! [B4: set_nat] :
( ( ord_less_set_nat @ B4 @ A4 )
=> ( P2 @ B4 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_332_finite__psubset__induct,axiom,
! [A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ! [B4: set_set_a] :
( ( ord_less_set_set_a @ B4 @ A4 )
=> ( P2 @ B4 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_333_finite__psubset__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ! [B4: set_a] :
( ( ord_less_set_a @ B4 @ A4 )
=> ( P2 @ B4 ) )
=> ( P2 @ A4 ) ) )
=> ( P2 @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_334_finite__incidence__system_Ofinite__sets,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( finite_finite_nat @ Point_set ) ) ).
% finite_incidence_system.finite_sets
thf(fact_335_finite__incidence__system_Ofinite__sets,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a] :
( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
=> ( finite_finite_set_a @ Point_set ) ) ).
% finite_incidence_system.finite_sets
thf(fact_336_finite__incidence__system_Ofinite__sets,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( finite_finite_a @ Point_set ) ) ).
% finite_incidence_system.finite_sets
thf(fact_337_finite__has__maximal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_338_finite__has__maximal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_339_finite__has__minimal,axiom,
! [A2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( A2 != bot_bot_set_nat )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A2 )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_340_finite__has__minimal,axiom,
! [A2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( A2 != bot_bot_set_set_a )
=> ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ! [Xa: set_a] :
( ( member_set_a @ Xa @ A2 )
=> ( ( ord_less_eq_set_a @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_341_diff__less,axiom,
! [N: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_nat @ ( minus_minus_nat @ M3 @ N ) @ M3 ) ) ) ).
% diff_less
thf(fact_342_less__diff__iff,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ K2 @ M3 )
=> ( ( ord_less_eq_nat @ K2 @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M3 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
= ( ord_less_nat @ M3 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_343_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_344_card__subset__eq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_345_card__subset__eq,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_346_card__subset__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_subset_eq
thf(fact_347_infinite__arbitrarily__large,axiom,
! [A2: set_nat,N: nat] :
( ~ ( finite_finite_nat @ A2 )
=> ? [B5: set_nat] :
( ( finite_finite_nat @ B5 )
& ( ( finite_card_nat @ B5 )
= N )
& ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_348_infinite__arbitrarily__large,axiom,
! [A2: set_set_a,N: nat] :
( ~ ( finite_finite_set_a @ A2 )
=> ? [B5: set_set_a] :
( ( finite_finite_set_a @ B5 )
& ( ( finite_card_set_a @ B5 )
= N )
& ( ord_le3724670747650509150_set_a @ B5 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_349_infinite__arbitrarily__large,axiom,
! [A2: set_a,N: nat] :
( ~ ( finite_finite_a @ A2 )
=> ? [B5: set_a] :
( ( finite_finite_a @ B5 )
& ( ( finite_card_a @ B5 )
= N )
& ( ord_less_eq_set_a @ B5 @ A2 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_350_finite__incidence__system_Ofinite__blocks,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ B @ ( set_mset_set_nat @ Block_collection ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_incidence_system.finite_blocks
thf(fact_351_finite__incidence__system_Ofinite__blocks,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,B: set_set_a] :
( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
=> ( ( member_set_set_a @ B @ ( set_mset_set_set_a @ Block_collection ) )
=> ( finite_finite_set_a @ B ) ) ) ).
% finite_incidence_system.finite_blocks
thf(fact_352_finite__incidence__system_Ofinite__blocks,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
=> ( finite_finite_a @ B ) ) ) ).
% finite_incidence_system.finite_blocks
thf(fact_353_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_354_card__eq__0__iff,axiom,
! [A2: set_set_a] :
( ( ( finite_card_set_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_set_a )
| ~ ( finite_finite_set_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_355_card__eq__0__iff,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_a )
| ~ ( finite_finite_a @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_356_card__eq__0__iff,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= zero_zero_nat )
= ( ( A2 = bot_bot_set_nat )
| ~ ( finite_finite_nat @ A2 ) ) ) ).
% card_eq_0_iff
thf(fact_357_card__subset__not__gt__card,axiom,
! [A2: set_nat,Ps: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ Ps ) )
=> ~ ( ord_less_eq_set_nat @ Ps @ A2 ) ) ) ).
% card_subset_not_gt_card
thf(fact_358_card__subset__not__gt__card,axiom,
! [A2: set_set_a,Ps: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ Ps ) )
=> ~ ( ord_le3724670747650509150_set_a @ Ps @ A2 ) ) ) ).
% card_subset_not_gt_card
thf(fact_359_card__subset__not__gt__card,axiom,
! [A2: set_a,Ps: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ Ps ) )
=> ~ ( ord_less_eq_set_a @ Ps @ A2 ) ) ) ).
% card_subset_not_gt_card
thf(fact_360_card__ge__0__finite,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
=> ( finite_finite_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_361_card__ge__0__finite,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
=> ( finite_finite_nat @ A2 ) ) ).
% card_ge_0_finite
thf(fact_362_card__ge__0__finite,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
=> ( finite_finite_set_a @ A2 ) ) ).
% card_ge_0_finite
thf(fact_363_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G: set_nat] :
( ( ord_less_eq_set_nat @ G @ F2 )
=> ( ( finite_finite_nat @ G )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_364_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_set_a,C2: nat] :
( ! [G: set_set_a] :
( ( ord_le3724670747650509150_set_a @ G @ F2 )
=> ( ( finite_finite_set_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_set_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_set_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_365_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_a,C2: nat] :
( ! [G: set_a] :
( ( ord_less_eq_set_a @ G @ F2 )
=> ( ( finite_finite_a @ G )
=> ( ord_less_eq_nat @ ( finite_card_a @ G ) @ C2 ) ) )
=> ( ( finite_finite_a @ F2 )
& ( ord_less_eq_nat @ ( finite_card_a @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_366_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_nat] :
( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S2 ) )
=> ~ ! [T3: set_nat] :
( ( ord_less_eq_set_nat @ T3 @ S2 )
=> ( ( ( finite_card_nat @ T3 )
= N )
=> ~ ( finite_finite_nat @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_367_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ S2 ) )
=> ~ ! [T3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ T3 @ S2 )
=> ( ( ( finite_card_set_a @ T3 )
= N )
=> ~ ( finite_finite_set_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_368_obtain__subset__with__card__n,axiom,
! [N: nat,S2: set_a] :
( ( ord_less_eq_nat @ N @ ( finite_card_a @ S2 ) )
=> ~ ! [T3: set_a] :
( ( ord_less_eq_set_a @ T3 @ S2 )
=> ( ( ( finite_card_a @ T3 )
= N )
=> ~ ( finite_finite_a @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_369_exists__subset__between,axiom,
! [A2: set_nat,N: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B5: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ C2 )
& ( ( finite_card_nat @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_370_exists__subset__between,axiom,
! [A2: set_set_a,N: nat,C2: set_set_a] :
( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_set_a @ C2 ) )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ C2 )
=> ( ( finite_finite_set_a @ C2 )
=> ? [B5: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B5 )
& ( ord_le3724670747650509150_set_a @ B5 @ C2 )
& ( ( finite_card_set_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_371_exists__subset__between,axiom,
! [A2: set_a,N: nat,C2: set_a] :
( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( finite_card_a @ C2 ) )
=> ( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( finite_finite_a @ C2 )
=> ? [B5: set_a] :
( ( ord_less_eq_set_a @ A2 @ B5 )
& ( ord_less_eq_set_a @ B5 @ C2 )
& ( ( finite_card_a @ B5 )
= N ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_372_card__seteq,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_373_card__seteq,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ B2 ) @ ( finite_card_set_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_374_card__seteq,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ B2 ) @ ( finite_card_a @ A2 ) )
=> ( A2 = B2 ) ) ) ) ).
% card_seteq
thf(fact_375_card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% card_mono
thf(fact_376_card__mono,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_377_card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% card_mono
thf(fact_378_psubset__card__mono,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_set_nat @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_379_psubset__card__mono,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_380_psubset__card__mono,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ).
% psubset_card_mono
thf(fact_381_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_382_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_383_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_384_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_385_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_386_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_387_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y2: set_a] :
~ ( member_set_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_388_equals0I,axiom,
! [A2: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_389_equals0I,axiom,
! [A2: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_390_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X3: set_a] : ( member_set_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_391_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_392_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_393_in__mono,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_394_in__mono,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ X @ B2 ) ) ) ).
% in_mono
thf(fact_395_in__mono,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B2 ) ) ) ).
% in_mono
thf(fact_396_subsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_397_subsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% subsetD
thf(fact_398_subsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% subsetD
thf(fact_399_equalityE,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ~ ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_400_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B6: set_set_a] :
! [X3: set_a] :
( ( member_set_a @ X3 @ A5 )
=> ( member_set_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_401_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_402_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
! [X3: a] :
( ( member_a @ X3 @ A5 )
=> ( member_a @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_403_equalityD1,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_404_equalityD2,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 = B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_405_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B6: set_set_a] :
! [T4: set_a] :
( ( member_set_a @ T4 @ A5 )
=> ( member_set_a @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_406_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
! [T4: nat] :
( ( member_nat @ T4 @ A5 )
=> ( member_nat @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_407_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
! [T4: a] :
( ( member_a @ T4 @ A5 )
=> ( member_a @ T4 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_408_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_409_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_410_subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ).
% subset_trans
thf(fact_411_set__eq__subset,axiom,
( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
& ( ord_less_eq_set_a @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_412_Collect__mono__iff,axiom,
! [P2: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
= ( ! [X3: a] :
( ( P2 @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_413_psubset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% psubset_trans
thf(fact_414_psubsetD,axiom,
! [A2: set_set_a,B2: set_set_a,C: set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_415_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_416_psubsetD,axiom,
! [A2: set_a,B2: set_a,C: a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_417_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_418_simple__design_Ointro,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( design1338723777345758283stem_a @ Point_set @ Block_collection )
=> ( design3982635895484621246sign_a @ Point_set @ Block_collection ) ) ) ).
% simple_design.intro
thf(fact_419_simple__design__def,axiom,
( design3982635895484621246sign_a
= ( ^ [Point_set2: set_a,Block_collection2: multiset_set_a] :
( ( design_design_a @ Point_set2 @ Block_collection2 )
& ( design1338723777345758283stem_a @ Point_set2 @ Block_collection2 ) ) ) ) ).
% simple_design_def
thf(fact_420_card__gt__0__iff,axiom,
! [A2: set_set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ A2 ) )
= ( ( A2 != bot_bot_set_set_a )
& ( finite_finite_set_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_421_card__gt__0__iff,axiom,
! [A2: set_a] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ A2 ) )
= ( ( A2 != bot_bot_set_a )
& ( finite_finite_a @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_422_card__gt__0__iff,axiom,
! [A2: set_nat] :
( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A2 ) )
= ( ( A2 != bot_bot_set_nat )
& ( finite_finite_nat @ A2 ) ) ) ).
% card_gt_0_iff
thf(fact_423_card__psubset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_set_nat @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_424_card__psubset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_set_set_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_425_card__psubset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ) ).
% card_psubset
thf(fact_426_design_Owf__design__iff,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Bl2: set_set_a] :
( ( design_design_set_a @ Point_set @ Block_collection )
=> ( ( member_set_set_a @ Bl2 @ ( set_mset_set_set_a @ Block_collection ) )
=> ( ( design_design_set_a @ Point_set @ Block_collection )
= ( ( ord_le3724670747650509150_set_a @ Bl2 @ Point_set )
& ( finite_finite_set_a @ Point_set )
& ( Bl2 != bot_bot_set_set_a ) ) ) ) ) ).
% design.wf_design_iff
thf(fact_427_design_Owf__design__iff,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( member_set_nat @ Bl2 @ ( set_mset_set_nat @ Block_collection ) )
=> ( ( design_design_nat @ Point_set @ Block_collection )
= ( ( ord_less_eq_set_nat @ Bl2 @ Point_set )
& ( finite_finite_nat @ Point_set )
& ( Bl2 != bot_bot_set_nat ) ) ) ) ) ).
% design.wf_design_iff
thf(fact_428_design_Owf__design__iff,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( design_design_a @ Point_set @ Block_collection )
= ( ( ord_less_eq_set_a @ Bl2 @ Point_set )
& ( finite_finite_a @ Point_set )
& ( Bl2 != bot_bot_set_a ) ) ) ) ) ).
% design.wf_design_iff
thf(fact_429_finite__incidence__system_Oblock__complement__size,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_nat @ B @ Point_set )
=> ( ( finite_card_nat @ ( design2875492832550762736nt_nat @ Point_set @ B ) )
= ( minus_minus_nat @ ( finite_card_nat @ Point_set ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% finite_incidence_system.block_complement_size
thf(fact_430_finite__incidence__system_Oblock__complement__size,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ B @ Point_set )
=> ( ( finite_card_a @ ( design6447616907850319326ment_a @ Point_set @ B ) )
= ( minus_minus_nat @ ( finite_card_a @ Point_set ) @ ( finite_card_a @ B ) ) ) ) ) ).
% finite_incidence_system.block_complement_size
thf(fact_431_simple__incidence__system_Osimple,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design1338723777345758283stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( count_set_a @ Block_collection @ Bl2 )
= one_one_nat ) ) ) ).
% simple_incidence_system.simple
thf(fact_432_simple__incidence__system_Osimple__alt__def__all,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1338723777345758283stem_a @ Point_set @ Block_collection )
=> ! [X4: set_a] :
( ( member_set_a @ X4 @ ( set_mset_set_a @ Block_collection ) )
=> ( ( count_set_a @ Block_collection @ X4 )
= one_one_nat ) ) ) ).
% simple_incidence_system.simple_alt_def_all
thf(fact_433_wf__design__implies,axiom,
! [B7: multiset_set_set_a,V: set_set_a] :
( ! [B3: set_set_a] :
( ( member_set_set_a @ B3 @ ( set_mset_set_set_a @ B7 ) )
=> ( ord_le3724670747650509150_set_a @ B3 @ V ) )
=> ( ! [B3: set_set_a] :
( ( member_set_set_a @ B3 @ ( set_mset_set_set_a @ B7 ) )
=> ( B3 != bot_bot_set_set_a ) )
=> ( ( finite_finite_set_a @ V )
=> ( ( B7 != zero_z6396401802697562811_set_a )
=> ( ( V != bot_bot_set_set_a )
=> ( design_design_set_a @ V @ B7 ) ) ) ) ) ) ).
% wf_design_implies
thf(fact_434_wf__design__implies,axiom,
! [B7: multiset_set_nat,V: set_nat] :
( ! [B3: set_nat] :
( ( member_set_nat @ B3 @ ( set_mset_set_nat @ B7 ) )
=> ( ord_less_eq_set_nat @ B3 @ V ) )
=> ( ! [B3: set_nat] :
( ( member_set_nat @ B3 @ ( set_mset_set_nat @ B7 ) )
=> ( B3 != bot_bot_set_nat ) )
=> ( ( finite_finite_nat @ V )
=> ( ( B7 != zero_z3157962936165190495et_nat )
=> ( ( V != bot_bot_set_nat )
=> ( design_design_nat @ V @ B7 ) ) ) ) ) ) ).
% wf_design_implies
thf(fact_435_wf__design__implies,axiom,
! [B7: multiset_set_a,V: set_a] :
( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ B7 ) )
=> ( ord_less_eq_set_a @ B3 @ V ) )
=> ( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ B7 ) )
=> ( B3 != bot_bot_set_a ) )
=> ( ( finite_finite_a @ V )
=> ( ( B7 != zero_z5079479921072680283_set_a )
=> ( ( V != bot_bot_set_a )
=> ( design_design_a @ V @ B7 ) ) ) ) ) ) ).
% wf_design_implies
thf(fact_436_not__psubset__empty,axiom,
! [A2: set_nat] :
~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% not_psubset_empty
thf(fact_437_not__psubset__empty,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_438_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_set_a @ A5 @ B6 )
| ( A5 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_439_subset__psubset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_440_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
& ~ ( ord_less_eq_set_a @ B6 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_441_psubset__subset__trans,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_set_a @ A2 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_442_psubset__imp__subset,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_443_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ( ord_less_eq_set_a @ A5 @ B6 )
& ( A5 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_444_psubsetE,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_445_add__block__design,axiom,
! [Bl2: set_a] :
( ( finite_finite_a @ Bl2 )
=> ( ( Bl2 != bot_bot_set_a )
=> ( design_design_a @ ( sup_sup_set_a @ point_set @ Bl2 ) @ ( design4001997691126659652lock_a @ block_collection @ Bl2 ) ) ) ) ).
% add_block_design
thf(fact_446_finite__incidence__system_Odel__point__order,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,P: set_a] :
( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ P @ Point_set )
=> ( ( finite_card_set_a @ ( design7586725432863044395_set_a @ Point_set @ P ) )
= ( minus_minus_nat @ ( finite_card_set_a @ Point_set ) @ one_one_nat ) ) ) ) ).
% finite_incidence_system.del_point_order
thf(fact_447_finite__incidence__system_Odel__point__order,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,P: nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ P @ Point_set )
=> ( ( finite_card_nat @ ( design4269233978287968195nt_nat @ Point_set @ P ) )
= ( minus_minus_nat @ ( finite_card_nat @ Point_set ) @ one_one_nat ) ) ) ) ).
% finite_incidence_system.del_point_order
thf(fact_448_finite__incidence__system_Odel__point__order,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ P @ Point_set )
=> ( ( finite_card_a @ ( design108908007054065099oint_a @ Point_set @ P ) )
= ( minus_minus_nat @ ( finite_card_a @ Point_set ) @ one_one_nat ) ) ) ) ).
% finite_incidence_system.del_point_order
thf(fact_449_add__block__fin,axiom,
! [B: set_a] :
( ( finite_finite_a @ B )
=> ( design9187838744727572296stem_a @ ( sup_sup_set_a @ point_set @ B ) @ ( design4001997691126659652lock_a @ block_collection @ B ) ) ) ).
% add_block_fin
thf(fact_450_multiple__not__simple,axiom,
! [N: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( block_collection != zero_z5079479921072680283_set_a )
=> ~ ( design1338723777345758283stem_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ) ).
% multiple_not_simple
thf(fact_451_design_Oadd__block__design__cond,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_nat @ Bl2 @ Point_set )
=> ( ( Bl2 != bot_bot_set_nat )
=> ( design_design_nat @ Point_set @ ( design4725324266511619850ck_nat @ Block_collection @ Bl2 ) ) ) ) ) ).
% design.add_block_design_cond
thf(fact_452_design_Oadd__block__design__cond,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ Bl2 @ Point_set )
=> ( ( Bl2 != bot_bot_set_a )
=> ( design_design_a @ Point_set @ ( design4001997691126659652lock_a @ Block_collection @ Bl2 ) ) ) ) ) ).
% design.add_block_design_cond
thf(fact_453_delete__point__index__eq,axiom,
! [Ps: set_a,P: a] :
( ( ord_less_eq_set_a @ Ps @ ( design108908007054065099oint_a @ point_set @ P ) )
=> ( ( design254580327166089565ndex_a @ ( design6411949732824333445ocks_a @ block_collection @ P ) @ Ps )
= ( design254580327166089565ndex_a @ block_collection @ Ps ) ) ) ).
% delete_point_index_eq
thf(fact_454_add__block__rep__number__in,axiom,
! [X: a,B: set_a] :
( ( member_a @ X @ B )
=> ( ( design6637022207325878697mber_a @ ( design4001997691126659652lock_a @ block_collection @ B ) @ X )
= ( plus_plus_nat @ ( design6637022207325878697mber_a @ block_collection @ X ) @ one_one_nat ) ) ) ).
% add_block_rep_number_in
thf(fact_455_block__complement__def,axiom,
! [B: set_a] :
( ( design6447616907850319326ment_a @ point_set @ B )
= ( minus_minus_set_a @ point_set @ B ) ) ).
% block_complement_def
thf(fact_456_finite__block__sizes,axiom,
finite_finite_nat @ ( design1769254222028858111izes_a @ block_collection ) ).
% finite_block_sizes
thf(fact_457_finite__design__support,axiom,
finite_finite_set_a @ ( design5397942185814921632port_a @ block_collection ) ).
% finite_design_support
thf(fact_458_multiple__1__same,axiom,
( ( repeat_mset_set_a @ one_one_nat @ block_collection )
= block_collection ) ).
% multiple_1_same
thf(fact_459_multiple__block__in__original,axiom,
! [B: set_a,N: nat] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ block_collection ) ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) ) ) ).
% multiple_block_in_original
thf(fact_460_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_461_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_462_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_463_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_464_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_465_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_466_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_467_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_468_finite__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_469_finite__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_470_finite__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff
thf(fact_471_finite__Diff2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( finite_finite_nat @ A2 ) ) ) ).
% finite_Diff2
thf(fact_472_finite__Diff2,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( finite_finite_set_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_473_finite__Diff2,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( finite_finite_a @ A2 ) ) ) ).
% finite_Diff2
thf(fact_474_replication__numbers__finite,axiom,
finite_finite_nat @ ( design8835372594653258411bers_a @ point_set @ block_collection ) ).
% replication_numbers_finite
thf(fact_475_UnCI,axiom,
! [C: set_a,B2: set_set_a,A2: set_set_a] :
( ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ A2 ) )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_476_UnCI,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ A2 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_477_UnCI,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ A2 ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnCI
thf(fact_478_Un__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
| ( member_set_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_479_Un__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
| ( member_a @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_480_Un__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
| ( member_nat @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_481_Un__Diff__cancel,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_482_Un__Diff__cancel,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_Diff_cancel
thf(fact_483_Un__Diff__cancel2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_484_Un__Diff__cancel2,axiom,
! [B2: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B2 @ A2 ) @ A2 )
= ( sup_sup_set_a @ B2 @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_485_add__block__index__not__in,axiom,
! [Ps: set_a,B: set_a] :
( ~ ( ord_less_eq_set_a @ Ps @ B )
=> ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ block_collection @ B ) @ Ps )
= ( design254580327166089565ndex_a @ block_collection @ Ps ) ) ) ).
% add_block_index_not_in
thf(fact_486_repeat__mset__block__point__rel,axiom,
! [B: set_a,N: nat,X: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ block_collection ) ) )
=> ( ( member_a @ X @ B )
=> ( member_a @ X @ point_set ) ) ) ).
% repeat_mset_block_point_rel
thf(fact_487_multiple__is__finite,axiom,
! [N: nat] : ( design9187838744727572296stem_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ).
% multiple_is_finite
thf(fact_488_multiple__is__design,axiom,
! [N: nat] : ( design_design_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ).
% multiple_is_design
thf(fact_489_multiple__block__in,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ block_collection ) )
=> ( member_set_a @ B @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ) ) ).
% multiple_block_in
thf(fact_490_multiple__block__sizes__same,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( design1769254222028858111izes_a @ block_collection )
= ( design1769254222028858111izes_a @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ) ).
% multiple_block_sizes_same
thf(fact_491_points__index__count__min,axiom,
! [N: nat,Bl2: set_a,Ps: set_a] :
( ( ord_less_eq_nat @ N @ ( count_set_a @ block_collection @ Bl2 ) )
=> ( ( ord_less_eq_set_a @ Ps @ Bl2 )
=> ( ord_less_eq_nat @ N @ ( design254580327166089565ndex_a @ block_collection @ Ps ) ) ) ) ).
% points_index_count_min
thf(fact_492_points__index__ps__nin,axiom,
! [Ps: set_a] :
( ~ ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ( design254580327166089565ndex_a @ block_collection @ Ps )
= zero_zero_nat ) ) ).
% points_index_ps_nin
thf(fact_493_multiple__blocks__gt,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ block_collection ) @ ( size_s6566526139600085008_set_a @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ) ).
% multiple_blocks_gt
thf(fact_494_block__original__count__le,axiom,
! [N: nat,B: set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ ( count_set_a @ block_collection @ B ) @ ( count_set_a @ ( repeat_mset_set_a @ N @ block_collection ) @ B ) ) ) ).
% block_original_count_le
thf(fact_495_add__block__index__in,axiom,
! [Ps: set_a,B: set_a] :
( ( ord_less_eq_set_a @ Ps @ B )
=> ( ( design254580327166089565ndex_a @ ( design4001997691126659652lock_a @ block_collection @ B ) @ Ps )
= ( plus_plus_nat @ ( design254580327166089565ndex_a @ block_collection @ Ps ) @ one_one_nat ) ) ) ).
% add_block_index_in
thf(fact_496_points__index__zero,axiom,
! [Ps: set_a] :
( ( ord_less_nat @ ( finite_card_a @ point_set ) @ ( finite_card_a @ Ps ) )
=> ( ( design254580327166089565ndex_a @ block_collection @ Ps )
= zero_zero_nat ) ) ).
% points_index_zero
thf(fact_497_multiple__proper__design,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( design7287791228148780576sign_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ) ).
% multiple_proper_design
thf(fact_498_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_499_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_500_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_501_add__0,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A )
= A ) ).
% add_0
thf(fact_502_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_503_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_504_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_505_add__cancel__right__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_right
thf(fact_506_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_507_add__cancel__right__left,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( A
= ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_right_left
thf(fact_508_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_509_add__cancel__left__right,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ A @ B )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_right
thf(fact_510_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_511_add__cancel__left__left,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ B @ A )
= A )
= ( B = zero_z5079479921072680283_set_a ) ) ).
% add_cancel_left_left
thf(fact_512_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_513_add_Oright__neutral,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% add.right_neutral
thf(fact_514_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_515_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_516_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_517_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_518_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_519_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_520_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_521_Diff__eq__empty__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( minus_minus_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% Diff_eq_empty_iff
thf(fact_522_Nat_Oadd__0__right,axiom,
! [M3: nat] :
( ( plus_plus_nat @ M3 @ zero_zero_nat )
= M3 ) ).
% Nat.add_0_right
thf(fact_523_add__is__0,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= zero_zero_nat )
= ( ( M3 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_524_Un__empty,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_525_Un__empty,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_526_nat__add__left__cancel__less,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_527_nat__add__left__cancel__le,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( ord_less_eq_nat @ M3 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_528_finite__Un,axiom,
! [F2: set_set_a,G2: set_set_a] :
( ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G2 ) )
= ( ( finite_finite_set_a @ F2 )
& ( finite_finite_set_a @ G2 ) ) ) ).
% finite_Un
thf(fact_529_finite__Un,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) )
= ( ( finite_finite_a @ F2 )
& ( finite_finite_a @ G2 ) ) ) ).
% finite_Un
thf(fact_530_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_531_Un__subset__iff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_nat @ A2 @ C2 )
& ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_532_Un__subset__iff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( ( ord_less_eq_set_a @ A2 @ C2 )
& ( ord_less_eq_set_a @ B2 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_533_diff__diff__left,axiom,
! [I: nat,J: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_534_count__diff,axiom,
! [M5: multiset_set_a,N5: multiset_set_a,A: set_a] :
( ( count_set_a @ ( minus_706656509937749387_set_a @ M5 @ N5 ) @ A )
= ( minus_minus_nat @ ( count_set_a @ M5 @ A ) @ ( count_set_a @ N5 @ A ) ) ) ).
% count_diff
thf(fact_535_repeat__mset__empty,axiom,
! [N: nat] :
( ( repeat_mset_set_a @ N @ zero_z5079479921072680283_set_a )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_empty
thf(fact_536_le__add__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% le_add_same_cancel2
thf(fact_537_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_538_le__add__same__cancel1,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% le_add_same_cancel1
thf(fact_539_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_540_add__le__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_le_same_cancel2
thf(fact_541_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_542_add__le__same__cancel1,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ B @ A ) @ B )
= ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_le_same_cancel1
thf(fact_543_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_544_add__less__same__cancel1,axiom,
! [B: multiset_set_a,A: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ B @ A ) @ B )
= ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_less_same_cancel1
thf(fact_545_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_546_add__less__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ B )
= ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a ) ) ).
% add_less_same_cancel2
thf(fact_547_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_548_less__add__same__cancel1,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% less_add_same_cancel1
thf(fact_549_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_550_less__add__same__cancel2,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ ( plus_p2331992037799027419_set_a @ B @ A ) )
= ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B ) ) ).
% less_add_same_cancel2
thf(fact_551_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_552_diff__add__zero,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A @ ( plus_p2331992037799027419_set_a @ A @ B ) )
= zero_z5079479921072680283_set_a ) ).
% diff_add_zero
thf(fact_553_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_554_add__gr__0,axiom,
! [M3: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_555_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_556_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_557_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_558_repeat__mset__0,axiom,
! [M5: multiset_set_a] :
( ( repeat_mset_set_a @ zero_zero_nat @ M5 )
= zero_z5079479921072680283_set_a ) ).
% repeat_mset_0
thf(fact_559_points__index__empty,axiom,
! [Ps: set_a] :
( ( design254580327166089565ndex_a @ zero_z5079479921072680283_set_a @ Ps )
= zero_zero_nat ) ).
% points_index_empty
thf(fact_560_point__indices__elem__in,axiom,
! [Ps: set_a,T: nat] :
( ( ord_less_eq_set_a @ Ps @ point_set )
=> ( ( ( finite_card_a @ Ps )
= T )
=> ( member_nat @ ( design254580327166089565ndex_a @ block_collection @ Ps ) @ ( design328527185268214962ices_a @ point_set @ block_collection @ T ) ) ) ) ).
% point_indices_elem_in
thf(fact_561_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_562_Diff__subset__conv,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_563_Diff__subset__conv,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ C2 )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Diff_subset_conv
thf(fact_564_Diff__partition,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_565_Diff__partition,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B2 @ A2 ) )
= B2 ) ) ).
% Diff_partition
thf(fact_566_UnE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) )
=> ( ~ ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_567_UnE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ~ ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% UnE
thf(fact_568_UnE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ~ ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% UnE
thf(fact_569_UnI1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_570_UnI1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_571_UnI1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI1
thf(fact_572_UnI2,axiom,
! [C: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( sup_sup_set_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_573_UnI2,axiom,
! [C: a,B2: set_a,A2: set_a] :
( ( member_a @ C @ B2 )
=> ( member_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_574_UnI2,axiom,
! [C: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% UnI2
thf(fact_575_bex__Un,axiom,
! [A2: set_a,B2: set_a,P2: a > $o] :
( ( ? [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: a] :
( ( member_a @ X3 @ A2 )
& ( P2 @ X3 ) )
| ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_576_bex__Un,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) )
& ( P2 @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P2 @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ( P2 @ X3 ) ) ) ) ).
% bex_Un
thf(fact_577_Un__Diff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_578_Un__Diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C2 ) @ ( minus_minus_set_a @ B2 @ C2 ) ) ) ).
% Un_Diff
thf(fact_579_ball__Un,axiom,
! [A2: set_a,B2: set_a,P2: a > $o] :
( ( ! [X3: a] :
( ( member_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( P2 @ X3 ) )
& ! [X3: a] :
( ( member_a @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_580_ball__Un,axiom,
! [A2: set_nat,B2: set_nat,P2: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( P2 @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( P2 @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( P2 @ X3 ) ) ) ) ).
% ball_Un
thf(fact_581_Un__assoc,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_582_Un__assoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_assoc
thf(fact_583_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_584_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_585_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_586_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B6: set_nat] : ( sup_sup_set_nat @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_587_Un__left__absorb,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_588_Un__left__absorb,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_left_absorb
thf(fact_589_Un__left__commute,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C2 ) )
= ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_590_Un__left__commute,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) )
= ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_591_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_592_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( K2 = L ) )
=> ( ( plus_plus_nat @ I @ K2 )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_593_group__cancel_Oadd1,axiom,
! [A2: nat,K2: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K2 @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_594_group__cancel_Oadd2,axiom,
! [B2: nat,K2: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K2 @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_595_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_596_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B8: nat] : ( plus_plus_nat @ B8 @ A3 ) ) ) ).
% add.commute
thf(fact_597_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_598_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_599_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_600_card__Un__le,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_nat @ ( finite_card_a @ ( sup_sup_set_a @ A2 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ).
% card_Un_le
thf(fact_601_card__Un__le,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A2 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ).
% card_Un_le
thf(fact_602_count__in__diffI,axiom,
! [N5: multiset_a,X: a,M5: multiset_a] :
( ! [N2: nat] :
( ( count_a @ N5 @ X )
!= ( plus_plus_nat @ N2 @ ( count_a @ M5 @ X ) ) )
=> ( member_a @ X @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ N5 ) ) ) ) ).
% count_in_diffI
thf(fact_603_count__in__diffI,axiom,
! [N5: multiset_nat,X: nat,M5: multiset_nat] :
( ! [N2: nat] :
( ( count_nat @ N5 @ X )
!= ( plus_plus_nat @ N2 @ ( count_nat @ M5 @ X ) ) )
=> ( member_nat @ X @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M5 @ N5 ) ) ) ) ).
% count_in_diffI
thf(fact_604_count__in__diffI,axiom,
! [N5: multiset_set_a,X: set_a,M5: multiset_set_a] :
( ! [N2: nat] :
( ( count_set_a @ N5 @ X )
!= ( plus_plus_nat @ N2 @ ( count_set_a @ M5 @ X ) ) )
=> ( member_set_a @ X @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M5 @ N5 ) ) ) ) ).
% count_in_diffI
thf(fact_605_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_606_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_607_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_608_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_609_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_610_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_611_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_612_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B8: nat] :
? [C4: nat] :
( B8
= ( plus_plus_nat @ A3 @ C4 ) ) ) ) ).
% le_iff_add
thf(fact_613_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_614_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_615_Un__empty__left,axiom,
! [B2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_616_Un__empty__left,axiom,
! [B2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_617_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_618_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_619_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_620_add_Ocomm__neutral,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ A @ zero_z5079479921072680283_set_a )
= A ) ).
% add.comm_neutral
thf(fact_621_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_622_comm__monoid__add__class_Oadd__0,axiom,
! [A: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_623_finite__UnI,axiom,
! [F2: set_set_a,G2: set_set_a] :
( ( finite_finite_set_a @ F2 )
=> ( ( finite_finite_set_a @ G2 )
=> ( finite_finite_set_a @ ( sup_sup_set_set_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_624_finite__UnI,axiom,
! [F2: set_a,G2: set_a] :
( ( finite_finite_a @ F2 )
=> ( ( finite_finite_a @ G2 )
=> ( finite_finite_a @ ( sup_sup_set_a @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_625_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_626_Un__infinite,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_627_Un__infinite,axiom,
! [S2: set_a,T2: set_a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( sup_sup_set_a @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_628_Un__infinite,axiom,
! [S2: set_nat,T2: set_nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) ) ).
% Un_infinite
thf(fact_629_infinite__Un,axiom,
! [S2: set_set_a,T2: set_set_a] :
( ( ~ ( finite_finite_set_a @ ( sup_sup_set_set_a @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_set_a @ S2 )
| ~ ( finite_finite_set_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_630_infinite__Un,axiom,
! [S2: set_a,T2: set_a] :
( ( ~ ( finite_finite_a @ ( sup_sup_set_a @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_a @ S2 )
| ~ ( finite_finite_a @ T2 ) ) ) ).
% infinite_Un
thf(fact_631_infinite__Un,axiom,
! [S2: set_nat,T2: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S2 @ T2 ) ) )
= ( ~ ( finite_finite_nat @ S2 )
| ~ ( finite_finite_nat @ T2 ) ) ) ).
% infinite_Un
thf(fact_632_Un__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_633_Un__mono,axiom,
! [A2: set_a,C2: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_634_Un__least,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_635_Un__least,axiom,
! [A2: set_a,C2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C2 ) ) ) ).
% Un_least
thf(fact_636_Un__upper1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_637_Un__upper1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper1
thf(fact_638_Un__upper2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_639_Un__upper2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% Un_upper2
thf(fact_640_Un__absorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_641_Un__absorb1,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_642_Un__absorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_643_Un__absorb2,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% Un_absorb2
thf(fact_644_subset__UnE,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
=> ~ ! [A6: set_nat] :
( ( ord_less_eq_set_nat @ A6 @ A2 )
=> ! [B9: set_nat] :
( ( ord_less_eq_set_nat @ B9 @ B2 )
=> ( C2
!= ( sup_sup_set_nat @ A6 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_645_subset__UnE,axiom,
! [C2: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A2 @ B2 ) )
=> ~ ! [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ A2 )
=> ! [B9: set_a] :
( ( ord_less_eq_set_a @ B9 @ B2 )
=> ( C2
!= ( sup_sup_set_a @ A6 @ B9 ) ) ) ) ) ).
% subset_UnE
thf(fact_646_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B6: set_nat] :
( ( sup_sup_set_nat @ A5 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_647_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ( sup_sup_set_a @ A5 @ B6 )
= B6 ) ) ) ).
% subset_Un_eq
thf(fact_648_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_649_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_650_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_651_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_652_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_653_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K2 = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_654_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_655_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_656_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_657_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_658_Diff__infinite__finite,axiom,
! [T2: set_nat,S2: set_nat] :
( ( finite_finite_nat @ T2 )
=> ( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_659_Diff__infinite__finite,axiom,
! [T2: set_set_a,S2: set_set_a] :
( ( finite_finite_set_a @ T2 )
=> ( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_660_Diff__infinite__finite,axiom,
! [T2: set_a,S2: set_a] :
( ( finite_finite_a @ T2 )
=> ( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ T2 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_661_Diff__mono,axiom,
! [A2: set_a,C2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( minus_minus_set_a @ C2 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_662_Diff__subset,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ A2 ) ).
% Diff_subset
thf(fact_663_double__diff,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ( minus_minus_set_a @ B2 @ ( minus_minus_set_a @ C2 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_664_repeat__mset__cancel1,axiom,
! [A: nat,A2: multiset_set_a,B2: multiset_set_a] :
( ( ( repeat_mset_set_a @ A @ A2 )
= ( repeat_mset_set_a @ A @ B2 ) )
= ( ( A2 = B2 )
| ( A = zero_zero_nat ) ) ) ).
% repeat_mset_cancel1
thf(fact_665_in__diffD,axiom,
! [A: a,M5: multiset_a,N5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ N5 ) ) )
=> ( member_a @ A @ ( set_mset_a @ M5 ) ) ) ).
% in_diffD
thf(fact_666_in__diffD,axiom,
! [A: nat,M5: multiset_nat,N5: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M5 @ N5 ) ) )
=> ( member_nat @ A @ ( set_mset_nat @ M5 ) ) ) ).
% in_diffD
thf(fact_667_in__diffD,axiom,
! [A: set_a,M5: multiset_set_a,N5: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M5 @ N5 ) ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ M5 ) ) ) ).
% in_diffD
thf(fact_668_diff__empty,axiom,
! [M5: multiset_set_a] :
( ( ( minus_706656509937749387_set_a @ M5 @ zero_z5079479921072680283_set_a )
= M5 )
& ( ( minus_706656509937749387_set_a @ zero_z5079479921072680283_set_a @ M5 )
= zero_z5079479921072680283_set_a ) ) ).
% diff_empty
thf(fact_669_Multiset_Odiff__cancel,axiom,
! [A2: multiset_set_a] :
( ( minus_706656509937749387_set_a @ A2 @ A2 )
= zero_z5079479921072680283_set_a ) ).
% Multiset.diff_cancel
thf(fact_670_add__eq__self__zero,axiom,
! [M3: nat,N: nat] :
( ( ( plus_plus_nat @ M3 @ N )
= M3 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_671_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_672_design_Omultiple__is__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ Point_set @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ).
% design.multiple_is_design
thf(fact_673_less__add__eq__less,axiom,
! [K2: nat,L: nat,M3: nat,N: nat] :
( ( ord_less_nat @ K2 @ L )
=> ( ( ( plus_plus_nat @ M3 @ L )
= ( plus_plus_nat @ K2 @ N ) )
=> ( ord_less_nat @ M3 @ N ) ) ) ).
% less_add_eq_less
thf(fact_674_trans__less__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_less_add2
thf(fact_675_trans__less__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_less_add1
thf(fact_676_add__less__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_677_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_678_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_679_add__less__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K2 @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_680_add__lessD1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
=> ( ord_less_nat @ I @ K2 ) ) ).
% add_lessD1
thf(fact_681_elem__in__repeat__in__original,axiom,
! [A: a,N: nat,A2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( repeat_mset_a @ N @ A2 ) ) )
=> ( member_a @ A @ ( set_mset_a @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_682_elem__in__repeat__in__original,axiom,
! [A: nat,N: nat,A2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( repeat_mset_nat @ N @ A2 ) ) )
=> ( member_nat @ A @ ( set_mset_nat @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_683_elem__in__repeat__in__original,axiom,
! [A: set_a,N: nat,A2: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ A2 ) ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ A2 ) ) ) ).
% elem_in_repeat_in_original
thf(fact_684_add__leE,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N )
=> ~ ( ( ord_less_eq_nat @ M3 @ N )
=> ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_685_le__add1,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M3 ) ) ).
% le_add1
thf(fact_686_le__add2,axiom,
! [N: nat,M3: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M3 @ N ) ) ).
% le_add2
thf(fact_687_add__leD1,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N )
=> ( ord_less_eq_nat @ M3 @ N ) ) ).
% add_leD1
thf(fact_688_add__leD2,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M3 @ K2 ) @ N )
=> ( ord_less_eq_nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_689_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq_nat @ K2 @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K2 @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_690_add__le__mono,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_691_add__le__mono1,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_692_trans__le__add1,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).
% trans_le_add1
thf(fact_693_trans__le__add2,axiom,
! [I: nat,J: nat,M3: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).
% trans_le_add2
thf(fact_694_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M4 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_695_repeat__mset__cancel2,axiom,
! [A: nat,A2: multiset_set_a,B: nat] :
( ( ( repeat_mset_set_a @ A @ A2 )
= ( repeat_mset_set_a @ B @ A2 ) )
= ( ( A = B )
| ( A2 = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_cancel2
thf(fact_696_finite__incidence__system_Omultiple__is__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ Point_set @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ).
% finite_incidence_system.multiple_is_finite
thf(fact_697_Nat_Odiff__cancel,axiom,
! [K2: nat,M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M3 ) @ ( plus_plus_nat @ K2 @ N ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% Nat.diff_cancel
thf(fact_698_diff__cancel2,axiom,
! [M3: nat,K2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
= ( minus_minus_nat @ M3 @ N ) ) ).
% diff_cancel2
thf(fact_699_diff__add__inverse,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M3 ) @ N )
= M3 ) ).
% diff_add_inverse
thf(fact_700_diff__add__inverse2,axiom,
! [M3: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M3 @ N ) @ N )
= M3 ) ).
% diff_add_inverse2
thf(fact_701_psubset__imp__ex__mem,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ B2 )
=> ? [B3: set_a] : ( member_set_a @ B3 @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_702_psubset__imp__ex__mem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_703_psubset__imp__ex__mem,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_704_repeat__mset__subset__in,axiom,
! [A2: multiset_set_set_a,B2: set_set_a,X5: set_set_a,N: nat,X: set_a] :
( ! [A7: set_set_a] :
( ( member_set_set_a @ A7 @ ( set_mset_set_set_a @ A2 ) )
=> ( ord_le3724670747650509150_set_a @ A7 @ B2 ) )
=> ( ( member_set_set_a @ X5 @ ( set_mset_set_set_a @ ( repeat3222187171979612824_set_a @ N @ A2 ) ) )
=> ( ( member_set_a @ X @ X5 )
=> ( member_set_a @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_705_repeat__mset__subset__in,axiom,
! [A2: multiset_set_nat,B2: set_nat,X5: set_nat,N: nat,X: nat] :
( ! [A7: set_nat] :
( ( member_set_nat @ A7 @ ( set_mset_set_nat @ A2 ) )
=> ( ord_less_eq_set_nat @ A7 @ B2 ) )
=> ( ( member_set_nat @ X5 @ ( set_mset_set_nat @ ( repeat_mset_set_nat @ N @ A2 ) ) )
=> ( ( member_nat @ X @ X5 )
=> ( member_nat @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_706_repeat__mset__subset__in,axiom,
! [A2: multiset_set_a,B2: set_a,X5: set_a,N: nat,X: a] :
( ! [A7: set_a] :
( ( member_set_a @ A7 @ ( set_mset_set_a @ A2 ) )
=> ( ord_less_eq_set_a @ A7 @ B2 ) )
=> ( ( member_set_a @ X5 @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ A2 ) ) )
=> ( ( member_a @ X @ X5 )
=> ( member_a @ X @ B2 ) ) ) ) ).
% repeat_mset_subset_in
thf(fact_707_add__nonpos__eq__0__iff,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ X @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ Y @ zero_z5079479921072680283_set_a )
=> ( ( ( plus_p2331992037799027419_set_a @ X @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_708_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_709_add__nonneg__eq__0__iff,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ X )
=> ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ Y )
=> ( ( ( plus_p2331992037799027419_set_a @ X @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_710_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_711_add__nonpos__nonpos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_nonpos_nonpos
thf(fact_712_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_713_add__nonneg__nonneg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_714_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_715_add__increasing2,axiom,
! [C: multiset_set_a,B: multiset_set_a,A: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ C )
=> ( ( ord_le7905258569527593284_set_a @ B @ A )
=> ( ord_le7905258569527593284_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_716_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_717_add__decreasing2,axiom,
! [C: multiset_set_a,A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ C @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ A @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_718_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_719_add__increasing,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ B @ C )
=> ( ord_le7905258569527593284_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_720_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_721_add__decreasing,axiom,
! [A: multiset_set_a,C: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ C @ B )
=> ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_722_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_723_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_724_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_725_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_726_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K2: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K2 @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_727_add__neg__neg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le5765082015083327056_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_neg_neg
thf(fact_728_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_729_add__pos__pos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_730_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_731_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_732_pos__add__strict,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ B @ C )
=> ( ord_le5765082015083327056_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_733_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_734_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_735_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_736_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_737_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_738_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_739_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_740_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_741_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_742_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_743_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_744_index__lt__rep__general,axiom,
! [X: set_a,Ps: set_set_a,B2: multiset_set_set_a] :
( ( member_set_a @ X @ Ps )
=> ( ord_less_eq_nat @ ( design88022138586678973_set_a @ B2 @ Ps ) @ ( design5008467512594872073_set_a @ B2 @ X ) ) ) ).
% index_lt_rep_general
thf(fact_745_index__lt__rep__general,axiom,
! [X: nat,Ps: set_nat,B2: multiset_set_nat] :
( ( member_nat @ X @ Ps )
=> ( ord_less_eq_nat @ ( design6574611146354332593ex_nat @ B2 @ Ps ) @ ( design3571518413069006949er_nat @ B2 @ X ) ) ) ).
% index_lt_rep_general
thf(fact_746_index__lt__rep__general,axiom,
! [X: a,Ps: set_a,B2: multiset_set_a] :
( ( member_a @ X @ Ps )
=> ( ord_less_eq_nat @ ( design254580327166089565ndex_a @ B2 @ Ps ) @ ( design6637022207325878697mber_a @ B2 @ X ) ) ) ).
% index_lt_rep_general
thf(fact_747_set__diff__non__empty__not__subset,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
=> ( ( C2 != bot_bot_set_nat )
=> ( ( A2 != bot_bot_set_nat )
=> ( ( B2 != bot_bot_set_nat )
=> ~ ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_748_set__diff__non__empty__not__subset,axiom,
! [A2: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
=> ( ( C2 != bot_bot_set_a )
=> ( ( A2 != bot_bot_set_a )
=> ( ( B2 != bot_bot_set_a )
=> ~ ( ord_less_eq_set_a @ A2 @ C2 ) ) ) ) ) ).
% set_diff_non_empty_not_subset
thf(fact_749_repeat__mset__eq__empty__iff,axiom,
! [N: nat,A2: multiset_set_a] :
( ( ( repeat_mset_set_a @ N @ A2 )
= zero_z5079479921072680283_set_a )
= ( ( N = zero_zero_nat )
| ( A2 = zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_eq_empty_iff
thf(fact_750_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
& ( ( plus_plus_nat @ I @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_751_mono__nat__linear__lb,axiom,
! [F: nat > nat,M3: nat,K2: nat] :
( ! [M7: nat,N2: nat] :
( ( ord_less_nat @ M7 @ N2 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_752_diff__add__0,axiom,
! [N: nat,M3: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M3 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_753_add__diff__inverse__nat,axiom,
! [M3: nat,N: nat] :
( ~ ( ord_less_nat @ M3 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M3 @ N ) )
= M3 ) ) ).
% add_diff_inverse_nat
thf(fact_754_less__diff__conv,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_755_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K2 )
= ( J
= ( plus_plus_nat @ K2 @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_756_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_757_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_758_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_759_le__diff__conv,axiom,
! [J: nat,K2: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).
% le_diff_conv
thf(fact_760_minus__multiset_Orep__eq,axiom,
! [X: multiset_set_a,Xa2: multiset_set_a] :
( ( count_set_a @ ( minus_706656509937749387_set_a @ X @ Xa2 ) )
= ( ^ [A3: set_a] : ( minus_minus_nat @ ( count_set_a @ X @ A3 ) @ ( count_set_a @ Xa2 @ A3 ) ) ) ) ).
% minus_multiset.rep_eq
thf(fact_761_finite__incidence__system_Oadd__block__fin,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,B: set_set_a] :
( ( design1749870844763721896_set_a @ Point_set @ Block_collection )
=> ( ( finite_finite_set_a @ B )
=> ( design1749870844763721896_set_a @ ( sup_sup_set_set_a @ Point_set @ B ) @ ( design7860908649167014820_set_a @ Block_collection @ B ) ) ) ) ).
% finite_incidence_system.add_block_fin
thf(fact_762_finite__incidence__system_Oadd__block__fin,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,B: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( finite_finite_nat @ B )
=> ( design5426232790142929158em_nat @ ( sup_sup_set_nat @ Point_set @ B ) @ ( design4725324266511619850ck_nat @ Block_collection @ B ) ) ) ) ).
% finite_incidence_system.add_block_fin
thf(fact_763_finite__incidence__system_Oadd__block__fin,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( finite_finite_a @ B )
=> ( design9187838744727572296stem_a @ ( sup_sup_set_a @ Point_set @ B ) @ ( design4001997691126659652lock_a @ Block_collection @ B ) ) ) ) ).
% finite_incidence_system.add_block_fin
thf(fact_764_finite__incidence__system_Oreplication__numbers__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( finite_finite_nat @ ( design8835372594653258411bers_a @ Point_set @ Block_collection ) ) ) ).
% finite_incidence_system.replication_numbers_finite
thf(fact_765_finite__incidence__system_Ofinite__design__support,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( finite_finite_set_a @ ( design5397942185814921632port_a @ Block_collection ) ) ) ).
% finite_incidence_system.finite_design_support
thf(fact_766_points__index__0__right__imp,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ B2 ) )
=> ~ ( ord_less_eq_set_a @ Ps @ B3 ) )
=> ( ( design254580327166089565ndex_a @ B2 @ Ps )
= zero_zero_nat ) ) ).
% points_index_0_right_imp
thf(fact_767_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_768_points__index__0__iff,axiom,
! [B2: multiset_set_a,Ps: set_a] :
( ( ( design254580327166089565ndex_a @ B2 @ Ps )
= zero_zero_nat )
= ( ! [B8: set_a] :
( ( member_set_a @ B8 @ ( set_mset_set_a @ B2 ) )
=> ~ ( ord_less_eq_set_a @ Ps @ B8 ) ) ) ) ).
% points_index_0_iff
thf(fact_769_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 )
& ! [Y3: set_a] :
( ( ( member_set_a @ Y3 @ ( set_mset_set_a @ B2 ) )
& ( ord_less_eq_set_a @ Ps @ Y3 ) )
=> ( Y3 = X2 ) ) ) ) ).
% points_index_one_unique_block
thf(fact_770_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_771_proper__design_Omultiple__proper__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat] :
( ( design7287791228148780576sign_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( design7287791228148780576sign_a @ Point_set @ ( repeat_mset_set_a @ N @ Block_collection ) ) ) ) ).
% proper_design.multiple_proper_design
thf(fact_772_add__strict__increasing2,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ B @ C )
=> ( ord_le5765082015083327056_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_773_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_774_add__strict__increasing,axiom,
! [A: multiset_set_a,B: multiset_set_a,C: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ B @ C )
=> ( ord_le5765082015083327056_set_a @ B @ ( plus_p2331992037799027419_set_a @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_775_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_776_add__pos__nonneg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_777_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_778_add__nonpos__neg,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le5765082015083327056_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_nonpos_neg
thf(fact_779_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_780_add__nonneg__pos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ A )
=> ( ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ B )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_781_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_782_add__neg__nonpos,axiom,
! [A: multiset_set_a,B: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ A @ zero_z5079479921072680283_set_a )
=> ( ( ord_le7905258569527593284_set_a @ B @ zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ ( plus_p2331992037799027419_set_a @ A @ B ) @ zero_z5079479921072680283_set_a ) ) ) ).
% add_neg_nonpos
thf(fact_783_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_784_design_Oadd__block__design,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,Bl2: set_set_a] :
( ( design_design_set_a @ Point_set @ Block_collection )
=> ( ( finite_finite_set_a @ Bl2 )
=> ( ( Bl2 != bot_bot_set_set_a )
=> ( design_design_set_a @ ( sup_sup_set_set_a @ Point_set @ Bl2 ) @ ( design7860908649167014820_set_a @ Block_collection @ Bl2 ) ) ) ) ) ).
% design.add_block_design
thf(fact_785_design_Oadd__block__design,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Bl2: set_nat] :
( ( design_design_nat @ Point_set @ Block_collection )
=> ( ( finite_finite_nat @ Bl2 )
=> ( ( Bl2 != bot_bot_set_nat )
=> ( design_design_nat @ ( sup_sup_set_nat @ Point_set @ Bl2 ) @ ( design4725324266511619850ck_nat @ Block_collection @ Bl2 ) ) ) ) ) ).
% design.add_block_design
thf(fact_786_design_Oadd__block__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( finite_finite_a @ Bl2 )
=> ( ( Bl2 != bot_bot_set_a )
=> ( design_design_a @ ( sup_sup_set_a @ Point_set @ Bl2 ) @ ( design4001997691126659652lock_a @ Block_collection @ Bl2 ) ) ) ) ) ).
% design.add_block_design
thf(fact_787_card__less__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_788_card__less__sym__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_789_card__less__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_790_card__le__sym__Diff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_791_card__le__sym__Diff,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_792_card__le__sym__Diff,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ B2 @ A2 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_793_elem__in__original__in__repeat,axiom,
! [N: nat,A: a,A2: multiset_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_a @ A @ ( set_mset_a @ A2 ) )
=> ( member_a @ A @ ( set_mset_a @ ( repeat_mset_a @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_794_elem__in__original__in__repeat,axiom,
! [N: nat,A: nat,A2: multiset_nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
=> ( member_nat @ A @ ( set_mset_nat @ ( repeat_mset_nat @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_795_elem__in__original__in__repeat,axiom,
! [N: nat,A: set_a,A2: multiset_set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( member_set_a @ A @ ( set_mset_set_a @ A2 ) )
=> ( member_set_a @ A @ ( set_mset_set_a @ ( repeat_mset_set_a @ N @ A2 ) ) ) ) ) ).
% elem_in_original_in_repeat
thf(fact_796_repeat__mset__not__empty,axiom,
! [N: nat,A2: multiset_set_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( A2 != zero_z5079479921072680283_set_a )
=> ( ( repeat_mset_set_a @ N @ A2 )
!= zero_z5079479921072680283_set_a ) ) ) ).
% repeat_mset_not_empty
thf(fact_797_in__diff__count,axiom,
! [A: a,M5: multiset_a,N5: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M5 @ N5 ) ) )
= ( ord_less_nat @ ( count_a @ N5 @ A ) @ ( count_a @ M5 @ A ) ) ) ).
% in_diff_count
thf(fact_798_in__diff__count,axiom,
! [A: nat,M5: multiset_nat,N5: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M5 @ N5 ) ) )
= ( ord_less_nat @ ( count_nat @ N5 @ A ) @ ( count_nat @ M5 @ A ) ) ) ).
% in_diff_count
thf(fact_799_in__diff__count,axiom,
! [A: set_a,M5: multiset_set_a,N5: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( minus_706656509937749387_set_a @ M5 @ N5 ) ) )
= ( ord_less_nat @ ( count_set_a @ N5 @ A ) @ ( count_set_a @ M5 @ A ) ) ) ).
% in_diff_count
thf(fact_800_diff__size__le__size__Diff,axiom,
! [M5: multiset_set_a,M8: multiset_set_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s6566526139600085008_set_a @ M5 ) @ ( size_s6566526139600085008_set_a @ M8 ) ) @ ( size_s6566526139600085008_set_a @ ( minus_706656509937749387_set_a @ M5 @ M8 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_801_nat__diff__split,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P2 @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P2 @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_802_nat__diff__split__asm,axiom,
! [P2: nat > $o,A: nat,B: nat] :
( ( P2 @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P2 @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P2 @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_803_less__diff__conv2,axiom,
! [K2: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K2 @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_804_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_805_finite__incidence__system_Opoints__index__zero,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Ps: set_nat] :
( ( design5426232790142929158em_nat @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ ( finite_card_nat @ Point_set ) @ ( finite_card_nat @ Ps ) )
=> ( ( design6574611146354332593ex_nat @ Block_collection @ Ps )
= zero_zero_nat ) ) ) ).
% finite_incidence_system.points_index_zero
thf(fact_806_finite__incidence__system_Opoints__index__zero,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Ps: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_nat @ ( finite_card_a @ Point_set ) @ ( finite_card_a @ Ps ) )
=> ( ( design254580327166089565ndex_a @ Block_collection @ Ps )
= zero_zero_nat ) ) ) ).
% finite_incidence_system.points_index_zero
thf(fact_807_incidence__system_Odel__block_Ocong,axiom,
design1146539425385464078lock_a = design1146539425385464078lock_a ).
% incidence_system.del_block.cong
thf(fact_808_incidence__system_Oadd__block_Ocong,axiom,
design4001997691126659652lock_a = design4001997691126659652lock_a ).
% incidence_system.add_block.cong
thf(fact_809_set__card__diff__ge__zero,axiom,
! [A2: set_nat,B2: set_nat] :
( ( finite_finite_nat @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ( A2 != B2 )
=> ( ( ( finite_card_nat @ A2 )
= ( finite_card_nat @ B2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_810_set__card__diff__ge__zero,axiom,
! [A2: set_set_a,B2: set_set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( finite_finite_set_a @ B2 )
=> ( ( A2 != B2 )
=> ( ( ( finite_card_set_a @ A2 )
= ( finite_card_set_a @ B2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_811_set__card__diff__ge__zero,axiom,
! [A2: set_a,B2: set_a] :
( ( finite_finite_a @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ( A2 != B2 )
=> ( ( ( finite_card_a @ A2 )
= ( finite_card_a @ B2 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ) ) ).
% set_card_diff_ge_zero
thf(fact_812_card__Diff__subset,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_813_card__Diff__subset,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ( ord_le3724670747650509150_set_a @ B2 @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_814_card__Diff__subset,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_815_design_Opoints__index__count__min,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,N: nat,Bl2: set_a,Ps: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_nat @ N @ ( count_set_a @ Block_collection @ Bl2 ) )
=> ( ( ord_less_eq_set_a @ Ps @ Bl2 )
=> ( ord_less_eq_nat @ N @ ( design254580327166089565ndex_a @ Block_collection @ Ps ) ) ) ) ) ).
% design.points_index_count_min
thf(fact_816_incidence__system_Odel__point_Ocong,axiom,
design108908007054065099oint_a = design108908007054065099oint_a ).
% incidence_system.del_point.cong
thf(fact_817_diff__card__le__card__Diff,axiom,
! [B2: set_nat,A2: set_nat] :
( ( finite_finite_nat @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_818_diff__card__le__card__Diff,axiom,
! [B2: set_set_a,A2: set_set_a] :
( ( finite_finite_set_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ ( finite_card_set_a @ B2 ) ) @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_819_diff__card__le__card__Diff,axiom,
! [B2: set_a,A2: set_a] :
( ( finite_finite_a @ B2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ B2 ) ) @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_820_incidence__system_Ostr__del__point__blocks_Ocong,axiom,
design5657747894866638574ocks_a = design5657747894866638574ocks_a ).
% incidence_system.str_del_point_blocks.cong
thf(fact_821_incidence__system_Odel__point__blocks_Ocong,axiom,
design6411949732824333445ocks_a = design6411949732824333445ocks_a ).
% incidence_system.del_point_blocks.cong
thf(fact_822_incidence__system_Oadd__point_Ocong,axiom,
design2964366272795260673oint_a = design2964366272795260673oint_a ).
% incidence_system.add_point.cong
thf(fact_823_incidence__system_Oadd__point__to__blocks_Ocong,axiom,
design2935547469388721088ocks_a = design2935547469388721088ocks_a ).
% incidence_system.add_point_to_blocks.cong
thf(fact_824_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_825_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_826_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_827_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_828_design_Odelete__block__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Bl2: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ Point_set @ ( design1146539425385464078lock_a @ Block_collection @ Bl2 ) ) ) ).
% design.delete_block_design
thf(fact_829_finite__incidence__system_Odelete__block__fin__incidence__sys,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ Point_set @ ( design1146539425385464078lock_a @ Block_collection @ B ) ) ) ).
% finite_incidence_system.delete_block_fin_incidence_sys
thf(fact_830_design_Oadd__point__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ Block_collection ) ) ).
% design.add_point_design
thf(fact_831_finite__incidence__system_Oadd__point__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ Block_collection ) ) ).
% finite_incidence_system.add_point_finite
thf(fact_832_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_833_finite__incidence__system_Oadd__block__fin__cond,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ B @ Point_set )
=> ( design9187838744727572296stem_a @ Point_set @ ( design4001997691126659652lock_a @ Block_collection @ B ) ) ) ) ).
% finite_incidence_system.add_block_fin_cond
thf(fact_834_design_Ostrong__del__point__design,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( design_design_a @ ( design108908007054065099oint_a @ Point_set @ P ) @ ( design5657747894866638574ocks_a @ Block_collection @ P ) ) ) ).
% design.strong_del_point_design
thf(fact_835_finite__incidence__system_Ostrong__del__point__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ Point_set @ P ) @ ( design5657747894866638574ocks_a @ Block_collection @ P ) ) ) ).
% finite_incidence_system.strong_del_point_finite
thf(fact_836_finite__incidence__system_Odelete__point__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( design108908007054065099oint_a @ Point_set @ P ) @ ( design6411949732824333445ocks_a @ Block_collection @ P ) ) ) ).
% finite_incidence_system.delete_point_finite
thf(fact_837_finite__incidence__system_Oadd__point__to__blocks__finite,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,P: a,Bs: set_set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( design2964366272795260673oint_a @ Point_set @ P ) @ ( design2935547469388721088ocks_a @ Block_collection @ P @ Bs ) ) ) ).
% finite_incidence_system.add_point_to_blocks_finite
thf(fact_838_strong__del__block__des,axiom,
! [B: set_a] :
( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ block_collection ) )
=> ~ ( ord_less_set_a @ Bl @ B ) )
=> ( design_design_a @ ( minus_minus_set_a @ point_set @ B ) @ ( design4241783006516448631lock_a @ block_collection @ B ) ) ) ).
% strong_del_block_des
thf(fact_839_strong__del__block__fin,axiom,
! [B: set_a] : ( design9187838744727572296stem_a @ ( minus_minus_set_a @ point_set @ B ) @ ( design4241783006516448631lock_a @ block_collection @ B ) ) ).
% strong_del_block_fin
thf(fact_840_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_841_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_842_delete__point__blocks__wf,axiom,
! [B: set_a,P: a] :
( ( member_set_a @ B @ ( set_mset_set_a @ ( design6411949732824333445ocks_a @ block_collection @ P ) ) )
=> ( ord_less_eq_set_a @ B @ ( minus_minus_set_a @ point_set @ ( insert_a @ P @ bot_bot_set_a ) ) ) ) ).
% delete_point_blocks_wf
thf(fact_843_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_844_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_845_add__point__def,axiom,
! [P: a] :
( ( design2964366272795260673oint_a @ point_set @ P )
= ( insert_a @ P @ point_set ) ) ).
% add_point_def
thf(fact_846_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_847_insert__absorb2,axiom,
! [X: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ X @ A2 ) )
= ( insert_nat @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_848_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_849_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_850_insert__iff,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
= ( ( A = B )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_851_insertCI,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B2 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_852_insertCI,axiom,
! [A: a,B2: set_a,B: a] :
( ( ~ ( member_a @ A @ B2 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertCI
thf(fact_853_insertCI,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( ~ ( member_nat @ A @ B2 )
=> ( A = B ) )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertCI
thf(fact_854_union__eq__empty,axiom,
! [M5: multiset_set_a,N5: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ M5 @ N5 )
= zero_z5079479921072680283_set_a )
= ( ( M5 = zero_z5079479921072680283_set_a )
& ( N5 = zero_z5079479921072680283_set_a ) ) ) ).
% union_eq_empty
thf(fact_855_empty__eq__union,axiom,
! [M5: multiset_set_a,N5: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ M5 @ N5 ) )
= ( ( M5 = zero_z5079479921072680283_set_a )
& ( N5 = zero_z5079479921072680283_set_a ) ) ) ).
% empty_eq_union
thf(fact_856_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( zero_z5079479921072680283_set_a
= ( plus_p2331992037799027419_set_a @ X @ Y ) )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_857_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [X: multiset_set_a,Y: multiset_set_a] :
( ( ( plus_p2331992037799027419_set_a @ X @ Y )
= zero_z5079479921072680283_set_a )
= ( ( X = zero_z5079479921072680283_set_a )
& ( Y = zero_z5079479921072680283_set_a ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_858_DiffI,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ A2 )
=> ( ~ ( member_set_a @ C @ B2 )
=> ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_859_DiffI,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ A2 )
=> ( ~ ( member_nat @ C @ B2 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_860_DiffI,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ A2 )
=> ( ~ ( member_a @ C @ B2 )
=> ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_861_Diff__iff,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
= ( ( member_set_a @ C @ A2 )
& ~ ( member_set_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_862_Diff__iff,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
= ( ( member_nat @ C @ A2 )
& ~ ( member_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_863_Diff__iff,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
= ( ( member_a @ C @ A2 )
& ~ ( member_a @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_864_Diff__idemp,axiom,
! [A2: set_a,B2: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ).
% Diff_idemp
thf(fact_865_repeat__mset__distrib2,axiom,
! [N: nat,A2: multiset_set_a,B2: multiset_set_a] :
( ( repeat_mset_set_a @ N @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ N @ A2 ) @ ( repeat_mset_set_a @ N @ B2 ) ) ) ).
% repeat_mset_distrib2
thf(fact_866_del__point__def,axiom,
! [P: a] :
( ( design108908007054065099oint_a @ point_set @ P )
= ( minus_minus_set_a @ point_set @ ( insert_a @ P @ bot_bot_set_a ) ) ) ).
% del_point_def
thf(fact_867_sup_Obounded__iff,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( ( ord_less_eq_set_nat @ B @ A )
& ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_868_sup_Obounded__iff,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
= ( ( ord_less_eq_nat @ B @ A )
& ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_869_sup_Obounded__iff,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.bounded_iff
thf(fact_870_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X @ Z2 )
& ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_871_le__sup__iff,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_872_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z2 )
= ( ( ord_less_eq_set_a @ X @ Z2 )
& ( ord_less_eq_set_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_873_str__del__block__del__point,axiom,
! [X: a] :
( ~ ( member_set_a @ ( insert_a @ X @ bot_bot_set_a ) @ ( set_mset_set_a @ block_collection ) )
=> ( ( design4241783006516448631lock_a @ block_collection @ ( insert_a @ X @ bot_bot_set_a ) )
= ( design6411949732824333445ocks_a @ block_collection @ X ) ) ) ).
% str_del_block_del_point
thf(fact_874_remove__invalid__point__block,axiom,
! [P: a,Bl2: set_a] :
( ~ ( member_a @ P @ point_set )
=> ( ( member_set_a @ Bl2 @ ( set_mset_set_a @ block_collection ) )
=> ( ( minus_minus_set_a @ Bl2 @ ( insert_a @ P @ bot_bot_set_a ) )
= Bl2 ) ) ) ).
% remove_invalid_point_block
thf(fact_875_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_876_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_877_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_878_finite__insert,axiom,
! [A: a,A2: set_a] :
( ( finite_finite_a @ ( insert_a @ A @ A2 ) )
= ( finite_finite_a @ A2 ) ) ).
% finite_insert
thf(fact_879_finite__insert,axiom,
! [A: nat,A2: set_nat] :
( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
= ( finite_finite_nat @ A2 ) ) ).
% finite_insert
thf(fact_880_finite__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) )
= ( finite_finite_set_a @ A2 ) ) ).
% finite_insert
thf(fact_881_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_882_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_883_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_884_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_885_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_886_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_887_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_888_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_889_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_890_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_891_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_892_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_893_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_894_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B ) )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_895_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( ( member_set_a @ X @ B2 )
& ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_896_insert__subset,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( ( member_nat @ X @ B2 )
& ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_897_insert__subset,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( ( member_a @ X @ B2 )
& ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% insert_subset
thf(fact_898_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_899_Diff__insert0,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_900_Diff__insert0,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% Diff_insert0
thf(fact_901_insert__Diff1,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_902_insert__Diff1,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_903_insert__Diff1,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% insert_Diff1
thf(fact_904_Un__insert__right,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_905_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_906_Un__insert__left,axiom,
! [A: a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B2 ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_907_Un__insert__left,axiom,
! [A: nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B2 ) @ C2 )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_908_set__mset__union,axiom,
! [M5: multiset_set_a,N5: multiset_set_a] :
( ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ M5 @ N5 ) )
= ( sup_sup_set_set_a @ ( set_mset_set_a @ M5 ) @ ( set_mset_set_a @ N5 ) ) ) ).
% set_mset_union
thf(fact_909_set__mset__union,axiom,
! [M5: multiset_a,N5: multiset_a] :
( ( set_mset_a @ ( plus_plus_multiset_a @ M5 @ N5 ) )
= ( sup_sup_set_a @ ( set_mset_a @ M5 ) @ ( set_mset_a @ N5 ) ) ) ).
% set_mset_union
thf(fact_910_set__mset__union,axiom,
! [M5: multiset_nat,N5: multiset_nat] :
( ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ M5 @ N5 ) )
= ( sup_sup_set_nat @ ( set_mset_nat @ M5 ) @ ( set_mset_nat @ N5 ) ) ) ).
% set_mset_union
thf(fact_911_size__union,axiom,
! [M5: multiset_set_a,N5: multiset_set_a] :
( ( size_s6566526139600085008_set_a @ ( plus_p2331992037799027419_set_a @ M5 @ N5 ) )
= ( plus_plus_nat @ ( size_s6566526139600085008_set_a @ M5 ) @ ( size_s6566526139600085008_set_a @ N5 ) ) ) ).
% size_union
thf(fact_912_count__union,axiom,
! [M5: multiset_set_a,N5: multiset_set_a,A: set_a] :
( ( count_set_a @ ( plus_p2331992037799027419_set_a @ M5 @ N5 ) @ A )
= ( plus_plus_nat @ ( count_set_a @ M5 @ A ) @ ( count_set_a @ N5 @ A ) ) ) ).
% count_union
thf(fact_913_point__rep__number__split,axiom,
! [A2: multiset_set_a,B2: multiset_set_a,X: a] :
( ( design6637022207325878697mber_a @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) @ X )
= ( plus_plus_nat @ ( design6637022207325878697mber_a @ A2 @ X ) @ ( design6637022207325878697mber_a @ B2 @ X ) ) ) ).
% point_rep_number_split
thf(fact_914_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ bot_bot_set_nat ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_915_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_916_singleton__insert__inj__eq,axiom,
! [B: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_917_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_918_insert__Diff__single,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= ( insert_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_919_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_920_finite__Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_921_finite__Diff__insert,axiom,
! [A2: set_set_a,A: set_a,B2: set_set_a] :
( ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B2 ) ) )
= ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_922_finite__Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( finite_finite_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( finite_finite_a @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ).
% finite_Diff_insert
thf(fact_923_card__Diff__insert,axiom,
! [A: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ A @ B2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_924_card__Diff__insert,axiom,
! [A: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ A @ B2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_925_card__Diff__insert,axiom,
! [A: a,A2: set_a,B2: set_a] :
( ( member_a @ A @ A2 )
=> ( ~ ( member_a @ A @ B2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) ) )
= ( minus_minus_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% card_Diff_insert
thf(fact_926_DiffE,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_927_DiffE,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_928_DiffE,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B2 ) ) ) ).
% DiffE
thf(fact_929_DiffD1,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ( member_set_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_930_DiffD1,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ( member_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_931_DiffD1,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ( member_a @ C @ A2 ) ) ).
% DiffD1
thf(fact_932_DiffD2,axiom,
! [C: set_a,A2: set_set_a,B2: set_set_a] :
( ( member_set_a @ C @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) )
=> ~ ( member_set_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_933_DiffD2,axiom,
! [C: nat,A2: set_nat,B2: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
=> ~ ( member_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_934_DiffD2,axiom,
! [C: a,A2: set_a,B2: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A2 @ B2 ) )
=> ~ ( member_a @ C @ B2 ) ) ).
% DiffD2
thf(fact_935_insert__Diff__if,axiom,
! [X: set_a,B2: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B2 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_936_insert__Diff__if,axiom,
! [X: nat,B2: set_nat,A2: set_nat] :
( ( ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( minus_minus_set_nat @ A2 @ B2 ) ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ B2 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_937_insert__Diff__if,axiom,
! [X: a,B2: set_a,A2: set_a] :
( ( ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( minus_minus_set_a @ A2 @ B2 ) ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B2 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B2 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_938_incidence__system_Opoint__indices_Ocong,axiom,
design328527185268214962ices_a = design328527185268214962ices_a ).
% incidence_system.point_indices.cong
thf(fact_939_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B5: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B5 ) )
& ~ ( member_set_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_940_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B5: set_a] :
( ( A2
= ( insert_a @ A @ B5 ) )
& ~ ( member_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_941_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B5: set_nat] :
( ( A2
= ( insert_nat @ A @ B5 ) )
& ~ ( member_nat @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_942_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_943_insert__commute,axiom,
! [X: nat,Y: nat,A2: set_nat] :
( ( insert_nat @ X @ ( insert_nat @ Y @ A2 ) )
= ( insert_nat @ Y @ ( insert_nat @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_944_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B: set_a,B2: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B @ B2 )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_set_a] :
( ( A2
= ( insert_set_a @ B @ C5 ) )
& ~ ( member_set_a @ B @ C5 )
& ( B2
= ( insert_set_a @ A @ C5 ) )
& ~ ( member_set_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_945_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B2: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B2 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_a] :
( ( A2
= ( insert_a @ B @ C5 ) )
& ~ ( member_a @ B @ C5 )
& ( B2
= ( insert_a @ A @ C5 ) )
& ~ ( member_a @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_946_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B @ B2 )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B @ B2 ) )
= ( ( ( A = B )
=> ( A2 = B2 ) )
& ( ( A != B )
=> ? [C5: set_nat] :
( ( A2
= ( insert_nat @ B @ C5 ) )
& ~ ( member_nat @ B @ C5 )
& ( B2
= ( insert_nat @ A @ C5 ) )
& ~ ( member_nat @ A @ C5 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_947_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_948_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_949_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_950_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_951_insert__ident,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B2 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_952_insert__ident,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ~ ( member_nat @ X @ B2 )
=> ( ( ( insert_nat @ X @ A2 )
= ( insert_nat @ X @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% insert_ident
thf(fact_953_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B5: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B5 ) )
=> ( member_set_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_954_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_955_Set_Oset__insert,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ~ ! [B5: set_nat] :
( ( A2
= ( insert_nat @ X @ B5 ) )
=> ( member_nat @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_956_insertI2,axiom,
! [A: set_a,B2: set_set_a,B: set_a] :
( ( member_set_a @ A @ B2 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_957_insertI2,axiom,
! [A: a,B2: set_a,B: a] :
( ( member_a @ A @ B2 )
=> ( member_a @ A @ ( insert_a @ B @ B2 ) ) ) ).
% insertI2
thf(fact_958_insertI2,axiom,
! [A: nat,B2: set_nat,B: nat] :
( ( member_nat @ A @ B2 )
=> ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% insertI2
thf(fact_959_insertI1,axiom,
! [A: set_a,B2: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B2 ) ) ).
% insertI1
thf(fact_960_insertI1,axiom,
! [A: a,B2: set_a] : ( member_a @ A @ ( insert_a @ A @ B2 ) ) ).
% insertI1
thf(fact_961_insertI1,axiom,
! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% insertI1
thf(fact_962_insertE,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_963_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_964_insertE,axiom,
! [A: nat,B: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
=> ( ( A != B )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_965_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_966_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_967_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_968_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_969_singleton__inject,axiom,
! [A: nat,B: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B @ bot_bot_set_nat ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_970_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_971_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_972_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_973_doubleton__eq__iff,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
= ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_974_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_975_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_976_singleton__iff,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_977_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_978_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_979_singletonD,axiom,
! [B: nat,A: nat] :
( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_980_Diff__insert__absorb,axiom,
! [X: set_a,A2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_981_Diff__insert__absorb,axiom,
! [X: nat,A2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_982_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_983_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_984_Diff__insert2,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B2 ) ) ).
% Diff_insert2
thf(fact_985_insert__Diff,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_986_insert__Diff,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_987_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_988_Diff__insert,axiom,
! [A2: set_nat,A: nat,B2: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_989_Diff__insert,axiom,
! [A2: set_a,A: a,B2: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B2 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B2 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_990_finite_OinsertI,axiom,
! [A2: set_a,A: a] :
( ( finite_finite_a @ A2 )
=> ( finite_finite_a @ ( insert_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_991_finite_OinsertI,axiom,
! [A2: set_nat,A: nat] :
( ( finite_finite_nat @ A2 )
=> ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_992_finite_OinsertI,axiom,
! [A2: set_set_a,A: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( finite_finite_set_a @ ( insert_set_a @ A @ A2 ) ) ) ).
% finite.insertI
thf(fact_993_subset__insertI2,axiom,
! [A2: set_nat,B2: set_nat,B: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_994_subset__insertI2,axiom,
! [A2: set_a,B2: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B2 ) ) ) ).
% subset_insertI2
thf(fact_995_subset__insertI,axiom,
! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% subset_insertI
thf(fact_996_subset__insertI,axiom,
! [B2: set_a,A: a] : ( ord_less_eq_set_a @ B2 @ ( insert_a @ A @ B2 ) ) ).
% subset_insertI
thf(fact_997_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_998_subset__insert,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ~ ( member_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_999_subset__insert,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ).
% subset_insert
thf(fact_1000_insert__mono,axiom,
! [C2: set_nat,D2: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C2 @ D2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1001_insert__mono,axiom,
! [C2: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C2 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C2 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_1002_subset__Diff__insert,axiom,
! [A2: set_set_a,B2: set_set_a,X: set_a,C2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ ( insert_set_a @ X @ C2 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B2 @ C2 ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1003_subset__Diff__insert,axiom,
! [A2: set_nat,B2: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C2 ) )
& ~ ( member_nat @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1004_subset__Diff__insert,axiom,
! [A2: set_a,B2: set_a,X: a,C2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ ( insert_a @ X @ C2 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B2 @ C2 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1005_union__iff,axiom,
! [A: a,A2: multiset_a,B2: multiset_a] :
( ( member_a @ A @ ( set_mset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) ) )
= ( ( member_a @ A @ ( set_mset_a @ A2 ) )
| ( member_a @ A @ ( set_mset_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_1006_union__iff,axiom,
! [A: nat,A2: multiset_nat,B2: multiset_nat] :
( ( member_nat @ A @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) )
= ( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
| ( member_nat @ A @ ( set_mset_nat @ B2 ) ) ) ) ).
% union_iff
thf(fact_1007_union__iff,axiom,
! [A: set_a,A2: multiset_set_a,B2: multiset_set_a] :
( ( member_set_a @ A @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ A2 @ B2 ) ) )
= ( ( member_set_a @ A @ ( set_mset_set_a @ A2 ) )
| ( member_set_a @ A @ ( set_mset_set_a @ B2 ) ) ) ) ).
% union_iff
thf(fact_1008_empty__neutral_I1_J,axiom,
! [X: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ zero_z5079479921072680283_set_a @ X )
= X ) ).
% empty_neutral(1)
thf(fact_1009_empty__neutral_I2_J,axiom,
! [X: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ X @ zero_z5079479921072680283_set_a )
= X ) ).
% empty_neutral(2)
thf(fact_1010_finite_Ocases,axiom,
! [A: set_set_a] :
( ( finite_finite_set_a @ A )
=> ( ( A != bot_bot_set_set_a )
=> ~ ! [A4: set_set_a] :
( ? [A7: set_a] :
( A
= ( insert_set_a @ A7 @ A4 ) )
=> ~ ( finite_finite_set_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_1011_finite_Ocases,axiom,
! [A: set_a] :
( ( finite_finite_a @ A )
=> ( ( A != bot_bot_set_a )
=> ~ ! [A4: set_a] :
( ? [A7: a] :
( A
= ( insert_a @ A7 @ A4 ) )
=> ~ ( finite_finite_a @ A4 ) ) ) ) ).
% finite.cases
thf(fact_1012_finite_Ocases,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ~ ! [A4: set_nat] :
( ? [A7: nat] :
( A
= ( insert_nat @ A7 @ A4 ) )
=> ~ ( finite_finite_nat @ A4 ) ) ) ) ).
% finite.cases
thf(fact_1013_finite_Osimps,axiom,
( finite_finite_set_a
= ( ^ [A3: set_set_a] :
( ( A3 = bot_bot_set_set_a )
| ? [A5: set_set_a,B8: set_a] :
( ( A3
= ( insert_set_a @ B8 @ A5 ) )
& ( finite_finite_set_a @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_1014_finite_Osimps,axiom,
( finite_finite_a
= ( ^ [A3: set_a] :
( ( A3 = bot_bot_set_a )
| ? [A5: set_a,B8: a] :
( ( A3
= ( insert_a @ B8 @ A5 ) )
& ( finite_finite_a @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_1015_finite_Osimps,axiom,
( finite_finite_nat
= ( ^ [A3: set_nat] :
( ( A3 = bot_bot_set_nat )
| ? [A5: set_nat,B8: nat] :
( ( A3
= ( insert_nat @ B8 @ A5 ) )
& ( finite_finite_nat @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_1016_finite__induct,axiom,
! [F2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1017_finite__induct,axiom,
! [F2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1018_finite__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ).
% finite_induct
thf(fact_1019_finite__ne__induct,axiom,
! [F2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( F2 != bot_bot_set_set_a )
=> ( ! [X2: set_a] : ( P2 @ ( insert_set_a @ X2 @ bot_bot_set_set_a ) )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( F3 != bot_bot_set_set_a )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1020_finite__ne__induct,axiom,
! [F2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( F2 != bot_bot_set_a )
=> ( ! [X2: a] : ( P2 @ ( insert_a @ X2 @ bot_bot_set_a ) )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( F3 != bot_bot_set_a )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1021_finite__ne__induct,axiom,
! [F2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( F2 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P2 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X2 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1022_infinite__finite__induct,axiom,
! [P2: set_set_a > $o,A2: set_set_a] :
( ! [A4: set_set_a] :
( ~ ( finite_finite_set_a @ A4 )
=> ( P2 @ A4 ) )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [X2: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ~ ( member_set_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1023_infinite__finite__induct,axiom,
! [P2: set_a > $o,A2: set_a] :
( ! [A4: set_a] :
( ~ ( finite_finite_a @ A4 )
=> ( P2 @ A4 ) )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [X2: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ~ ( member_a @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1024_infinite__finite__induct,axiom,
! [P2: set_nat > $o,A2: set_nat] :
( ! [A4: set_nat] :
( ~ ( finite_finite_nat @ A4 )
=> ( P2 @ A4 ) )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [X2: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ~ ( member_nat @ X2 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ X2 @ F3 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% infinite_finite_induct
thf(fact_1025_infinite__remove,axiom,
! [S2: set_set_a,A: set_a] :
( ~ ( finite_finite_set_a @ S2 )
=> ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ S2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1026_infinite__remove,axiom,
! [S2: set_nat,A: nat] :
( ~ ( finite_finite_nat @ S2 )
=> ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% infinite_remove
thf(fact_1027_infinite__remove,axiom,
! [S2: set_a,A: a] :
( ~ ( finite_finite_a @ S2 )
=> ~ ( finite_finite_a @ ( minus_minus_set_a @ S2 @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% infinite_remove
thf(fact_1028_infinite__coinduct,axiom,
! [X5: set_set_a > $o,A2: set_set_a] :
( ( X5 @ A2 )
=> ( ! [A4: set_set_a] :
( ( X5 @ A4 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
& ( ( X5 @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) )
| ~ ( finite_finite_set_a @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) ) ) )
=> ~ ( finite_finite_set_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1029_infinite__coinduct,axiom,
! [X5: set_nat > $o,A2: set_nat] :
( ( X5 @ A2 )
=> ( ! [A4: set_nat] :
( ( X5 @ A4 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ A4 )
& ( ( X5 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
=> ~ ( finite_finite_nat @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1030_infinite__coinduct,axiom,
! [X5: set_a > $o,A2: set_a] :
( ( X5 @ A2 )
=> ( ! [A4: set_a] :
( ( X5 @ A4 )
=> ? [X4: a] :
( ( member_a @ X4 @ A4 )
& ( ( X5 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ~ ( finite_finite_a @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) )
=> ~ ( finite_finite_a @ A2 ) ) ) ).
% infinite_coinduct
thf(fact_1031_finite__empty__induct,axiom,
! [A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A7: set_a,A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( member_set_a @ A7 @ A4 )
=> ( ( P2 @ A4 )
=> ( P2 @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ A7 @ bot_bot_set_set_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1032_finite__empty__induct,axiom,
! [A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A7: nat,A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( member_nat @ A7 @ A4 )
=> ( ( P2 @ A4 )
=> ( P2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A7 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P2 @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1033_finite__empty__induct,axiom,
! [A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ( P2 @ A2 )
=> ( ! [A7: a,A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( member_a @ A7 @ A4 )
=> ( ( P2 @ A4 )
=> ( P2 @ ( minus_minus_set_a @ A4 @ ( insert_a @ A7 @ bot_bot_set_a ) ) ) ) ) )
=> ( P2 @ bot_bot_set_a ) ) ) ) ).
% finite_empty_induct
thf(fact_1034_subset__singleton__iff,axiom,
! [X5: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X5 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X5 = bot_bot_set_nat )
| ( X5
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_1035_subset__singleton__iff,axiom,
! [X5: set_a,A: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_1036_subset__singletonD,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_1037_subset__singletonD,axiom,
! [A2: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_1038_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1039_subset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1040_subset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ).
% subset_insert_iff
thf(fact_1041_Diff__single__insert,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1042_Diff__single__insert,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B2 ) ) ) ).
% Diff_single_insert
thf(fact_1043_singleton__Un__iff,axiom,
! [X: a,A2: set_a,B2: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1044_singleton__Un__iff,axiom,
! [X: nat,A2: set_nat,B2: set_nat] :
( ( ( insert_nat @ X @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_1045_Un__singleton__iff,axiom,
! [A2: set_a,B2: set_a,X: a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2 = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B2
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1046_Un__singleton__iff,axiom,
! [A2: set_nat,B2: set_nat,X: nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= ( insert_nat @ X @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2 = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X @ bot_bot_set_nat ) )
& ( B2
= ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_1047_insert__is__Un,axiom,
( insert_a
= ( ^ [A3: a] : ( sup_sup_set_a @ ( insert_a @ A3 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_1048_insert__is__Un,axiom,
( insert_nat
= ( ^ [A3: nat] : ( sup_sup_set_nat @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_1049_card__insert__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ A2 ) @ ( finite_card_a @ ( insert_a @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_1050_card__insert__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A2 ) @ ( finite_card_nat @ ( insert_nat @ X @ A2 ) ) ) ).
% card_insert_le
thf(fact_1051_plus__multiset_Orep__eq,axiom,
! [X: multiset_set_a,Xa2: multiset_set_a] :
( ( count_set_a @ ( plus_p2331992037799027419_set_a @ X @ Xa2 ) )
= ( ^ [A3: set_a] : ( plus_plus_nat @ ( count_set_a @ X @ A3 ) @ ( count_set_a @ Xa2 @ A3 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_1052_left__add__mult__distrib__mset,axiom,
! [I: nat,U: multiset_set_a,J: nat,K2: multiset_set_a] :
( ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I @ U ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J @ U ) @ K2 ) )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( plus_plus_nat @ I @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib_mset
thf(fact_1053_repeat__mset__distrib,axiom,
! [M3: nat,N: nat,A2: multiset_set_a] :
( ( repeat_mset_set_a @ ( plus_plus_nat @ M3 @ N ) @ A2 )
= ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ M3 @ A2 ) @ ( repeat_mset_set_a @ N @ A2 ) ) ) ).
% repeat_mset_distrib
thf(fact_1054_point__index__distrib,axiom,
! [B12: multiset_set_a,B23: multiset_set_a,Ps: set_a] :
( ( design254580327166089565ndex_a @ ( plus_p2331992037799027419_set_a @ B12 @ B23 ) @ Ps )
= ( plus_plus_nat @ ( design254580327166089565ndex_a @ B12 @ Ps ) @ ( design254580327166089565ndex_a @ B23 @ Ps ) ) ) ).
% point_index_distrib
thf(fact_1055_point__index__diff,axiom,
! [B23: multiset_set_a] :
( design254580327166089565ndex_a
= ( ^ [B13: multiset_set_a,Ps2: set_a] : ( minus_minus_nat @ ( design254580327166089565ndex_a @ ( plus_p2331992037799027419_set_a @ B13 @ B23 ) @ Ps2 ) @ ( design254580327166089565ndex_a @ B23 @ Ps2 ) ) ) ) ).
% point_index_diff
thf(fact_1056_finite__incidence__system_Ostrong__del__block__fin,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design9187838744727572296stem_a @ Point_set @ Block_collection )
=> ( design9187838744727572296stem_a @ ( minus_minus_set_a @ Point_set @ B ) @ ( design4241783006516448631lock_a @ Block_collection @ B ) ) ) ).
% finite_incidence_system.strong_del_block_fin
thf(fact_1057_finite__subset__induct_H,axiom,
! [F2: set_set_a,A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [A7: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A7 @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ F3 @ A2 )
=> ( ~ ( member_set_a @ A7 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ A7 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1058_finite__subset__induct_H,axiom,
! [F2: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A7: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A7 @ A2 )
=> ( ( ord_less_eq_set_nat @ F3 @ A2 )
=> ( ~ ( member_nat @ A7 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A7 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1059_finite__subset__induct_H,axiom,
! [F2: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A7: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A7 @ A2 )
=> ( ( ord_less_eq_set_a @ F3 @ A2 )
=> ( ~ ( member_a @ A7 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ A7 @ F3 ) ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1060_finite__subset__induct,axiom,
! [F2: set_set_a,A2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ F2 )
=> ( ( ord_le3724670747650509150_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [A7: set_a,F3: set_set_a] :
( ( finite_finite_set_a @ F3 )
=> ( ( member_set_a @ A7 @ A2 )
=> ( ~ ( member_set_a @ A7 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_set_a @ A7 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1061_finite__subset__induct,axiom,
! [F2: set_nat,A2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ F2 )
=> ( ( ord_less_eq_set_nat @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A7: nat,F3: set_nat] :
( ( finite_finite_nat @ F3 )
=> ( ( member_nat @ A7 @ A2 )
=> ( ~ ( member_nat @ A7 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_nat @ A7 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1062_finite__subset__induct,axiom,
! [F2: set_a,A2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ F2 )
=> ( ( ord_less_eq_set_a @ F2 @ A2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A7: a,F3: set_a] :
( ( finite_finite_a @ F3 )
=> ( ( member_a @ A7 @ A2 )
=> ( ~ ( member_a @ A7 @ F3 )
=> ( ( P2 @ F3 )
=> ( P2 @ ( insert_a @ A7 @ F3 ) ) ) ) ) )
=> ( P2 @ F2 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1063_remove__induct,axiom,
! [P2: set_set_a > $o,B2: set_set_a] :
( ( P2 @ bot_bot_set_set_a )
=> ( ( ~ ( finite_finite_set_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( P2 @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1064_remove__induct,axiom,
! [P2: set_nat > $o,B2: set_nat] :
( ( P2 @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( P2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1065_remove__induct,axiom,
! [P2: set_a > $o,B2: set_a] :
( ( P2 @ bot_bot_set_a )
=> ( ( ~ ( finite_finite_a @ B2 )
=> ( P2 @ B2 ) )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( P2 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% remove_induct
thf(fact_1066_finite__remove__induct,axiom,
! [B2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ B2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [A4: set_set_a] :
( ( finite_finite_set_a @ A4 )
=> ( ( A4 != bot_bot_set_set_a )
=> ( ( ord_le3724670747650509150_set_a @ A4 @ B2 )
=> ( ! [X4: set_a] :
( ( member_set_a @ X4 @ A4 )
=> ( P2 @ ( minus_5736297505244876581_set_a @ A4 @ ( insert_set_a @ X4 @ bot_bot_set_set_a ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1067_finite__remove__induct,axiom,
! [B2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ B2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [A4: set_nat] :
( ( finite_finite_nat @ A4 )
=> ( ( A4 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A4 @ B2 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A4 )
=> ( P2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1068_finite__remove__induct,axiom,
! [B2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ B2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [A4: set_a] :
( ( finite_finite_a @ A4 )
=> ( ( A4 != bot_bot_set_a )
=> ( ( ord_less_eq_set_a @ A4 @ B2 )
=> ( ! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( P2 @ ( minus_minus_set_a @ A4 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) )
=> ( P2 @ A4 ) ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% finite_remove_induct
thf(fact_1069_card__Diff1__le,axiom,
! [A2: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ).
% card_Diff1_le
thf(fact_1070_card__Diff1__le,axiom,
! [A2: set_a,X: a] : ( ord_less_eq_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ).
% card_Diff1_le
thf(fact_1071_card__1__singletonE,axiom,
! [A2: set_a] :
( ( ( finite_card_a @ A2 )
= one_one_nat )
=> ~ ! [X2: a] :
( A2
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% card_1_singletonE
thf(fact_1072_card__1__singletonE,axiom,
! [A2: set_nat] :
( ( ( finite_card_nat @ A2 )
= one_one_nat )
=> ~ ! [X2: nat] :
( A2
!= ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% card_1_singletonE
thf(fact_1073_finite__induct__select,axiom,
! [S2: set_set_a,P2: set_set_a > $o] :
( ( finite_finite_set_a @ S2 )
=> ( ( P2 @ bot_bot_set_set_a )
=> ( ! [T3: set_set_a] :
( ( ord_less_set_set_a @ T3 @ S2 )
=> ( ( P2 @ T3 )
=> ? [X4: set_a] :
( ( member_set_a @ X4 @ ( minus_5736297505244876581_set_a @ S2 @ T3 ) )
& ( P2 @ ( insert_set_a @ X4 @ T3 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1074_finite__induct__select,axiom,
! [S2: set_nat,P2: set_nat > $o] :
( ( finite_finite_nat @ S2 )
=> ( ( P2 @ bot_bot_set_nat )
=> ( ! [T3: set_nat] :
( ( ord_less_set_nat @ T3 @ S2 )
=> ( ( P2 @ T3 )
=> ? [X4: nat] :
( ( member_nat @ X4 @ ( minus_minus_set_nat @ S2 @ T3 ) )
& ( P2 @ ( insert_nat @ X4 @ T3 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1075_finite__induct__select,axiom,
! [S2: set_a,P2: set_a > $o] :
( ( finite_finite_a @ S2 )
=> ( ( P2 @ bot_bot_set_a )
=> ( ! [T3: set_a] :
( ( ord_less_set_a @ T3 @ S2 )
=> ( ( P2 @ T3 )
=> ? [X4: a] :
( ( member_a @ X4 @ ( minus_minus_set_a @ S2 @ T3 ) )
& ( P2 @ ( insert_a @ X4 @ T3 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_1076_psubset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B2: set_set_a] :
( ( ord_less_set_set_a @ A2 @ ( insert_set_a @ X @ B2 ) )
= ( ( ( member_set_a @ X @ B2 )
=> ( ord_less_set_set_a @ A2 @ B2 ) )
& ( ~ ( member_set_a @ X @ B2 )
=> ( ( ( member_set_a @ X @ A2 )
=> ( ord_less_set_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B2 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1077_psubset__insert__iff,axiom,
! [A2: set_nat,X: nat,B2: set_nat] :
( ( ord_less_set_nat @ A2 @ ( insert_nat @ X @ B2 ) )
= ( ( ( member_nat @ X @ B2 )
=> ( ord_less_set_nat @ A2 @ B2 ) )
& ( ~ ( member_nat @ X @ B2 )
=> ( ( ( member_nat @ X @ A2 )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B2 ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1078_psubset__insert__iff,axiom,
! [A2: set_a,X: a,B2: set_a] :
( ( ord_less_set_a @ A2 @ ( insert_a @ X @ B2 ) )
= ( ( ( member_a @ X @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) )
& ( ~ ( member_a @ X @ B2 )
=> ( ( ( member_a @ X @ A2 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B2 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1079_points__index__single__rep__num,axiom,
! [B2: multiset_set_nat,X: nat] :
( ( design6574611146354332593ex_nat @ B2 @ ( insert_nat @ X @ bot_bot_set_nat ) )
= ( design3571518413069006949er_nat @ B2 @ X ) ) ).
% points_index_single_rep_num
thf(fact_1080_points__index__single__rep__num,axiom,
! [B2: multiset_set_a,X: a] :
( ( design254580327166089565ndex_a @ B2 @ ( insert_a @ X @ bot_bot_set_a ) )
= ( design6637022207325878697mber_a @ B2 @ X ) ) ).
% points_index_single_rep_num
thf(fact_1081_card__Diff1__less__iff,axiom,
! [A2: set_set_a,X: set_a] :
( ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) )
= ( ( finite_finite_set_a @ A2 )
& ( member_set_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1082_card__Diff1__less__iff,axiom,
! [A2: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) )
= ( ( finite_finite_nat @ A2 )
& ( member_nat @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1083_card__Diff1__less__iff,axiom,
! [A2: set_a,X: a] :
( ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) )
= ( ( finite_finite_a @ A2 )
& ( member_a @ X @ A2 ) ) ) ).
% card_Diff1_less_iff
thf(fact_1084_card__Diff2__less,axiom,
! [A2: set_set_a,X: set_a,Y: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ( member_set_a @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1085_card__Diff2__less,axiom,
! [A2: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1086_card__Diff2__less,axiom,
! [A2: set_a,X: a,Y: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ( member_a @ Y @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ ( insert_a @ Y @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1087_card__Diff1__less,axiom,
! [A2: set_set_a,X: set_a] :
( ( finite_finite_set_a @ A2 )
=> ( ( member_set_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) ) @ ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1088_card__Diff1__less,axiom,
! [A2: set_nat,X: nat] :
( ( finite_finite_nat @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1089_card__Diff1__less,axiom,
! [A2: set_a,X: a] :
( ( finite_finite_a @ A2 )
=> ( ( member_a @ X @ A2 )
=> ( ord_less_nat @ ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) ) @ ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff1_less
thf(fact_1090_card__Diff__singleton__if,axiom,
! [X: set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( finite_card_set_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1091_card__Diff__singleton__if,axiom,
! [X: nat,A2: set_nat] :
( ( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1092_card__Diff__singleton__if,axiom,
! [X: a,A2: set_a] :
( ( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( finite_card_a @ A2 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1093_card__Diff__singleton,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ( finite_card_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_set_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1094_card__Diff__singleton,axiom,
! [X: nat,A2: set_nat] :
( ( member_nat @ X @ A2 )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1095_card__Diff__singleton,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ( ( finite_card_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) )
= ( minus_minus_nat @ ( finite_card_a @ A2 ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1096_points__index__pair__rep__num,axiom,
! [B2: multiset_set_set_a,X: set_a,Y: set_a] :
( ! [B3: set_set_a] :
( ( member_set_set_a @ B3 @ ( set_mset_set_set_a @ B2 ) )
=> ( member_set_a @ X @ B3 ) )
=> ( ( design88022138586678973_set_a @ B2 @ ( insert_set_a @ X @ ( insert_set_a @ Y @ bot_bot_set_set_a ) ) )
= ( design5008467512594872073_set_a @ B2 @ Y ) ) ) ).
% points_index_pair_rep_num
thf(fact_1097_points__index__pair__rep__num,axiom,
! [B2: multiset_set_nat,X: nat,Y: nat] :
( ! [B3: set_nat] :
( ( member_set_nat @ B3 @ ( set_mset_set_nat @ B2 ) )
=> ( member_nat @ X @ B3 ) )
=> ( ( design6574611146354332593ex_nat @ B2 @ ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
= ( design3571518413069006949er_nat @ B2 @ Y ) ) ) ).
% points_index_pair_rep_num
thf(fact_1098_points__index__pair__rep__num,axiom,
! [B2: multiset_set_a,X: a,Y: a] :
( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ B2 ) )
=> ( member_a @ X @ B3 ) )
=> ( ( design254580327166089565ndex_a @ B2 @ ( insert_a @ X @ ( insert_a @ Y @ bot_bot_set_a ) ) )
= ( design6637022207325878697mber_a @ B2 @ Y ) ) ) ).
% points_index_pair_rep_num
thf(fact_1099_card__insert__le__m1,axiom,
! [N: nat,Y: set_a,X: a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_a @ ( insert_a @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1100_card__insert__le__m1,axiom,
! [N: nat,Y: set_nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_1101_design_Ostrong__del__block__des,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design_design_a @ Point_set @ Block_collection )
=> ( ! [Bl: set_a] :
( ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) )
=> ~ ( ord_less_set_a @ Bl @ B ) )
=> ( design_design_a @ ( minus_minus_set_a @ Point_set @ B ) @ ( design4241783006516448631lock_a @ Block_collection @ B ) ) ) ) ).
% design.strong_del_block_des
thf(fact_1102_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_1103_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1104_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1105_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_1106_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1107_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1108_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_1109_le__supE,axiom,
! [A: set_nat,B: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A @ X )
=> ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1110_le__supE,axiom,
! [A: nat,B: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_nat @ A @ X )
=> ~ ( ord_less_eq_nat @ B @ X ) ) ) ).
% le_supE
thf(fact_1111_le__supE,axiom,
! [A: set_a,B: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A @ X )
=> ~ ( ord_less_eq_set_a @ B @ X ) ) ) ).
% le_supE
thf(fact_1112_le__supI,axiom,
! [A: set_nat,X: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ X )
=> ( ( ord_less_eq_set_nat @ B @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1113_le__supI,axiom,
! [A: nat,X: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X )
=> ( ( ord_less_eq_nat @ B @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1114_le__supI,axiom,
! [A: set_a,X: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ X )
=> ( ( ord_less_eq_set_a @ B @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ X ) ) ) ).
% le_supI
thf(fact_1115_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_1116_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_1117_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_1118_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1119_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_1120_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_1121_le__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_1122_le__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X @ A )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI1
thf(fact_1123_le__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ X @ A )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI1
thf(fact_1124_le__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X @ B )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_1125_le__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ X @ B )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% le_supI2
thf(fact_1126_le__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X @ B )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% le_supI2
thf(fact_1127_sup_Omono,axiom,
! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_1128_sup_Omono,axiom,
! [C: nat,A: nat,D: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ( ord_less_eq_nat @ D @ B )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_1129_sup_Omono,axiom,
! [C: set_a,A: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C @ D ) @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.mono
thf(fact_1130_sup__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1131_sup__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1132_sup__mono,axiom,
! [A: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_1133_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1134_sup__least,axiom,
! [Y: nat,X: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1135_sup__least,axiom,
! [Y: set_a,X: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z2 ) @ X ) ) ) ).
% sup_least
thf(fact_1136_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X3: set_nat,Y5: set_nat] :
( ( sup_sup_set_nat @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_1137_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( sup_sup_nat @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_1138_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X3: set_a,Y5: set_a] :
( ( sup_sup_set_a @ X3 @ Y5 )
= Y5 ) ) ) ).
% le_iff_sup
thf(fact_1139_sup_OorderE,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_1140_sup_OorderE,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( A
= ( sup_sup_nat @ A @ B ) ) ) ).
% sup.orderE
thf(fact_1141_sup_OorderE,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( A
= ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.orderE
thf(fact_1142_sup_OorderI,axiom,
! [A: set_nat,B: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B ) )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_1143_sup_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( sup_sup_nat @ A @ B ) )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% sup.orderI
thf(fact_1144_sup_OorderI,axiom,
! [A: set_a,B: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
=> ( ord_less_eq_set_a @ B @ A ) ) ).
% sup.orderI
thf(fact_1145_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
( ( ord_less_eq_set_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z3 @ X2 )
=> ( ord_less_eq_set_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1146_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X2 )
=> ( ( ord_less_eq_nat @ Z3 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1147_sup__unique,axiom,
! [F: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ X2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_a,Y2: set_a] : ( ord_less_eq_set_a @ Y2 @ ( F @ X2 @ Y2 ) )
=> ( ! [X2: set_a,Y2: set_a,Z3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X2 )
=> ( ( ord_less_eq_set_a @ Z3 @ X2 )
=> ( ord_less_eq_set_a @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_1148_sup_Oabsorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_1149_sup_Oabsorb1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_1150_sup_Oabsorb1,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb1
thf(fact_1151_sup_Oabsorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1152_sup_Oabsorb2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1153_sup_Oabsorb2,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb2
thf(fact_1154_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1155_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1156_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_1157_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1158_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1159_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_1160_sup_OboundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B @ A )
=> ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1161_sup_OboundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_nat @ B @ A )
=> ~ ( ord_less_eq_nat @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1162_sup_OboundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B @ A )
=> ~ ( ord_less_eq_set_a @ C @ A ) ) ) ).
% sup.boundedE
thf(fact_1163_sup_OboundedI,axiom,
! [B: set_nat,A: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1164_sup_OboundedI,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1165_sup_OboundedI,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B @ A )
=> ( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C ) @ A ) ) ) ).
% sup.boundedI
thf(fact_1166_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B8: set_nat,A3: set_nat] :
( A3
= ( sup_sup_set_nat @ A3 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_1167_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B8: nat,A3: nat] :
( A3
= ( sup_sup_nat @ A3 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_1168_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B8: set_a,A3: set_a] :
( A3
= ( sup_sup_set_a @ A3 @ B8 ) ) ) ) ).
% sup.order_iff
thf(fact_1169_sup_Ocobounded1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1170_sup_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1171_sup_Ocobounded1,axiom,
! [A: set_a,B: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded1
thf(fact_1172_sup_Ocobounded2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1173_sup_Ocobounded2,axiom,
! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1174_sup_Ocobounded2,axiom,
! [B: set_a,A: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A @ B ) ) ).
% sup.cobounded2
thf(fact_1175_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B8: set_nat,A3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B8 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1176_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B8: nat,A3: nat] :
( ( sup_sup_nat @ A3 @ B8 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1177_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B8: set_a,A3: set_a] :
( ( sup_sup_set_a @ A3 @ B8 )
= A3 ) ) ) ).
% sup.absorb_iff1
thf(fact_1178_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B8: set_nat] :
( ( sup_sup_set_nat @ A3 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_1179_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B8: nat] :
( ( sup_sup_nat @ A3 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_1180_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B8: set_a] :
( ( sup_sup_set_a @ A3 @ B8 )
= B8 ) ) ) ).
% sup.absorb_iff2
thf(fact_1181_sup_OcoboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1182_sup_OcoboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1183_sup_OcoboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ A )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI1
thf(fact_1184_sup_OcoboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1185_sup_OcoboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1186_sup_OcoboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C @ B )
=> ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.coboundedI2
thf(fact_1187_less__supI1,axiom,
! [X: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ X @ A )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_1188_less__supI1,axiom,
! [X: nat,A: nat,B: nat] :
( ( ord_less_nat @ X @ A )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI1
thf(fact_1189_less__supI1,axiom,
! [X: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ X @ A )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_1190_less__supI2,axiom,
! [X: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ X @ B )
=> ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_1191_less__supI2,axiom,
! [X: nat,B: nat,A: nat] :
( ( ord_less_nat @ X @ B )
=> ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).
% less_supI2
thf(fact_1192_less__supI2,axiom,
! [X: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ X @ B )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_1193_sup_Oabsorb3,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1194_sup_Oabsorb3,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( sup_sup_nat @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1195_sup_Oabsorb3,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( sup_sup_set_a @ A @ B )
= A ) ) ).
% sup.absorb3
thf(fact_1196_sup_Oabsorb4,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1197_sup_Oabsorb4,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( sup_sup_nat @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1198_sup_Oabsorb4,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( sup_sup_set_a @ A @ B )
= B ) ) ).
% sup.absorb4
thf(fact_1199_sup_Ostrict__boundedE,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_set_nat @ B @ A )
=> ~ ( ord_less_set_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1200_sup_Ostrict__boundedE,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B @ C ) @ A )
=> ~ ( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1201_sup_Ostrict__boundedE,axiom,
! [B: set_a,C: set_a,A: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B @ C ) @ A )
=> ~ ( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ C @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_1202_sup_Ostrict__order__iff,axiom,
( ord_less_set_nat
= ( ^ [B8: set_nat,A3: set_nat] :
( ( A3
= ( sup_sup_set_nat @ A3 @ B8 ) )
& ( A3 != B8 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1203_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B8: nat,A3: nat] :
( ( A3
= ( sup_sup_nat @ A3 @ B8 ) )
& ( A3 != B8 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1204_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B8: set_a,A3: set_a] :
( ( A3
= ( sup_sup_set_a @ A3 @ B8 ) )
& ( A3 != B8 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_1205_sup_Ostrict__coboundedI1,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_set_nat @ C @ A )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1206_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ C @ A )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1207_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ C @ A )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_1208_sup_Ostrict__coboundedI2,axiom,
! [C: set_nat,B: set_nat,A: set_nat] :
( ( ord_less_set_nat @ C @ B )
=> ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1209_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1210_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_1211_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_1212_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1213_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_1214_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1215_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1216_add__le__add__imp__diff__le,axiom,
! [I: nat,K2: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1217_add__le__imp__le__diff,axiom,
! [I: nat,K2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).
% add_le_imp_le_diff
thf(fact_1218_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1219_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1220_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1221_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1222_add__point__sys__rep__numbers,axiom,
! [P: a] :
( ( design8835372594653258411bers_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ block_collection )
= ( sup_sup_set_nat @ ( design8835372594653258411bers_a @ point_set @ block_collection ) @ ( insert_nat @ ( design6637022207325878697mber_a @ block_collection @ P ) @ bot_bot_set_nat ) ) ) ).
% add_point_sys_rep_numbers
thf(fact_1223_ex__gt__count__imp__le__multiset,axiom,
! [M5: multiset_nat,N5: multiset_nat,X: nat] :
( ! [Y2: nat] :
( ( member_nat @ Y2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ M5 @ N5 ) ) )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ord_less_nat @ ( count_nat @ M5 @ X ) @ ( count_nat @ N5 @ X ) )
=> ( ord_le5777773500796000884et_nat @ M5 @ N5 ) ) ) ).
% ex_gt_count_imp_le_multiset
thf(fact_1224_ex__gt__count__imp__le__multiset,axiom,
! [M5: multiset_set_a,N5: multiset_set_a,X: set_a] :
( ! [Y2: set_a] :
( ( member_set_a @ Y2 @ ( set_mset_set_a @ ( plus_p2331992037799027419_set_a @ M5 @ N5 ) ) )
=> ( ord_less_eq_set_a @ Y2 @ X ) )
=> ( ( ord_less_nat @ ( count_set_a @ M5 @ X ) @ ( count_set_a @ N5 @ X ) )
=> ( ord_le5765082015083327056_set_a @ M5 @ N5 ) ) ) ).
% ex_gt_count_imp_le_multiset
thf(fact_1225_less__eq__multiset__empty__left,axiom,
! [M5: multiset_set_a] : ( ord_le7905258569527593284_set_a @ zero_z5079479921072680283_set_a @ M5 ) ).
% less_eq_multiset_empty_left
thf(fact_1226_less__eq__multiset__empty__right,axiom,
! [M5: multiset_set_a] :
( ( M5 != zero_z5079479921072680283_set_a )
=> ~ ( ord_le7905258569527593284_set_a @ M5 @ zero_z5079479921072680283_set_a ) ) ).
% less_eq_multiset_empty_right
thf(fact_1227_le__multiset__empty__right,axiom,
! [M5: multiset_set_a] :
~ ( ord_le5765082015083327056_set_a @ M5 @ zero_z5079479921072680283_set_a ) ).
% le_multiset_empty_right
thf(fact_1228_le__multiset__empty__left,axiom,
! [M5: multiset_set_a] :
( ( M5 != zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ zero_z5079479921072680283_set_a @ M5 ) ) ).
% le_multiset_empty_left
thf(fact_1229_bot__multiset__def,axiom,
bot_bo4176661893541381648_set_a = zero_z5079479921072680283_set_a ).
% bot_multiset_def
thf(fact_1230_le__multiset__plus__right__nonempty,axiom,
! [N5: multiset_set_a,M5: multiset_set_a] :
( ( N5 != zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ M5 @ ( plus_p2331992037799027419_set_a @ M5 @ N5 ) ) ) ).
% le_multiset_plus_right_nonempty
thf(fact_1231_le__multiset__plus__left__nonempty,axiom,
! [M5: multiset_set_a,N5: multiset_set_a] :
( ( M5 != zero_z5079479921072680283_set_a )
=> ( ord_le5765082015083327056_set_a @ N5 @ ( plus_p2331992037799027419_set_a @ M5 @ N5 ) ) ) ).
% le_multiset_plus_left_nonempty
thf(fact_1232_ex__gt__imp__less__multiset,axiom,
! [N5: multiset_nat,M5: multiset_nat] :
( ? [Y3: nat] :
( ( member_nat @ Y3 @ ( set_mset_nat @ N5 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ ( set_mset_nat @ M5 ) )
=> ( ord_less_nat @ X2 @ Y3 ) ) )
=> ( ord_le5777773500796000884et_nat @ M5 @ N5 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_1233_ex__gt__imp__less__multiset,axiom,
! [N5: multiset_set_a,M5: multiset_set_a] :
( ? [Y3: set_a] :
( ( member_set_a @ Y3 @ ( set_mset_set_a @ N5 ) )
& ! [X2: set_a] :
( ( member_set_a @ X2 @ ( set_mset_set_a @ M5 ) )
=> ( ord_less_set_a @ X2 @ Y3 ) ) )
=> ( ord_le5765082015083327056_set_a @ M5 @ N5 ) ) ).
% ex_gt_imp_less_multiset
thf(fact_1234_lt__imp__ex__count__lt,axiom,
! [M5: multiset_set_a,N5: multiset_set_a] :
( ( ord_le5765082015083327056_set_a @ M5 @ N5 )
=> ? [Y2: set_a] : ( ord_less_nat @ ( count_set_a @ M5 @ Y2 ) @ ( count_set_a @ N5 @ Y2 ) ) ) ).
% lt_imp_ex_count_lt
thf(fact_1235_less__eq__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le6602235886369790592et_nat
= ( ^ [M6: multiset_nat,N4: multiset_nat] :
! [Y5: nat] :
( ( ord_less_nat @ ( count_nat @ N4 @ Y5 ) @ ( count_nat @ M6 @ Y5 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ Y5 @ X3 )
& ( ord_less_nat @ ( count_nat @ M6 @ X3 ) @ ( count_nat @ N4 @ X3 ) ) ) ) ) ) ).
% less_eq_multiset\<^sub>H\<^sub>O
thf(fact_1236_less__eq__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le7905258569527593284_set_a
= ( ^ [M6: multiset_set_a,N4: multiset_set_a] :
! [Y5: set_a] :
( ( ord_less_nat @ ( count_set_a @ N4 @ Y5 ) @ ( count_set_a @ M6 @ Y5 ) )
=> ? [X3: set_a] :
( ( ord_less_set_a @ Y5 @ X3 )
& ( ord_less_nat @ ( count_set_a @ M6 @ X3 ) @ ( count_set_a @ N4 @ X3 ) ) ) ) ) ) ).
% less_eq_multiset\<^sub>H\<^sub>O
thf(fact_1237_less__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le5777773500796000884et_nat
= ( ^ [M6: multiset_nat,N4: multiset_nat] :
( ( M6 != N4 )
& ! [Y5: nat] :
( ( ord_less_nat @ ( count_nat @ N4 @ Y5 ) @ ( count_nat @ M6 @ Y5 ) )
=> ? [X3: nat] :
( ( ord_less_nat @ Y5 @ X3 )
& ( ord_less_nat @ ( count_nat @ M6 @ X3 ) @ ( count_nat @ N4 @ X3 ) ) ) ) ) ) ) ).
% less_multiset\<^sub>H\<^sub>O
thf(fact_1238_less__multiset_092_060_094sub_062H_092_060_094sub_062O,axiom,
( ord_le5765082015083327056_set_a
= ( ^ [M6: multiset_set_a,N4: multiset_set_a] :
( ( M6 != N4 )
& ! [Y5: set_a] :
( ( ord_less_nat @ ( count_set_a @ N4 @ Y5 ) @ ( count_set_a @ M6 @ Y5 ) )
=> ? [X3: set_a] :
( ( ord_less_set_a @ Y5 @ X3 )
& ( ord_less_nat @ ( count_set_a @ M6 @ X3 ) @ ( count_set_a @ N4 @ X3 ) ) ) ) ) ) ) ).
% less_multiset\<^sub>H\<^sub>O
thf(fact_1239_mset__le__add__iff2,axiom,
! [I: nat,J: nat,U: multiset_set_a,M3: multiset_set_a,N: multiset_set_a] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I @ U ) @ M3 ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J @ U ) @ N ) )
= ( ord_le7905258569527593284_set_a @ M3 @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% mset_le_add_iff2
thf(fact_1240_mset__le__add__iff1,axiom,
! [J: nat,I: nat,U: multiset_set_a,M3: multiset_set_a,N: multiset_set_a] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ I @ U ) @ M3 ) @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ J @ U ) @ N ) )
= ( ord_le7905258569527593284_set_a @ ( plus_p2331992037799027419_set_a @ ( repeat_mset_set_a @ ( minus_minus_nat @ I @ J ) @ U ) @ M3 ) @ N ) ) ) ).
% mset_le_add_iff1
thf(fact_1241_add__block__wf,axiom,
! [B: set_a] : ( design1863209521793301785stem_a @ ( sup_sup_set_a @ point_set @ B ) @ ( design4001997691126659652lock_a @ block_collection @ B ) ) ).
% add_block_wf
thf(fact_1242_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_1243_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_1244_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_1245_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_1246_incidence__system__axioms,axiom,
design1863209521793301785stem_a @ point_set @ block_collection ).
% incidence_system_axioms
thf(fact_1247_multiple__is__wellformed,axiom,
! [N: nat] : ( design1863209521793301785stem_a @ point_set @ ( repeat_mset_set_a @ N @ block_collection ) ) ).
% multiple_is_wellformed
thf(fact_1248_delete__block__wf,axiom,
! [B: set_a] : ( design1863209521793301785stem_a @ point_set @ ( design1146539425385464078lock_a @ block_collection @ B ) ) ).
% delete_block_wf
thf(fact_1249_add__point__wf,axiom,
! [P: a] : ( design1863209521793301785stem_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ block_collection ) ).
% add_point_wf
thf(fact_1250_add__block__wf__cond,axiom,
! [B: set_a] :
( ( ord_less_eq_set_a @ B @ point_set )
=> ( design1863209521793301785stem_a @ point_set @ ( design4001997691126659652lock_a @ block_collection @ B ) ) ) ).
% add_block_wf_cond
thf(fact_1251_strong__del__point__incidence__wf,axiom,
! [P: a] : ( design1863209521793301785stem_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design5657747894866638574ocks_a @ block_collection @ P ) ) ).
% strong_del_point_incidence_wf
thf(fact_1252_delete__point__wf,axiom,
! [P: a] : ( design1863209521793301785stem_a @ ( design108908007054065099oint_a @ point_set @ P ) @ ( design6411949732824333445ocks_a @ block_collection @ P ) ) ).
% delete_point_wf
thf(fact_1253_strong__del__block__wf,axiom,
! [B: set_a] : ( design1863209521793301785stem_a @ ( minus_minus_set_a @ point_set @ B ) @ ( design4241783006516448631lock_a @ block_collection @ B ) ) ).
% strong_del_block_wf
thf(fact_1254_add__point__blocks__wf,axiom,
! [P: a,Bs: set_set_a] : ( design1863209521793301785stem_a @ ( design2964366272795260673oint_a @ point_set @ P ) @ ( design2935547469388721088ocks_a @ block_collection @ P @ Bs ) ) ).
% add_point_blocks_wf
thf(fact_1255_complement__wf,axiom,
design1863209521793301785stem_a @ point_set @ ( design8640656491286871389ocks_a @ point_set @ block_collection ) ).
% complement_wf
thf(fact_1256_incidence__system_Ointro,axiom,
! [Block_collection: multiset_set_a,Point_set: set_a] :
( ! [B3: set_a] :
( ( member_set_a @ B3 @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_eq_set_a @ B3 @ Point_set ) )
=> ( design1863209521793301785stem_a @ Point_set @ Block_collection ) ) ).
% incidence_system.intro
thf(fact_1257_incidence__system_Owellformed,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,B: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ B @ ( set_mset_set_a @ Block_collection ) )
=> ( ord_less_eq_set_a @ B @ Point_set ) ) ) ).
% incidence_system.wellformed
thf(fact_1258_incidence__system__def,axiom,
( design1863209521793301785stem_a
= ( ^ [Point_set2: set_a,Block_collection2: multiset_set_a] :
! [B8: set_a] :
( ( member_set_a @ B8 @ ( set_mset_set_a @ Block_collection2 ) )
=> ( ord_less_eq_set_a @ B8 @ Point_set2 ) ) ) ) ).
% incidence_system_def
thf(fact_1259_incidence__system_Opoint__rep__non__existance,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ~ ( member_set_a @ X @ Point_set )
=> ( ( design5008467512594872073_set_a @ Block_collection @ X )
= zero_zero_nat ) ) ) ).
% incidence_system.point_rep_non_existance
thf(fact_1260_incidence__system_Opoint__rep__non__existance,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ~ ( member_nat @ X @ Point_set )
=> ( ( design3571518413069006949er_nat @ Block_collection @ X )
= zero_zero_nat ) ) ) ).
% incidence_system.point_rep_non_existance
thf(fact_1261_incidence__system_Opoint__rep__non__existance,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ~ ( member_a @ X @ Point_set )
=> ( ( design6637022207325878697mber_a @ Block_collection @ X )
= zero_zero_nat ) ) ) ).
% incidence_system.point_rep_non_existance
thf(fact_1262_incidence__system_Ofinite__sysI,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( finite_finite_nat @ Point_set )
=> ( design5426232790142929158em_nat @ Point_set @ Block_collection ) ) ) ).
% incidence_system.finite_sysI
thf(fact_1263_incidence__system_Ofinite__sysI,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( finite_finite_set_a @ Point_set )
=> ( design1749870844763721896_set_a @ Point_set @ Block_collection ) ) ) ).
% incidence_system.finite_sysI
thf(fact_1264_incidence__system_Ofinite__sysI,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( finite_finite_a @ Point_set )
=> ( design9187838744727572296stem_a @ Point_set @ Block_collection ) ) ) ).
% incidence_system.finite_sysI
thf(fact_1265_incidence__system_Oblock__set__nempty__imp__block__ex,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( Block_collection != zero_z5079479921072680283_set_a )
=> ? [Bl: set_a] : ( member_set_a @ Bl @ ( set_mset_set_a @ Block_collection ) ) ) ) ).
% incidence_system.block_set_nempty_imp_block_ex
thf(fact_1266_incidence__system_Oblock__comp__elem__alt__left,axiom,
! [Point_set: set_set_a,Block_collection: multiset_set_set_a,X: set_a,Bl2: set_set_a,Ps: set_set_a] :
( ( design9013482484999600761_set_a @ Point_set @ Block_collection )
=> ( ( member_set_a @ X @ Bl2 )
=> ( ( ord_le3724670747650509150_set_a @ Ps @ ( design4243878040612417342_set_a @ Point_set @ Bl2 ) )
=> ~ ( member_set_a @ X @ Ps ) ) ) ) ).
% incidence_system.block_comp_elem_alt_left
thf(fact_1267_incidence__system_Oblock__comp__elem__alt__left,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,X: nat,Bl2: set_nat,Ps: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( member_nat @ X @ Bl2 )
=> ( ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ Point_set @ Bl2 ) )
=> ~ ( member_nat @ X @ Ps ) ) ) ) ).
% incidence_system.block_comp_elem_alt_left
thf(fact_1268_incidence__system_Oblock__comp__elem__alt__left,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,X: a,Bl2: set_a,Ps: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( member_a @ X @ Bl2 )
=> ( ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ Point_set @ Bl2 ) )
=> ~ ( member_a @ X @ Ps ) ) ) ) ).
% incidence_system.block_comp_elem_alt_left
thf(fact_1269_incidence__system_Oblock__comp__elem__alt__right,axiom,
! [Point_set: set_nat,Block_collection: multiset_set_nat,Ps: set_nat,Bl2: set_nat] :
( ( design3753904077504641269em_nat @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_nat @ Ps @ Point_set )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ Ps )
=> ~ ( member_nat @ X2 @ Bl2 ) )
=> ( ord_less_eq_set_nat @ Ps @ ( design2875492832550762736nt_nat @ Point_set @ Bl2 ) ) ) ) ) ).
% incidence_system.block_comp_elem_alt_right
thf(fact_1270_incidence__system_Oblock__comp__elem__alt__right,axiom,
! [Point_set: set_a,Block_collection: multiset_set_a,Ps: set_a,Bl2: set_a] :
( ( design1863209521793301785stem_a @ Point_set @ Block_collection )
=> ( ( ord_less_eq_set_a @ Ps @ Point_set )
=> ( ! [X2: a] :
( ( member_a @ X2 @ Ps )
=> ~ ( member_a @ X2 @ Bl2 ) )
=> ( ord_less_eq_set_a @ Ps @ ( design6447616907850319326ment_a @ Point_set @ Bl2 ) ) ) ) ) ).
% incidence_system.block_comp_elem_alt_right
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_eq_nat @ ( size_s6566526139600085008_set_a @ block_collection ) @ ( finite_card_a @ point_set ) ).
%------------------------------------------------------------------------------