TPTP Problem File: SLH0527^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% 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 : Eval_FO/0005_Ailamazyan/prob_04677_199579__16242616_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1516 ( 766 unt; 233 typ; 0 def)
% Number of atoms : 3184 (1475 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9550 ( 275 ~; 61 |; 261 &;8003 @)
% ( 0 <=>; 950 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 1163 (1163 >; 0 *; 0 +; 0 <<)
% Number of symbols : 204 ( 201 usr; 39 con; 0-6 aty)
% Number of variables : 3332 ( 410 ^;2887 !; 35 ?;3332 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:13:08.038
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
thf(ty_n_t__Product____Type__Oprod_It__Set__Oset_Itf__a_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
produc8208987855206359447_a_nat: $tType ).
thf(ty_n_t__Mapping__Omapping_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
mappin5678568201568998833_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
produc5986856040925105290_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_P5056861408695629236_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
list_o8492589577666311121_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
set_op4410912819191416235_a_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
option4559388950256476203_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
set_se5822283258546872870_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
set_se4330304633200676677_a_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
option2159618704653257803_a_nat: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_n989787106983797996_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_na3699693778330250182_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_l4703314356710769291_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_li6526943997496501093_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
list_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__b_Mt__Nat__Onat_J,type,
product_prod_b_nat: $tType ).
thf(ty_n_t__List__Olist_It__FO__Ofo____term_Itf__a_J_J,type,
list_fo_term_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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__FO__Ofo____fmla_Itf__a_Mtf__b_J,type,
fo_fmla_a_b: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__List__Olist_I_Eo_J,type,
list_o: $tType ).
thf(ty_n_t__Set__Oset_I_Eo_J,type,
set_o: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (201)
thf(sy_c_Ailamazyan_Oact__edom_001tf__a_001tf__b,type,
act_edom_a_b: fo_fmla_a_b > ( product_prod_b_nat > set_list_a ) > set_a ).
thf(sy_c_Ailamazyan_Oad__agr__close__set_001tf__a,type,
ad_agr_close_set_a: set_a > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Oad__agr__list_001tf__a_001t__Nat__Onat,type,
ad_agr_list_a_nat: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Oad__agr__sets_001tf__a_001t__Nat__Onat,type,
ad_agr_sets_a_nat: set_nat > set_nat > set_a > ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o ).
thf(sy_c_Ailamazyan_Oesat_001tf__a_001tf__b,type,
esat_a_b: fo_fmla_a_b > ( product_prod_b_nat > set_list_a ) > ( nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Oeval__ajoin_001tf__a,type,
eval_ajoin_a: list_nat > produc8208987855206359447_a_nat > list_nat > produc8208987855206359447_a_nat > produc8208987855206359447_a_nat ).
thf(sy_c_Ailamazyan_Oeval__conj__set_001tf__a,type,
eval_conj_set_a: set_a > list_nat > set_li6526943997496501093_a_nat > list_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Oeval__eterms_001tf__a_001t__Nat__Onat,type,
eval_eterms_a_nat: ( nat > sum_sum_a_nat ) > list_fo_term_a > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Oeval__table_001tf__a_001t__Nat__Onat,type,
eval_table_a_nat: list_fo_term_a > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Oext__tuple__set_001tf__a,type,
ext_tuple_set_a: set_a > list_nat > list_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Ofo__nmlz_001tf__a,type,
fo_nmlz_a: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Ofo__wf_001tf__a_001tf__b,type,
fo_wf_a_b: fo_fmla_a_b > ( product_prod_b_nat > set_list_a ) > produc8208987855206359447_a_nat > $o ).
thf(sy_c_Ailamazyan_Oidx__join_001tf__a,type,
idx_join_a: set_a > list_nat > list_nat > set_li6526943997496501093_a_nat > list_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Ailamazyan_Oproj__tuple_001tf__a,type,
proj_tuple_a: list_nat > list_P5056861408695629236_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Ailamazyan_Oproj__vals_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
proj_v3643391342904276326_a_nat: set_na3699693778330250182_a_nat > list_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Cluster_Ocluster_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
cluste8110478419151344300_a_nat: ( list_Sum_sum_a_nat > option2159618704653257803_a_nat ) > set_li6526943997496501093_a_nat > mappin5678568201568998833_a_nat ).
thf(sy_c_FO_Ofv__fo__fmla_001tf__a_001tf__b,type,
fv_fo_fmla_a_b: fo_fmla_a_b > set_nat ).
thf(sy_c_FO_Ofv__fo__fmla__list_001tf__a_001tf__b,type,
fv_fo_fmla_list_a_b: fo_fmla_a_b > list_nat ).
thf(sy_c_FO_Ofv__fo__terms__list_001tf__a,type,
fv_fo_terms_list_a: list_fo_term_a > list_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
comp_l6861429754633178466_a_nat: ( list_l4703314356710769291_a_nat > list_o8492589577666311121_a_nat ) > ( list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ) > list_l4703314356710769291_a_nat > list_o8492589577666311121_a_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
comp_l4895937012117054562st_nat: ( list_nat > list_Sum_sum_a_nat ) > ( list_nat > list_nat ) > list_nat > list_Sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
comp_l2590854360456707st_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > ( list_nat > list_Sum_sum_a_nat ) > list_nat > list_Sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_l3689981812903912354_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_l8558361960953920476_a_nat: ( list_Sum_sum_a_nat > option2159618704653257803_a_nat ) > ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat > option2159618704653257803_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
comp_n1522318729830440540at_nat: ( nat > sum_sum_a_nat ) > ( nat > nat ) > nat > sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Option__Ooption_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_o2481550590722179362_a_nat: ( option2159618704653257803_a_nat > option2159618704653257803_a_nat ) > ( list_Sum_sum_a_nat > option2159618704653257803_a_nat ) > list_Sum_sum_a_nat > option2159618704653257803_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
comp_S2395038772458240499at_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
minus_7403279044699189488_nat_o: ( ( nat > sum_sum_a_nat ) > $o ) > ( ( nat > sum_sum_a_nat ) > $o ) > ( nat > sum_sum_a_nat ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
minus_5799240027661600017_nat_o: ( list_Sum_sum_a_nat > $o ) > ( list_Sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
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_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
minus_5517490076408937517_a_nat: set_na3699693778330250182_a_nat > set_na3699693778330250182_a_nat > set_na3699693778330250182_a_nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
minus_minus_set_o: set_o > set_o > set_o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
minus_7395159227704179404_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_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_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
uminus8220868946613220448_nat_o: ( ( nat > sum_sum_a_nat ) > $o ) > ( nat > sum_sum_a_nat ) > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
uminus4331925969656920193_nat_o: ( list_Sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
uminus3259373730256538813_a_nat: set_na3699693778330250182_a_nat > set_na3699693778330250182_a_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_Eo_J,type,
uminus_uminus_set_o: set_o > set_o ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
uminus2192744996606729052_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
uminus_uminus_set_a: set_a > set_a ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
inf_in8399021836546144180_a_nat: set_na3699693778330250182_a_nat > set_na3699693778330250182_a_nat > set_na3699693778330250182_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
inf_in3249246906714053971_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
sup_su3226716170639437251_nat_o: ( ( nat > sum_sum_a_nat ) > $o ) > ( ( nat > sum_sum_a_nat ) > $o ) > ( nat > sum_sum_a_nat ) > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
sup_su1334248866174809316_nat_o: ( list_Sum_sum_a_nat > $o ) > ( list_Sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Set__Oset_I_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_set_o_set_o: ( set_o > set_o ) > set_set_o > set_set_o ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
image_3472601871771700037_a_nat: ( set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ) > set_se4330304633200676677_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_Itf__a_J_001t__Set__Oset_Itf__a_J,type,
image_set_a_set_a: ( set_a > set_a ) > set_set_a > set_set_a ).
thf(sy_c_Set_Oimage_001tf__a_001_Eo,type,
image_a_o: ( a > $o ) > set_a > set_o ).
thf(sy_c_Set_Oimage_001tf__a_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_7897140031816760844_a_nat: ( a > list_Sum_sum_a_nat ) > set_a > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
insert5265011953798106934_a_nat: ( nat > sum_sum_a_nat ) > set_na3699693778330250182_a_nat > set_na3699693778330250182_a_nat ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
insert2950094090816004437_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_member_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member8690443509505302927_a_nat: ( nat > sum_sum_a_nat ) > set_na3699693778330250182_a_nat > $o ).
thf(sy_c_member_001_Eo,type,
member_o: $o > set_o > $o ).
thf(sy_c_member_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member408289922725080238_a_nat: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
member3060896489619847151_a_nat: set_na3699693778330250182_a_nat > set_se5822283258546872870_a_nat > $o ).
thf(sy_c_member_001t__Set__Oset_I_Eo_J,type,
member_set_o: set_o > set_set_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
member5553968465346197646_a_nat: set_li6526943997496501093_a_nat > set_se4330304633200676677_a_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_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_AD_092_060Delta_062_092_060phi_062____,type,
aD_Delta_phi: set_a ).
thf(sy_v_AD_092_060Delta_062_092_060psi_062____,type,
aD_Delta_psi: set_a ).
thf(sy_v_AD_092_060phi_062____,type,
aD_phi: set_a ).
thf(sy_v_AD_092_060psi_062____,type,
aD_psi: set_a ).
thf(sy_v_AD____,type,
ad: set_a ).
thf(sy_v_I,type,
i: product_prod_b_nat > set_list_a ).
thf(sy_v_S_092_060phi_062____,type,
s_phi: set_na3699693778330250182_a_nat ).
thf(sy_v_S_092_060psi_062____,type,
s_psi: set_na3699693778330250182_a_nat ).
thf(sy_v_X_092_060phi_062_H____,type,
x_phi: set_li6526943997496501093_a_nat ).
thf(sy_v_X_092_060phi_062____,type,
x_phi2: set_li6526943997496501093_a_nat ).
thf(sy_v_X_092_060psi_062____,type,
x_psi: set_li6526943997496501093_a_nat ).
thf(sy_v_X____,type,
x: set_li6526943997496501093_a_nat ).
thf(sy_v_Y____,type,
y: set_li6526943997496501093_a_nat ).
thf(sy_v__092_060phi_062,type,
phi: fo_fmla_a_b ).
thf(sy_v__092_060psi_062,type,
psi: fo_fmla_a_b ).
thf(sy_v__092_060sigma_062_H_H,type,
sigma: nat > sum_sum_a_nat ).
thf(sy_v__092_060sigma_062_H____,type,
sigma2: nat > sum_sum_a_nat ).
thf(sy_v__092_060sigma_062____,type,
sigma3: nat > sum_sum_a_nat ).
thf(sy_v__092_060tau_062,type,
tau: nat > sum_sum_a_nat ).
thf(sy_v_both____,type,
both: list_nat ).
thf(sy_v_idx_092_060phi_062____,type,
idx_phi: mappin5678568201568998833_a_nat ).
thf(sy_v_idx_092_060psi_062____,type,
idx_psi: mappin5678568201568998833_a_nat ).
thf(sy_v_n_092_060phi_062____,type,
n_phi: nat ).
thf(sy_v_n_092_060psi_062____,type,
n_psi: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_ns_092_060phi_062_H____,type,
ns_phi: list_nat ).
thf(sy_v_ns_092_060phi_062____,type,
ns_phi2: list_nat ).
thf(sy_v_ns_092_060psi_062_H____,type,
ns_psi: list_nat ).
thf(sy_v_ns_092_060psi_062____,type,
ns_psi2: list_nat ).
thf(sy_v_ns____,type,
ns: list_nat ).
thf(sy_v_res____,type,
res: mappin5678568201568998833_a_nat ).
thf(sy_v_t_092_060phi_062,type,
t_phi: produc8208987855206359447_a_nat ).
thf(sy_v_t_092_060psi_062,type,
t_psi: produc8208987855206359447_a_nat ).
thf(sy_v_x____,type,
x2: list_Sum_sum_a_nat ).
% Relevant facts (1278)
thf(fact_0_ad__agr__S_092_060psi_062,axiom,
! [Tau: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Tau @ s_psi )
=> ( ( ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ Tau @ ns_psi2 ) @ ( map_na823391071729141993_a_nat @ Tau2 @ ns_psi2 ) )
=> ( member8690443509505302927_a_nat @ Tau2 @ s_psi ) ) ) ).
% ad_agr_S\<psi>
thf(fact_1_ad__agr__list__comm,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X @ Xs @ Ys )
=> ( ad_agr_list_a_nat @ X @ Ys @ Xs ) ) ).
% ad_agr_list_comm
thf(fact_2_ad__agr__list__refl,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ X @ Xs @ Xs ) ).
% ad_agr_list_refl
thf(fact_3_ad__agr__list__trans,axiom,
! [X: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X @ Xs @ Ys )
=> ( ( ad_agr_list_a_nat @ X @ Ys @ Zs )
=> ( ad_agr_list_a_nat @ X @ Xs @ Zs ) ) ) ).
% ad_agr_list_trans
thf(fact_4_AD__def_I4_J,axiom,
ord_less_eq_set_a @ aD_psi @ ad ).
% AD_def(4)
thf(fact_5_AD_092_060Delta_062_092_060psi_062__def,axiom,
( aD_Delta_psi
= ( minus_minus_set_a @ ad @ aD_psi ) ) ).
% AD\<Delta>\<psi>_def
thf(fact_6_AD__delta_I2_J,axiom,
( ad
= ( sup_sup_set_a @ aD_psi @ aD_Delta_psi ) ) ).
% AD_delta(2)
thf(fact_7_AD__def_I5_J,axiom,
( ad
= ( sup_sup_set_a @ aD_phi @ aD_psi ) ) ).
% AD_def(5)
thf(fact_8_ns__sd_I5_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ ns_psi2 ) ).
% ns_sd(5)
thf(fact_9__092_060open_062fo__nmlz_AAD_092_060psi_062_A_096_Aproj__vals_A_123_092_060sigma_062_A_092_060in_062_AS_092_060psi_062_O_Afo__nmlz_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_Ans_J_A_061_Afo__nmlz_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_H_Ans_J_125_Ans_092_060psi_062_A_061_Afo__nmlz_AAD_092_060psi_062_A_096_Aproj__vals_A_123_092_060sigma_062_A_092_060in_062_AS_092_060psi_062_O_Aad__agr__list_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_Ans_J_A_Imap_A_092_060sigma_062_H_Ans_J_125_Ans_092_060psi_062_092_060close_062,axiom,
( ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma @ s_psi )
& ( ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) )
= ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) ) )
@ ns_psi2 ) )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma @ s_psi )
& ( ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) )
@ ns_psi2 ) ) ) ).
% \<open>fo_nmlz AD\<psi> ` proj_vals {\<sigma> \<in> S\<psi>. fo_nmlz AD\<psi> (map \<sigma> ns) = fo_nmlz AD\<psi> (map \<sigma>' ns)} ns\<psi> = fo_nmlz AD\<psi> ` proj_vals {\<sigma> \<in> S\<psi>. ad_agr_list AD\<psi> (map \<sigma> ns) (map \<sigma>' ns)} ns\<psi>\<close>
thf(fact_10_Y__def,axiom,
( y
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma @ s_psi )
& ( ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) )
@ ns_psi2 ) ) ) ).
% Y_def
thf(fact_11_x__ns,axiom,
( ( proj_tuple_a @ ns @ ( zip_na2013496608136855606_a_nat @ ns_phi2 @ x2 ) )
= ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ).
% x_ns
thf(fact_12__092_060open_062ad__agr__close__set_AAD_092_060Delta_062_092_060psi_062_A_Ifo__nmlz_AAD_092_060psi_062_A_096_Aproj__vals_A_123_092_060sigma_062_A_092_060in_062_A_N_AS_092_060psi_062_O_Aad__agr__list_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_Ans_J_A_Imap_A_092_060sigma_062_H_Ans_J_125_Ans_092_060psi_062_J_A_061_Afo__nmlz_AAD_A_096_Aproj__vals_A_123_092_060sigma_062_A_092_060in_062_A_N_AS_092_060psi_062_O_Aad__agr__list_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_Ans_J_A_Imap_A_092_060sigma_062_H_Ans_J_125_Ans_092_060psi_062_092_060close_062,axiom,
( ( ad_agr_close_set_a @ aD_Delta_psi
@ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma @ ( uminus3259373730256538813_a_nat @ s_psi ) )
& ( ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) )
@ ns_psi2 ) ) )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma @ ( uminus3259373730256538813_a_nat @ s_psi ) )
& ( ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) )
@ ns_psi2 ) ) ) ).
% \<open>ad_agr_close_set AD\<Delta>\<psi> (fo_nmlz AD\<psi> ` proj_vals {\<sigma> \<in> - S\<psi>. ad_agr_list AD\<psi> (map \<sigma> ns) (map \<sigma>' ns)} ns\<psi>) = fo_nmlz AD ` proj_vals {\<sigma> \<in> - S\<psi>. ad_agr_list AD\<psi> (map \<sigma> ns) (map \<sigma>' ns)} ns\<psi>\<close>
thf(fact_13_AD__def_I3_J,axiom,
ord_less_eq_set_a @ aD_phi @ ad ).
% AD_def(3)
thf(fact_14_map__eq__map__tailrec,axiom,
map_na823391071729141993_a_nat = map_ta1136998156224711455_a_nat ).
% map_eq_map_tailrec
thf(fact_15_ad__agr__list__fv__list,axiom,
! [X: set_a,Sigma2: nat > sum_sum_a_nat,Ts: list_fo_term_a,Tau3: nat > sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X @ ( eval_eterms_a_nat @ Sigma2 @ Ts ) @ ( eval_eterms_a_nat @ Tau3 @ Ts ) )
=> ( ad_agr_list_a_nat @ X @ ( map_na823391071729141993_a_nat @ Sigma2 @ ( fv_fo_terms_list_a @ Ts ) ) @ ( map_na823391071729141993_a_nat @ Tau3 @ ( fv_fo_terms_list_a @ Ts ) ) ) ) ).
% ad_agr_list_fv_list
thf(fact_16_ns__sd_I4_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ ns_phi2 ) ).
% ns_sd(4)
thf(fact_17_map__eq__conv,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= ( G @ X2 ) ) ) ) ) ).
% map_eq_conv
thf(fact_18_list_Oset__map,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F @ V ) )
= ( image_5081948215111134021_a_nat @ F @ ( set_li2392974972034027290_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_19_list_Oset__map,axiom,
! [F: nat > sum_sum_a_nat,V: list_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ V ) )
= ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_20_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_21__092_060open_062_123ys_A_092_060in_062_Afo__nmlz_AAD_092_060psi_062_A_096_Aproj__vals_AS_092_060psi_062_Ans_092_060psi_062_O_Afo__nmlz_AAD_092_060psi_062_A_Iproj__tuple_Ans_A_Izip_Ans_092_060psi_062_Ays_J_J_A_061_Afo__nmlz_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_H_Ans_J_125_A_061_Afo__nmlz_AAD_092_060psi_062_A_096_Aproj__vals_A_123_092_060sigma_062_A_092_060in_062_AS_092_060psi_062_O_Afo__nmlz_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_Ans_J_A_061_Afo__nmlz_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_H_Ans_J_125_Ans_092_060psi_062_092_060close_062,axiom,
( ( collec7555443234367654128_a_nat
@ ^ [Ys2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Ys2 @ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi ) @ ( proj_v3643391342904276326_a_nat @ s_psi @ ns_psi2 ) ) )
& ( ( fo_nmlz_a @ aD_psi @ ( proj_tuple_a @ ns @ ( zip_na2013496608136855606_a_nat @ ns_psi2 @ Ys2 ) ) )
= ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) ) )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma @ s_psi )
& ( ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) )
= ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) ) )
@ ns_psi2 ) ) ) ).
% \<open>{ys \<in> fo_nmlz AD\<psi> ` proj_vals S\<psi> ns\<psi>. fo_nmlz AD\<psi> (proj_tuple ns (zip ns\<psi> ys)) = fo_nmlz AD\<psi> (map \<sigma>' ns)} = fo_nmlz AD\<psi> ` proj_vals {\<sigma> \<in> S\<psi>. fo_nmlz AD\<psi> (map \<sigma> ns) = fo_nmlz AD\<psi> (map \<sigma>' ns)} ns\<psi>\<close>
thf(fact_22__092_060open_062Y_A_061_A_123ys_A_092_060in_062_Afo__nmlz_AAD_092_060psi_062_A_096_Aproj__vals_AS_092_060psi_062_Ans_092_060psi_062_O_Afo__nmlz_AAD_092_060psi_062_A_Iproj__tuple_Ans_A_Izip_Ans_092_060psi_062_Ays_J_J_A_061_Afo__nmlz_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_H_Ans_J_125_092_060close_062,axiom,
( y
= ( collec7555443234367654128_a_nat
@ ^ [Ys2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Ys2 @ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi ) @ ( proj_v3643391342904276326_a_nat @ s_psi @ ns_psi2 ) ) )
& ( ( fo_nmlz_a @ aD_psi @ ( proj_tuple_a @ ns @ ( zip_na2013496608136855606_a_nat @ ns_psi2 @ Ys2 ) ) )
= ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) ) ) ) ).
% \<open>Y = {ys \<in> fo_nmlz AD\<psi> ` proj_vals S\<psi> ns\<psi>. fo_nmlz AD\<psi> (proj_tuple ns (zip ns\<psi> ys)) = fo_nmlz AD\<psi> (map \<sigma>' ns)}\<close>
thf(fact_23_subset__code_I1_J,axiom,
! [Xs: list_n989787106983797996_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ ( set_na645604395003041787_a_nat @ Xs ) @ B )
= ( ! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( member8690443509505302927_a_nat @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_24_subset__code_I1_J,axiom,
! [Xs: list_l4703314356710769291_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ B )
= ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( member408289922725080238_a_nat @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_25_subset__code_I1_J,axiom,
! [Xs: list_a,B: set_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B )
= ( ! [X2: a] :
( ( member_a @ X2 @ ( set_a2 @ Xs ) )
=> ( member_a @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_26_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X2 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_27_image__set,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( set_li2392974972034027290_a_nat @ Xs ) )
= ( set_li2392974972034027290_a_nat @ ( map_li6507455427659069316_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_28_image__set,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_29_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_30_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ? [Xs2: list_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F @ Xs2 ) ) )
= ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y: nat] :
( X2
= ( F @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_31_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_32_map__idI,axiom,
! [Xs: list_n989787106983797996_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ! [X3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_na722425985163074756_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_33_map__idI,axiom,
! [Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_34_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= X3 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_35_map__ext,axiom,
! [Xs: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_36_list_Omap__ident__strong,axiom,
! [T: list_n989787106983797996_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ! [Z: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Z @ ( set_na645604395003041787_a_nat @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_na722425985163074756_a_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_37_list_Omap__ident__strong,axiom,
! [T: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [Z: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Z @ ( set_li2392974972034027290_a_nat @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_38_list_Omap__ident__strong,axiom,
! [T: list_nat,F: nat > nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_nat_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_39_list_Oinj__map__strong,axiom,
! [X4: list_nat,Xa: list_nat,F: nat > sum_sum_a_nat,Fa: nat > sum_sum_a_nat] :
( ! [Z: nat,Za: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X4 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_na823391071729141993_a_nat @ F @ X4 )
= ( map_na823391071729141993_a_nat @ Fa @ Xa ) )
=> ( X4 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_40_list_Omap__cong0,axiom,
! [X4: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X4 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X4 )
= ( map_na823391071729141993_a_nat @ G @ X4 ) ) ) ).
% list.map_cong0
thf(fact_41_list_Omap__cong,axiom,
! [X4: list_nat,Ya: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( X4 = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X4 )
= ( map_na823391071729141993_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_42_ad__agr__list__subset,axiom,
! [Ms: list_a,Ns: list_a,X: set_a,Sigma2: a > sum_sum_a_nat,Sigma3: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ms ) @ ( set_a2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ns ) @ ( map_a_Sum_sum_a_nat @ Sigma3 @ Ns ) )
=> ( ad_agr_list_a_nat @ X @ ( map_a_Sum_sum_a_nat @ Sigma2 @ Ms ) @ ( map_a_Sum_sum_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_43_ad__agr__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,X: set_a,Sigma2: nat > sum_sum_a_nat,Sigma3: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ns ) )
=> ( ad_agr_list_a_nat @ X @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma3 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_44_fo__nmlz__map,axiom,
! [AD: set_a,Sigma2: nat > sum_sum_a_nat,Ns: list_nat] :
? [Tau4: nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) )
= ( map_na823391071729141993_a_nat @ Tau4 @ Ns ) ) ).
% fo_nmlz_map
thf(fact_45_fo__nmlz__ad__agr,axiom,
! [AD: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ AD @ Xs @ ( fo_nmlz_a @ AD @ Xs ) ) ).
% fo_nmlz_ad_agr
thf(fact_46_fo__nmlz__eqI,axiom,
! [AD: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ AD @ Vs @ Vs2 )
=> ( ( fo_nmlz_a @ AD @ Vs )
= ( fo_nmlz_a @ AD @ Vs2 ) ) ) ).
% fo_nmlz_eqI
thf(fact_47_fo__nmlz__eqD,axiom,
! [AD: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ( fo_nmlz_a @ AD @ Vs )
= ( fo_nmlz_a @ AD @ Vs2 ) )
=> ( ad_agr_list_a_nat @ AD @ Vs @ Vs2 ) ) ).
% fo_nmlz_eqD
thf(fact_48_fo__nmlz__eq,axiom,
! [AD: set_a,Vs: list_Sum_sum_a_nat,Vs2: list_Sum_sum_a_nat] :
( ( ( fo_nmlz_a @ AD @ Vs )
= ( fo_nmlz_a @ AD @ Vs2 ) )
= ( ad_agr_list_a_nat @ AD @ Vs @ Vs2 ) ) ).
% fo_nmlz_eq
thf(fact_49_ad__agr__list__mono,axiom,
! [X: set_a,Y2: set_a,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X @ Y2 )
=> ( ( ad_agr_list_a_nat @ Y2 @ Ys @ Xs )
=> ( ad_agr_list_a_nat @ X @ Ys @ Xs ) ) ) ).
% ad_agr_list_mono
thf(fact_50_AD_092_060Delta_062_092_060phi_062__def,axiom,
( aD_Delta_phi
= ( minus_minus_set_a @ ad @ aD_phi ) ) ).
% AD\<Delta>\<phi>_def
thf(fact_51_AD__delta_I1_J,axiom,
( ad
= ( sup_sup_set_a @ aD_phi @ aD_Delta_phi ) ) ).
% AD_delta(1)
thf(fact_52_ad__agr__S_092_060phi_062,axiom,
! [Tau: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Tau @ s_phi )
=> ( ( ad_agr_list_a_nat @ aD_phi @ ( map_na823391071729141993_a_nat @ Tau @ ns_phi2 ) @ ( map_na823391071729141993_a_nat @ Tau2 @ ns_phi2 ) )
=> ( member8690443509505302927_a_nat @ Tau2 @ s_phi ) ) ) ).
% ad_agr_S\<phi>
thf(fact_53_ns__sd_I8_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns_psi2 ) @ ( set_nat2 @ both ) ).
% ns_sd(8)
thf(fact_54_ns__sd_I6_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ both ) ).
% ns_sd(6)
thf(fact_55_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_56_mem__Collect__eq,axiom,
! [A: nat > sum_sum_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ A @ ( collec5629555741568564177_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_57_mem__Collect__eq,axiom,
! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_58_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_61_Collect__cong,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec5629555741568564177_a_nat @ P )
= ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_62_Collect__cong,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collec7555443234367654128_a_nat @ P )
= ( collec7555443234367654128_a_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_63_Compl__Diff__eq,axiom,
! [A2: set_nat,B: set_nat] :
( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_64_Compl__Diff__eq,axiom,
! [A2: set_a,B: set_a] :
( ( uminus_uminus_set_a @ ( minus_minus_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ A2 ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_65_Compl__Diff__eq,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( uminus2192744996606729052_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( uminus2192744996606729052_a_nat @ A2 ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_66_Compl__Diff__eq,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( uminus3259373730256538813_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ B ) )
= ( sup_su3329769938372955546_a_nat @ ( uminus3259373730256538813_a_nat @ A2 ) @ B ) ) ).
% Compl_Diff_eq
thf(fact_67_Compl__anti__mono,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ B )
=> ( ord_le8108555184339247974_a_nat @ ( uminus3259373730256538813_a_nat @ B ) @ ( uminus3259373730256538813_a_nat @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_68_Compl__anti__mono,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_69_Compl__anti__mono,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B ) @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_70_Compl__subset__Compl__iff,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ ( uminus3259373730256538813_a_nat @ A2 ) @ ( uminus3259373730256538813_a_nat @ B ) )
= ( ord_le8108555184339247974_a_nat @ B @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_71_Compl__subset__Compl__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B ) )
= ( ord_less_eq_set_a @ B @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_72_Compl__subset__Compl__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( uminus5710092332889474511et_nat @ B ) )
= ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_73_Un__Diff__cancel,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_74_Un__Diff__cancel,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_75_Un__Diff__cancel,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ ( minus_7395159227704179404_a_nat @ B @ A2 ) )
= ( sup_su4083067149120280889_a_nat @ A2 @ B ) ) ).
% Un_Diff_cancel
thf(fact_76_Un__Diff__cancel2,axiom,
! [B: set_nat,A2: set_nat] :
( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B @ A2 ) @ A2 )
= ( sup_sup_set_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_77_Un__Diff__cancel2,axiom,
! [B: set_a,A2: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B @ A2 ) @ A2 )
= ( sup_sup_set_a @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_78_Un__Diff__cancel2,axiom,
! [B: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( minus_7395159227704179404_a_nat @ B @ A2 ) @ A2 )
= ( sup_su4083067149120280889_a_nat @ B @ A2 ) ) ).
% Un_Diff_cancel2
thf(fact_79_Un__subset__iff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_a @ A2 @ C )
& ( ord_less_eq_set_a @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_80_Un__subset__iff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( ( ord_less_eq_set_nat @ A2 @ C )
& ( ord_less_eq_set_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_81_image__eqI,axiom,
! [B2: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( member8690443509505302927_a_nat @ B2 @ ( image_6222892899998961285_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_82_image__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6721470456781115300_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_83_image__eqI,axiom,
! [B2: nat,F: ( nat > sum_sum_a_nat ) > nat,X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( member_nat @ B2 @ ( image_5786201776793816049at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_84_image__eqI,axiom,
! [B2: nat > sum_sum_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( member8690443509505302927_a_nat @ B2 @ ( image_701559317304863014_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_85_image__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( member408289922725080238_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_86_image__eqI,axiom,
! [B2: nat,F: list_Sum_sum_a_nat > nat,X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( member_nat @ B2 @ ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_87_image__eqI,axiom,
! [B2: nat > sum_sum_a_nat,F: nat > nat > sum_sum_a_nat,X4: nat,A2: set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member8690443509505302927_a_nat @ B2 @ ( image_1051037728736664655_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_88_image__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,X4: nat,A2: set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_89_image__eqI,axiom,
! [B2: nat,F: nat > nat,X4: nat,A2: set_nat] :
( ( B2
= ( F @ X4 ) )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_90_subset__antisym,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_91_subset__antisym,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_92_subsetI,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ! [X3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ X3 @ B ) )
=> ( ord_le8108555184339247974_a_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_93_subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ X3 @ B ) )
=> ( ord_le1147066620699065093_a_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_94_subsetI,axiom,
! [A2: set_a,B: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B ) )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% subsetI
thf(fact_95_subsetI,axiom,
! [A2: set_nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ X3 @ B ) )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% subsetI
thf(fact_96_Un__iff,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A2 @ B ) )
= ( ( member8690443509505302927_a_nat @ C2 @ A2 )
| ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_97_Un__iff,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) )
= ( ( member408289922725080238_a_nat @ C2 @ A2 )
| ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_98_Un__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
| ( member_a @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_99_Un__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_100_UnCI,axiom,
! [C2: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( ~ ( member8690443509505302927_a_nat @ C2 @ B )
=> ( member8690443509505302927_a_nat @ C2 @ A2 ) )
=> ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_101_UnCI,axiom,
! [C2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ C2 @ B )
=> ( member408289922725080238_a_nat @ C2 @ A2 ) )
=> ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_102_UnCI,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ A2 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnCI
thf(fact_103_UnCI,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A2 ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnCI
thf(fact_104_Diff__idemp,axiom,
! [A2: set_a,B: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_105_Diff__idemp,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ B ) @ B )
= ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ).
% Diff_idemp
thf(fact_106_Diff__iff,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( minus_5517490076408937517_a_nat @ A2 @ B ) )
= ( ( member8690443509505302927_a_nat @ C2 @ A2 )
& ~ ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_107_Diff__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ~ ( member_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_108_Diff__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ~ ( member_a @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_109_Diff__iff,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A2 @ B ) )
= ( ( member408289922725080238_a_nat @ C2 @ A2 )
& ~ ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% Diff_iff
thf(fact_110_DiffI,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ A2 )
=> ( ~ ( member8690443509505302927_a_nat @ C2 @ B )
=> ( member8690443509505302927_a_nat @ C2 @ ( minus_5517490076408937517_a_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_111_DiffI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_112_DiffI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ~ ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_113_DiffI,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ C2 @ B )
=> ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ) ).
% DiffI
thf(fact_114_Compl__eq__Compl__iff,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ( uminus3259373730256538813_a_nat @ A2 )
= ( uminus3259373730256538813_a_nat @ B ) )
= ( A2 = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_115_Compl__iff,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( uminus2192744996606729052_a_nat @ A2 ) )
= ( ~ ( member408289922725080238_a_nat @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_116_Compl__iff,axiom,
! [C2: nat,A2: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( ~ ( member_nat @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_117_Compl__iff,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( uminus3259373730256538813_a_nat @ A2 ) )
= ( ~ ( member8690443509505302927_a_nat @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_118_ComplI,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( member408289922725080238_a_nat @ C2 @ ( uminus2192744996606729052_a_nat @ A2 ) ) ) ).
% ComplI
thf(fact_119_ComplI,axiom,
! [C2: nat,A2: set_nat] :
( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% ComplI
thf(fact_120_ComplI,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ C2 @ A2 )
=> ( member8690443509505302927_a_nat @ C2 @ ( uminus3259373730256538813_a_nat @ A2 ) ) ) ).
% ComplI
thf(fact_121_image__ident,axiom,
! [Y2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X2: list_Sum_sum_a_nat] : X2
@ Y2 )
= Y2 ) ).
% image_ident
thf(fact_122_minus__set__def,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( minus_minus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_123_minus__set__def,axiom,
( minus_5517490076408937517_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ( minus_7403279044699189488_nat_o
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ A3 )
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_124_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_125_minus__set__def,axiom,
( minus_7395159227704179404_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( minus_5799240027661600017_nat_o
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ A3 )
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_126_uminus__set__def,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A3: set_nat] :
( collect_nat
@ ( uminus_uminus_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_127_uminus__set__def,axiom,
( uminus2192744996606729052_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( uminus4331925969656920193_nat_o
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_128_uminus__set__def,axiom,
( uminus3259373730256538813_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ( uminus8220868946613220448_nat_o
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ A3 ) ) ) ) ) ).
% uminus_set_def
thf(fact_129_proj__vals__def,axiom,
( proj_v3643391342904276326_a_nat
= ( ^ [R: set_na3699693778330250182_a_nat,Ns2: list_nat] :
( image_6721470456781115300_a_nat
@ ^ [Tau5: nat > sum_sum_a_nat] : ( map_na823391071729141993_a_nat @ Tau5 @ Ns2 )
@ R ) ) ) ).
% proj_vals_def
thf(fact_130_diff__right__commute,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B2 ) @ C2 ) ) ).
% diff_right_commute
thf(fact_131_rev__image__eqI,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member8690443509505302927_a_nat @ B2 @ ( image_6222892899998961285_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_132_rev__image__eqI,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: list_Sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6721470456781115300_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_133_rev__image__eqI,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: nat,F: ( nat > sum_sum_a_nat ) > nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_nat @ B2 @ ( image_5786201776793816049at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_134_rev__image__eqI,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: nat > sum_sum_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member8690443509505302927_a_nat @ B2 @ ( image_701559317304863014_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_135_rev__image__eqI,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_136_rev__image__eqI,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_nat @ B2 @ ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_137_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B2: nat > sum_sum_a_nat,F: nat > nat > sum_sum_a_nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member8690443509505302927_a_nat @ B2 @ ( image_1051037728736664655_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_138_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_139_rev__image__eqI,axiom,
! [X4: nat,A2: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( B2
= ( F @ X4 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_140_ball__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
=> ( P @ X3 ) )
=> ! [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ A2 )
=> ( P @ ( F @ X5 ) ) ) ) ).
% ball_imageD
thf(fact_141_image__cong,axiom,
! [M: set_li6526943997496501093_a_nat,N: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( M = N )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ N )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_5081948215111134021_a_nat @ F @ M )
= ( image_5081948215111134021_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_142_bex__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ? [X5: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X5 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
& ( P @ X5 ) )
=> ? [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_143_image__iff,axiom,
! [Z2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ Z2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( ? [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
& ( Z2
= ( F @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_144_imageI,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F @ X4 ) @ ( image_6222892899998961285_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_145_imageI,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X4 ) @ ( image_6721470456781115300_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_146_imageI,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_5786201776793816049at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_147_imageI,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F @ X4 ) @ ( image_701559317304863014_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_148_imageI,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X4 ) @ ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_149_imageI,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_2535339886381165584at_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_150_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > nat > sum_sum_a_nat] :
( ( member_nat @ X4 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F @ X4 ) @ ( image_1051037728736664655_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_151_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X4 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X4 ) @ ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_152_imageI,axiom,
! [X4: nat,A2: set_nat,F: nat > nat] :
( ( member_nat @ X4 @ A2 )
=> ( member_nat @ ( F @ X4 ) @ ( image_nat_nat @ F @ A2 ) ) ) ).
% imageI
thf(fact_153_Collect__mono__iff,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) )
= ( ! [X2: nat > sum_sum_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_154_Collect__mono__iff,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ ( collec7555443234367654128_a_nat @ P ) @ ( collec7555443234367654128_a_nat @ Q ) )
= ( ! [X2: list_Sum_sum_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_155_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_156_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_157_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z3: set_a] : ( Y3 = Z3 ) )
= ( ^ [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
& ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_158_set__eq__subset,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A3: set_nat,B3: set_nat] :
( ( ord_less_eq_set_nat @ A3 @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_159_subset__trans,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_160_subset__trans,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% subset_trans
thf(fact_161_Collect__mono,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le8108555184339247974_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_162_Collect__mono,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_le1147066620699065093_a_nat @ ( collec7555443234367654128_a_nat @ P ) @ ( collec7555443234367654128_a_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_163_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_164_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_165_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_166_subset__refl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% subset_refl
thf(fact_167_subset__iff,axiom,
( ord_le8108555184339247974_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
! [T2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ T2 @ A3 )
=> ( member8690443509505302927_a_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_168_subset__iff,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
! [T2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ T2 @ A3 )
=> ( member408289922725080238_a_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_169_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [T2: a] :
( ( member_a @ T2 @ A3 )
=> ( member_a @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_170_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_171_Set_OequalityD2,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ B @ A2 ) ) ).
% Set.equalityD2
thf(fact_172_Set_OequalityD2,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ B @ A2 ) ) ).
% Set.equalityD2
thf(fact_173_equalityD1,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ).
% equalityD1
thf(fact_174_equalityD1,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% equalityD1
thf(fact_175_subset__eq,axiom,
( ord_le8108555184339247974_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
! [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A3 )
=> ( member8690443509505302927_a_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_176_subset__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A3 )
=> ( member408289922725080238_a_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_177_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_178_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A3 )
=> ( member_nat @ X2 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_179_equalityE,axiom,
! [A2: set_a,B: set_a] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B )
=> ~ ( ord_less_eq_set_a @ B @ A2 ) ) ) ).
% equalityE
thf(fact_180_equalityE,axiom,
! [A2: set_nat,B: set_nat] :
( ( A2 = B )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A2 ) ) ) ).
% equalityE
thf(fact_181_subsetD,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,C2: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ B )
=> ( ( member8690443509505302927_a_nat @ C2 @ A2 )
=> ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_182_subsetD,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B )
=> ( ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_183_subsetD,axiom,
! [A2: set_a,B: set_a,C2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% subsetD
thf(fact_184_subsetD,axiom,
! [A2: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_185_in__mono,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,X4: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ B )
=> ( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( member8690443509505302927_a_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_186_in__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B )
=> ( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( member408289922725080238_a_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_187_in__mono,axiom,
! [A2: set_a,B: set_a,X4: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( member_a @ X4 @ A2 )
=> ( member_a @ X4 @ B ) ) ) ).
% in_mono
thf(fact_188_in__mono,axiom,
! [A2: set_nat,B: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( member_nat @ X4 @ A2 )
=> ( member_nat @ X4 @ B ) ) ) ).
% in_mono
thf(fact_189_Un__left__commute,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) )
= ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_190_Un__left__commute,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_191_Un__left__absorb,axiom,
! [A2: set_a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_192_Un__left__absorb,axiom,
! [A2: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_193_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_194_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_195_Un__absorb,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_196_Un__absorb,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_197_Un__assoc,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_assoc
thf(fact_198_Un__assoc,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_199_ball__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_200_ball__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_201_bex__Un,axiom,
! [A2: set_a,B: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_202_bex__Un,axiom,
! [A2: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X2: nat] :
( ( member_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_203_UnI2,axiom,
! [C2: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ B )
=> ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_204_UnI2,axiom,
! [C2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ B )
=> ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_205_UnI2,axiom,
! [C2: a,B: set_a,A2: set_a] :
( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI2
thf(fact_206_UnI2,axiom,
! [C2: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI2
thf(fact_207_UnI1,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ A2 )
=> ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_208_UnI1,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_209_UnI1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% UnI1
thf(fact_210_UnI1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% UnI1
thf(fact_211_UnE,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A2 @ B ) )
=> ( ~ ( member8690443509505302927_a_nat @ C2 @ A2 )
=> ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_212_UnE,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) )
=> ( ~ ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_213_UnE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A2 @ B ) )
=> ( ~ ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% UnE
thf(fact_214_UnE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B ) )
=> ( ~ ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_215_DiffD2,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( minus_5517490076408937517_a_nat @ A2 @ B ) )
=> ~ ( member8690443509505302927_a_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_216_DiffD2,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( member_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_217_DiffD2,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( member_a @ C2 @ B ) ) ).
% DiffD2
thf(fact_218_DiffD2,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A2 @ B ) )
=> ~ ( member408289922725080238_a_nat @ C2 @ B ) ) ).
% DiffD2
thf(fact_219_DiffD1,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( minus_5517490076408937517_a_nat @ A2 @ B ) )
=> ( member8690443509505302927_a_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_220_DiffD1,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ( member_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_221_DiffD1,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ( member_a @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_222_DiffD1,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A2 @ B ) )
=> ( member408289922725080238_a_nat @ C2 @ A2 ) ) ).
% DiffD1
thf(fact_223_DiffE,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( minus_5517490076408937517_a_nat @ A2 @ B ) )
=> ~ ( ( member8690443509505302927_a_nat @ C2 @ A2 )
=> ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_224_DiffE,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( minus_minus_set_nat @ A2 @ B ) )
=> ~ ( ( member_nat @ C2 @ A2 )
=> ( member_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_225_DiffE,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A2 @ B ) )
=> ~ ( ( member_a @ C2 @ A2 )
=> ( member_a @ C2 @ B ) ) ) ).
% DiffE
thf(fact_226_DiffE,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( minus_7395159227704179404_a_nat @ A2 @ B ) )
=> ~ ( ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% DiffE
thf(fact_227_double__complement,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( uminus3259373730256538813_a_nat @ ( uminus3259373730256538813_a_nat @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_228_ComplD,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( uminus2192744996606729052_a_nat @ A2 ) )
=> ~ ( member408289922725080238_a_nat @ C2 @ A2 ) ) ).
% ComplD
thf(fact_229_ComplD,axiom,
! [C2: nat,A2: set_nat] :
( ( member_nat @ C2 @ ( uminus5710092332889474511et_nat @ A2 ) )
=> ~ ( member_nat @ C2 @ A2 ) ) ).
% ComplD
thf(fact_230_ComplD,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( uminus3259373730256538813_a_nat @ A2 ) )
=> ~ ( member8690443509505302927_a_nat @ C2 @ A2 ) ) ).
% ComplD
thf(fact_231_Compr__image__eq,axiom,
! [F: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_232_Compr__image__eq,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat,A2: set_na3699693778330250182_a_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_5786201776793816049at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_5786201776793816049at_nat @ F
@ ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_233_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_2535339886381165584at_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_2535339886381165584at_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_234_Compr__image__eq,axiom,
! [F: nat > nat > sum_sum_a_nat,A2: set_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ ( image_1051037728736664655_a_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_1051037728736664655_a_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_235_Compr__image__eq,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ ( image_6222892899998961285_a_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_6222892899998961285_a_nat @ F
@ ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_236_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ ( image_701559317304863014_a_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_701559317304863014_a_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_237_Compr__image__eq,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: set_nat,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( image_6262589752765146990_a_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_6262589752765146990_a_nat @ F
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_238_Compr__image__eq,axiom,
! [F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( image_6721470456781115300_a_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_6721470456781115300_a_nat @ F
@ ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_239_Compr__image__eq,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
& ( P @ X2 ) ) )
= ( image_5081948215111134021_a_nat @ F
@ ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_240_image__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ A2 ) )
= ( image_5081948215111134021_a_nat
@ ^ [X2: list_Sum_sum_a_nat] : ( F @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_241_imageE,axiom,
! [B2: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ B2 @ ( image_6222892899998961285_a_nat @ F @ A2 ) )
=> ~ ! [X3: nat > sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member8690443509505302927_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_242_imageE,axiom,
! [B2: nat > sum_sum_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member8690443509505302927_a_nat @ B2 @ ( image_701559317304863014_a_nat @ F @ A2 ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member408289922725080238_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_243_imageE,axiom,
! [B2: nat > sum_sum_a_nat,F: nat > nat > sum_sum_a_nat,A2: set_nat] :
( ( member8690443509505302927_a_nat @ B2 @ ( image_1051037728736664655_a_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_244_imageE,axiom,
! [B2: list_Sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( image_6721470456781115300_a_nat @ F @ A2 ) )
=> ~ ! [X3: nat > sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member8690443509505302927_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_245_imageE,axiom,
! [B2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member408289922725080238_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_246_imageE,axiom,
! [B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,A2: set_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_247_imageE,axiom,
! [B2: nat,F: ( nat > sum_sum_a_nat ) > nat,A2: set_na3699693778330250182_a_nat] :
( ( member_nat @ B2 @ ( image_5786201776793816049at_nat @ F @ A2 ) )
=> ~ ! [X3: nat > sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member8690443509505302927_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_248_imageE,axiom,
! [B2: nat,F: list_Sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat] :
( ( member_nat @ B2 @ ( image_2535339886381165584at_nat @ F @ A2 ) )
=> ~ ! [X3: list_Sum_sum_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member408289922725080238_a_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_249_imageE,axiom,
! [B2: nat,F: nat > nat,A2: set_nat] :
( ( member_nat @ B2 @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [X3: nat] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_250_Collect__subset,axiom,
! [A2: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ord_le8108555184339247974_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_251_Collect__subset,axiom,
! [A2: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_252_Collect__subset,axiom,
! [A2: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_253_Collect__subset,axiom,
! [A2: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_254_Collect__disj__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_su3329769938372955546_a_nat @ ( collec5629555741568564177_a_nat @ P ) @ ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_255_Collect__disj__eq,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_su4083067149120280889_a_nat @ ( collec7555443234367654128_a_nat @ P ) @ ( collec7555443234367654128_a_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_256_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_257_Collect__disj__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_258_Un__def,axiom,
( sup_su3329769938372955546_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A3 )
| ( member8690443509505302927_a_nat @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_259_Un__def,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A3 )
| ( member408289922725080238_a_nat @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_260_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
| ( member_a @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_261_Un__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
| ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% Un_def
thf(fact_262_set__diff__eq,axiom,
( minus_minus_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A3 )
& ~ ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_263_set__diff__eq,axiom,
( minus_5517490076408937517_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ A3 )
& ~ ( member8690443509505302927_a_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_264_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A3 )
& ~ ( member_a @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_265_set__diff__eq,axiom,
( minus_7395159227704179404_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A3 )
& ~ ( member408289922725080238_a_nat @ X2 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_266_Collect__neg__eq,axiom,
! [P: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
~ ( P @ X2 ) )
= ( uminus2192744996606729052_a_nat @ ( collec7555443234367654128_a_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_267_Collect__neg__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
~ ( P @ X2 ) )
= ( uminus3259373730256538813_a_nat @ ( collec5629555741568564177_a_nat @ P ) ) ) ).
% Collect_neg_eq
thf(fact_268_Compl__eq,axiom,
( uminus5710092332889474511et_nat
= ( ^ [A3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
~ ( member_nat @ X2 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_269_Compl__eq,axiom,
( uminus2192744996606729052_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X2 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_270_Compl__eq,axiom,
( uminus3259373730256538813_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ X2 @ A3 ) ) ) ) ).
% Compl_eq
thf(fact_271_subset__image__iff,axiom,
! [B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( ? [AA: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ AA @ A2 )
& ( B
= ( image_5081948215111134021_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_272_subset__image__iff,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_273_subset__image__iff,axiom,
! [B: set_a,F: nat > a,A2: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_a @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_274_subset__image__iff,axiom,
! [B: set_nat,F: a > nat,A2: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A2 ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A2 )
& ( B
= ( image_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_275_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A2 )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_276_image__subset__iff,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ B )
= ( ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X2 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_277_subset__imageE,axiom,
! [B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C3 @ A2 )
=> ( B
!= ( image_5081948215111134021_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_278_subset__imageE,axiom,
! [B: set_a,F: a > a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ ( image_a_a @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B
!= ( image_a_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_279_subset__imageE,axiom,
! [B: set_a,F: nat > a,A2: set_nat] :
( ( ord_less_eq_set_a @ B @ ( image_nat_a @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B
!= ( image_nat_a @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_280_subset__imageE,axiom,
! [B: set_nat,F: a > nat,A2: set_a] :
( ( ord_less_eq_set_nat @ B @ ( image_a_nat @ F @ A2 ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( B
!= ( image_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_281_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A2 ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A2 )
=> ( B
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_282_image__subsetI,axiom,
! [A2: set_nat,F: nat > a,B: set_a] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_283_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat,B: set_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_284_image__subsetI,axiom,
! [A2: set_nat,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_285_image__subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > a,B: set_a] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_4685209397977471678_nat_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_286_image__subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat,B: set_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_2535339886381165584at_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_287_image__subsetI,axiom,
! [A2: set_nat,F: nat > nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member8690443509505302927_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le8108555184339247974_a_nat @ ( image_1051037728736664655_a_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_288_image__subsetI,axiom,
! [A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > a,B: set_a] :
( ! [X3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_7126633039417159965_nat_a @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_289_image__subsetI,axiom,
! [A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > nat,B: set_nat] :
( ! [X3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_5786201776793816049at_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_290_image__subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_291_image__subsetI,axiom,
! [A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6721470456781115300_a_nat @ F @ A2 ) @ B ) ) ).
% image_subsetI
thf(fact_292_image__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_293_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_mono
thf(fact_294_image__mono,axiom,
! [A2: set_a,B: set_a,F: a > nat] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A2 ) @ ( image_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_295_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > a] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_a @ ( image_nat_a @ F @ A2 ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_mono
thf(fact_296_image__mono,axiom,
! [A2: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_297_image__Un,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_298_image__Un,axiom,
! [F: a > a,A2: set_a,B: set_a] :
( ( image_a_a @ F @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) ) ).
% image_Un
thf(fact_299_image__Un,axiom,
! [F: a > nat,A2: set_a,B: set_a] :
( ( image_a_nat @ F @ ( sup_sup_set_a @ A2 @ B ) )
= ( sup_sup_set_nat @ ( image_a_nat @ F @ A2 ) @ ( image_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_300_image__Un,axiom,
! [F: nat > a,A2: set_nat,B: set_nat] :
( ( image_nat_a @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_a @ ( image_nat_a @ F @ A2 ) @ ( image_nat_a @ F @ B ) ) ) ).
% image_Un
thf(fact_301_image__Un,axiom,
! [F: nat > nat,A2: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A2 @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A2 ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_302_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_303_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_304_subset__UnE,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( sup_sup_set_a @ A2 @ B ) )
=> ~ ! [A4: set_a] :
( ( ord_less_eq_set_a @ A4 @ A2 )
=> ! [B4: set_a] :
( ( ord_less_eq_set_a @ B4 @ B )
=> ( C
!= ( sup_sup_set_a @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_305_subset__UnE,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B ) )
=> ~ ! [A4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ A2 )
=> ! [B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ B )
=> ( C
!= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ) ).
% subset_UnE
thf(fact_306_Un__absorb2,axiom,
! [B: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_307_Un__absorb2,axiom,
! [B: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B )
= A2 ) ) ).
% Un_absorb2
thf(fact_308_Un__absorb1,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_309_Un__absorb1,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ B )
= B ) ) ).
% Un_absorb1
thf(fact_310_Un__upper2,axiom,
! [B: set_a,A2: set_a] : ( ord_less_eq_set_a @ B @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper2
thf(fact_311_Un__upper2,axiom,
! [B: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper2
thf(fact_312_Un__upper1,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B ) ) ).
% Un_upper1
thf(fact_313_Un__upper1,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B ) ) ).
% Un_upper1
thf(fact_314_Un__least,axiom,
! [A2: set_a,C: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_315_Un__least,axiom,
! [A2: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C ) ) ) ).
% Un_least
thf(fact_316_Un__mono,axiom,
! [A2: set_a,C: set_a,B: set_a,D: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ B @ D )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B ) @ ( sup_sup_set_a @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_317_Un__mono,axiom,
! [A2: set_nat,C: set_nat,B: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% Un_mono
thf(fact_318_double__diff,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C )
=> ( ( minus_7395159227704179404_a_nat @ B @ ( minus_7395159227704179404_a_nat @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_319_double__diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( ord_less_eq_set_a @ B @ C )
=> ( ( minus_minus_set_a @ B @ ( minus_minus_set_a @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_320_double__diff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( minus_minus_set_nat @ B @ ( minus_minus_set_nat @ C @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_321_Diff__subset,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_322_Diff__subset,axiom,
! [A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_323_Diff__subset,axiom,
! [A2: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ A2 ) ).
% Diff_subset
thf(fact_324_Diff__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,D: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ C )
=> ( ( ord_le1147066620699065093_a_nat @ D @ B )
=> ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ B ) @ ( minus_7395159227704179404_a_nat @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_325_Diff__mono,axiom,
! [A2: set_a,C: set_a,D: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ C )
=> ( ( ord_less_eq_set_a @ D @ B )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( minus_minus_set_a @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_326_Diff__mono,axiom,
! [A2: set_nat,C: set_nat,D: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ D @ B )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( minus_minus_set_nat @ C @ D ) ) ) ) ).
% Diff_mono
thf(fact_327_Un__Diff,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A2 @ B ) @ C )
= ( sup_sup_set_nat @ ( minus_minus_set_nat @ A2 @ C ) @ ( minus_minus_set_nat @ B @ C ) ) ) ).
% Un_Diff
thf(fact_328_Un__Diff,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A2 @ B ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A2 @ C ) @ ( minus_minus_set_a @ B @ C ) ) ) ).
% Un_Diff
thf(fact_329_Un__Diff,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) @ C )
= ( sup_su4083067149120280889_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ C ) @ ( minus_7395159227704179404_a_nat @ B @ C ) ) ) ).
% Un_Diff
thf(fact_330_Collect__imp__eq,axiom,
! [P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_su4083067149120280889_a_nat @ ( uminus2192744996606729052_a_nat @ ( collec7555443234367654128_a_nat @ P ) ) @ ( collec7555443234367654128_a_nat @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_331_Collect__imp__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup_set_a @ ( uminus_uminus_set_a @ ( collect_a @ P ) ) @ ( collect_a @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_332_Collect__imp__eq,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_333_Collect__imp__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_su3329769938372955546_a_nat @ ( uminus3259373730256538813_a_nat @ ( collec5629555741568564177_a_nat @ P ) ) @ ( collec5629555741568564177_a_nat @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_334_image__diff__subset,axiom,
! [F: a > list_Sum_sum_a_nat,A2: set_a,B: set_a] : ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ ( image_7897140031816760844_a_nat @ F @ A2 ) @ ( image_7897140031816760844_a_nat @ F @ B ) ) @ ( image_7897140031816760844_a_nat @ F @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_335_image__diff__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ ( image_5081948215111134021_a_nat @ F @ A2 ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) @ ( image_5081948215111134021_a_nat @ F @ ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_336_image__diff__subset,axiom,
! [F: a > a,A2: set_a,B: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_a_a @ F @ A2 ) @ ( image_a_a @ F @ B ) ) @ ( image_a_a @ F @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_337_image__diff__subset,axiom,
! [F: list_Sum_sum_a_nat > a,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_4685209397977471678_nat_a @ F @ A2 ) @ ( image_4685209397977471678_nat_a @ F @ B ) ) @ ( image_4685209397977471678_nat_a @ F @ ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_338_image__diff__subset,axiom,
! [F: a > nat,A2: set_a,B: set_a] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_a_nat @ F @ A2 ) @ ( image_a_nat @ F @ B ) ) @ ( image_a_nat @ F @ ( minus_minus_set_a @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_339_image__diff__subset,axiom,
! [F: list_Sum_sum_a_nat > nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ ( image_2535339886381165584at_nat @ F @ A2 ) @ ( image_2535339886381165584at_nat @ F @ B ) ) @ ( image_2535339886381165584at_nat @ F @ ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ).
% image_diff_subset
thf(fact_340_Diff__subset__conv,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ B ) @ C )
= ( ord_le1147066620699065093_a_nat @ A2 @ ( sup_su4083067149120280889_a_nat @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_341_Diff__subset__conv,axiom,
! [A2: set_a,B: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B ) @ C )
= ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_342_Diff__subset__conv,axiom,
! [A2: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ C )
= ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% Diff_subset_conv
thf(fact_343_Diff__partition,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B )
=> ( ( sup_su4083067149120280889_a_nat @ A2 @ ( minus_7395159227704179404_a_nat @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_344_Diff__partition,axiom,
! [A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ( sup_sup_set_a @ A2 @ ( minus_minus_set_a @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_345_Diff__partition,axiom,
! [A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ( sup_sup_set_nat @ A2 @ ( minus_minus_set_nat @ B @ A2 ) )
= B ) ) ).
% Diff_partition
thf(fact_346_aux6_I2_J,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ns_psi2 ) @ ( set_nat2 @ ns_psi ) )
= ( set_nat2 @ both ) ) ).
% aux6(2)
thf(fact_347_sup_Obounded__iff,axiom,
! [B2: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B2 @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_348_sup_Obounded__iff,axiom,
! [B2: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_set_nat @ B2 @ A )
& ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_349_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_350_le__sup__iff,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X4 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X4 @ Z2 )
& ( ord_less_eq_set_a @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_351_le__sup__iff,axiom,
! [X4: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X4 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X4 @ Z2 )
& ( ord_less_eq_set_nat @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_352_le__sup__iff,axiom,
! [X4: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X4 @ Y4 ) @ Z2 )
= ( ( ord_less_eq_nat @ X4 @ Z2 )
& ( ord_less_eq_nat @ Y4 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_353_compl__le__compl__iff,axiom,
! [X4: set_na3699693778330250182_a_nat,Y4: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ ( uminus3259373730256538813_a_nat @ X4 ) @ ( uminus3259373730256538813_a_nat @ Y4 ) )
= ( ord_le8108555184339247974_a_nat @ Y4 @ X4 ) ) ).
% compl_le_compl_iff
thf(fact_354_compl__le__compl__iff,axiom,
! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X4 ) @ ( uminus_uminus_set_a @ Y4 ) )
= ( ord_less_eq_set_a @ Y4 @ X4 ) ) ).
% compl_le_compl_iff
thf(fact_355_compl__le__compl__iff,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ ( uminus5710092332889474511et_nat @ Y4 ) )
= ( ord_less_eq_set_nat @ Y4 @ X4 ) ) ).
% compl_le_compl_iff
thf(fact_356_ns__def,axiom,
( ns
= ( filter_nat
@ ^ [N2: nat] : ( member_nat @ N2 @ ( set_nat2 @ ns_psi2 ) )
@ ns_phi2 ) ) ).
% ns_def
thf(fact_357_set__union,axiom,
! [Xs: list_a,Ys: list_a] :
( ( set_a2 @ ( union_a @ Xs @ Ys ) )
= ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).
% set_union
thf(fact_358_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_359_eval__table,axiom,
( eval_table_a_nat
= ( ^ [Ts2: list_fo_term_a,X6: set_li6526943997496501093_a_nat] :
( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( member408289922725080238_a_nat @ ( eval_eterms_a_nat @ Sigma @ Ts2 ) @ X6 ) )
@ ( fv_fo_terms_list_a @ Ts2 ) ) ) ) ).
% eval_table
thf(fact_360_aux_I5_J,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ns_phi2 ) @ ( set_nat2 @ ns_phi ) )
= ( set_nat2 @ both ) ) ).
% aux(5)
thf(fact_361_Some,axiom,
( ( lookup8968130772531298012_a_nat @ idx_psi @ ( fo_nmlz_a @ aD_psi @ ( proj_tuple_a @ ns @ ( zip_na2013496608136855606_a_nat @ ns_phi2 @ x2 ) ) ) )
= ( some_s2697817922134046282_a_nat @ y ) ) ).
% Some
thf(fact_362_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [X4: set_na3699693778330250182_a_nat] :
( ( uminus3259373730256538813_a_nat @ ( uminus3259373730256538813_a_nat @ X4 ) )
= X4 ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_363_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [X4: set_na3699693778330250182_a_nat,Y4: set_na3699693778330250182_a_nat] :
( ( ( uminus3259373730256538813_a_nat @ X4 )
= ( uminus3259373730256538813_a_nat @ Y4 ) )
= ( X4 = Y4 ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_364_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_365_sup_Oidem,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% sup.idem
thf(fact_366_sup__idem,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_367_sup__idem,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ X4 )
= X4 ) ).
% sup_idem
thf(fact_368_sup_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_369_sup_Oleft__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_370_sup__left__idem,axiom,
! [X4: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y4 ) )
= ( sup_sup_set_a @ X4 @ Y4 ) ) ).
% sup_left_idem
thf(fact_371_sup__left__idem,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y4 ) )
= ( sup_sup_set_nat @ X4 @ Y4 ) ) ).
% sup_left_idem
thf(fact_372_sup_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ B2 )
= ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_373_sup_Oright__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_374_ns__sd_I7_J,axiom,
ord_less_eq_set_nat @ ( set_nat2 @ ns_phi ) @ ( set_nat2 @ ns_psi2 ) ).
% ns_sd(7)
thf(fact_375_filter__filter,axiom,
! [P: nat > $o,Q: nat > $o,Xs: list_nat] :
( ( filter_nat @ P @ ( filter_nat @ Q @ Xs ) )
= ( filter_nat
@ ^ [X2: nat] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_376_aux2_I2_J,axiom,
( ns_phi2
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ ns_phi ) )
@ ns_phi2 ) ) ).
% aux2(2)
thf(fact_377_aux2_I1_J,axiom,
( ns_phi
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ ns_phi2 ) )
@ ns_phi ) ) ).
% aux2(1)
thf(fact_378_aux4_I2_J,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ ns_phi ) )
= ( set_nat2 @ ns_psi2 ) ) ).
% aux4(2)
thf(fact_379_aux3_I2_J,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ ns_phi ) @ ( set_nat2 @ ns ) )
= ( set_nat2 @ ns_psi2 ) ) ).
% aux3(2)
thf(fact_380_aux5_I1_J,axiom,
( ns_phi
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ ns_phi2 ) )
@ ns_psi2 ) ) ).
% aux5(1)
thf(fact_381_filter__True,axiom,
! [Xs: list_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( P @ X3 ) )
=> ( ( filter_nat @ P @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_382_aux5_I2_J,axiom,
( ns_psi
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( set_nat2 @ ns_psi2 ) )
@ ns_phi2 ) ) ).
% aux5(2)
thf(fact_383_set__filter,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,Xs: list_n989787106983797996_a_nat] :
( ( set_na645604395003041787_a_nat @ ( filter7153797121941418979_a_nat @ P @ Xs ) )
= ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ ( set_na645604395003041787_a_nat @ Xs ) )
& ( P @ X2 ) ) ) ) ).
% set_filter
thf(fact_384_set__filter,axiom,
! [P: list_Sum_sum_a_nat > $o,Xs: list_l4703314356710769291_a_nat] :
( ( set_li2392974972034027290_a_nat @ ( filter5373755100216644354_a_nat @ P @ Xs ) )
= ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( set_li2392974972034027290_a_nat @ Xs ) )
& ( P @ X2 ) ) ) ) ).
% set_filter
thf(fact_385_set__filter,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( set_nat2 @ ( filter_nat @ P @ Xs ) )
= ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
& ( P @ X2 ) ) ) ) ).
% set_filter
thf(fact_386_filter__cong,axiom,
! [Xs: list_n989787106983797996_a_nat,Ys: list_n989787106983797996_a_nat,P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( Xs = Ys )
=> ( ! [X3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter7153797121941418979_a_nat @ P @ Xs )
= ( filter7153797121941418979_a_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_387_filter__cong,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter5373755100216644354_a_nat @ P @ Xs )
= ( filter5373755100216644354_a_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_388_filter__cong,axiom,
! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
( ( Xs = Ys )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
=> ( ( P @ X3 )
= ( Q @ X3 ) ) )
=> ( ( filter_nat @ P @ Xs )
= ( filter_nat @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_389_filter__id__conv,axiom,
! [P: nat > $o,Xs: list_nat] :
( ( ( filter_nat @ P @ Xs )
= Xs )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( P @ X2 ) ) ) ) ).
% filter_id_conv
thf(fact_390_sup__set__def,axiom,
( sup_su3329769938372955546_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ( sup_su3226716170639437251_nat_o
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ A3 )
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_391_sup__set__def,axiom,
( sup_su4083067149120280889_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ( sup_su1334248866174809316_nat_o
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ A3 )
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_392_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A3: set_a,B3: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_393_sup__set__def,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( collect_nat
@ ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% sup_set_def
thf(fact_394_less__eq__set__def,axiom,
( ord_le8108555184339247974_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
( ord_le6982925868732858103_nat_o
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ A3 )
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_395_less__eq__set__def,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
( ord_le8737610411969296920_nat_o
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ A3 )
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_396_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A3: set_a,B3: set_a] :
( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A3 )
@ ^ [X2: a] : ( member_a @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_397_less__eq__set__def,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A3 )
@ ^ [X2: nat] : ( member_nat @ X2 @ B3 ) ) ) ) ).
% less_eq_set_def
thf(fact_398_filter__is__subset,axiom,
! [P: a > $o,Xs: list_a] : ( ord_less_eq_set_a @ ( set_a2 @ ( filter_a @ P @ Xs ) ) @ ( set_a2 @ Xs ) ) ).
% filter_is_subset
thf(fact_399_filter__is__subset,axiom,
! [P: nat > $o,Xs: list_nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( filter_nat @ P @ Xs ) ) @ ( set_nat2 @ Xs ) ) ).
% filter_is_subset
thf(fact_400_inf__sup__aci_I8_J,axiom,
! [X4: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y4 ) )
= ( sup_sup_set_a @ X4 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_401_inf__sup__aci_I8_J,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y4 ) )
= ( sup_sup_set_nat @ X4 @ Y4 ) ) ).
% inf_sup_aci(8)
thf(fact_402_inf__sup__aci_I7_J,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y4 @ Z2 ) )
= ( sup_sup_set_a @ Y4 @ ( sup_sup_set_a @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_403_inf__sup__aci_I7_J,axiom,
! [X4: set_nat,Y4: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y4 @ Z2 ) )
= ( sup_sup_set_nat @ Y4 @ ( sup_sup_set_nat @ X4 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_404_inf__sup__aci_I6_J,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X4 @ Y4 ) @ Z2 )
= ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_405_inf__sup__aci_I6_J,axiom,
! [X4: set_nat,Y4: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X4 @ Y4 ) @ Z2 )
= ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_406_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_407_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_408_sup_Oassoc,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B2 ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_409_sup_Oassoc,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_410_sup__assoc,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X4 @ Y4 ) @ Z2 )
= ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_411_sup__assoc,axiom,
! [X4: set_nat,Y4: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X4 @ Y4 ) @ Z2 )
= ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y4 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_412_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_413_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_414_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_415_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X2: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_416_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_a,K: set_a,A: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_417_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_nat,K: set_nat,A: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ K @ A ) )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_418_boolean__algebra__cancel_Osup2,axiom,
! [B: set_a,K: set_a,B2: set_a,A: set_a] :
( ( B
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_419_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_420_sup_Oleft__commute,axiom,
! [B2: set_a,A: set_a,C2: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A @ C2 ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_421_sup_Oleft__commute,axiom,
! [B2: set_nat,A: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A @ C2 ) )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_422_sup__left__commute,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X4 @ ( sup_sup_set_a @ Y4 @ Z2 ) )
= ( sup_sup_set_a @ Y4 @ ( sup_sup_set_a @ X4 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_423_sup__left__commute,axiom,
! [X4: set_nat,Y4: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( sup_sup_set_nat @ Y4 @ Z2 ) )
= ( sup_sup_set_nat @ Y4 @ ( sup_sup_set_nat @ X4 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_424_compl__mono,axiom,
! [X4: set_na3699693778330250182_a_nat,Y4: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ X4 @ Y4 )
=> ( ord_le8108555184339247974_a_nat @ ( uminus3259373730256538813_a_nat @ Y4 ) @ ( uminus3259373730256538813_a_nat @ X4 ) ) ) ).
% compl_mono
thf(fact_425_compl__mono,axiom,
! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y4 ) @ ( uminus_uminus_set_a @ X4 ) ) ) ).
% compl_mono
thf(fact_426_compl__mono,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y4 ) @ ( uminus5710092332889474511et_nat @ X4 ) ) ) ).
% compl_mono
thf(fact_427_compl__le__swap1,axiom,
! [Y4: set_na3699693778330250182_a_nat,X4: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ Y4 @ ( uminus3259373730256538813_a_nat @ X4 ) )
=> ( ord_le8108555184339247974_a_nat @ X4 @ ( uminus3259373730256538813_a_nat @ Y4 ) ) ) ).
% compl_le_swap1
thf(fact_428_compl__le__swap1,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ ( uminus_uminus_set_a @ X4 ) )
=> ( ord_less_eq_set_a @ X4 @ ( uminus_uminus_set_a @ Y4 ) ) ) ).
% compl_le_swap1
thf(fact_429_compl__le__swap1,axiom,
! [Y4: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ ( uminus5710092332889474511et_nat @ X4 ) )
=> ( ord_less_eq_set_nat @ X4 @ ( uminus5710092332889474511et_nat @ Y4 ) ) ) ).
% compl_le_swap1
thf(fact_430_compl__le__swap2,axiom,
! [Y4: set_na3699693778330250182_a_nat,X4: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ ( uminus3259373730256538813_a_nat @ Y4 ) @ X4 )
=> ( ord_le8108555184339247974_a_nat @ ( uminus3259373730256538813_a_nat @ X4 ) @ Y4 ) ) ).
% compl_le_swap2
thf(fact_431_compl__le__swap2,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ Y4 ) @ X4 )
=> ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ X4 ) @ Y4 ) ) ).
% compl_le_swap2
thf(fact_432_compl__le__swap2,axiom,
! [Y4: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y4 ) @ X4 )
=> ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X4 ) @ Y4 ) ) ).
% compl_le_swap2
thf(fact_433_inf__sup__ord_I4_J,axiom,
! [Y4: set_a,X4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( sup_sup_set_a @ X4 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_434_inf__sup__ord_I4_J,axiom,
! [Y4: set_nat,X4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( sup_sup_set_nat @ X4 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_435_inf__sup__ord_I4_J,axiom,
! [Y4: nat,X4: nat] : ( ord_less_eq_nat @ Y4 @ ( sup_sup_nat @ X4 @ Y4 ) ) ).
% inf_sup_ord(4)
thf(fact_436_inf__sup__ord_I3_J,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_437_inf__sup__ord_I3_J,axiom,
! [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_438_inf__sup__ord_I3_J,axiom,
! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y4 ) ) ).
% inf_sup_ord(3)
thf(fact_439_le__supE,axiom,
! [A: set_a,B2: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_set_a @ A @ X4 )
=> ~ ( ord_less_eq_set_a @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_440_le__supE,axiom,
! [A: set_nat,B2: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_set_nat @ A @ X4 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_441_le__supE,axiom,
! [A: nat,B2: nat,X4: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X4 )
=> ~ ( ( ord_less_eq_nat @ A @ X4 )
=> ~ ( ord_less_eq_nat @ B2 @ X4 ) ) ) ).
% le_supE
thf(fact_442_le__supI,axiom,
! [A: set_a,X4: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ X4 )
=> ( ( ord_less_eq_set_a @ B2 @ X4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_443_le__supI,axiom,
! [A: set_nat,X4: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ X4 )
=> ( ( ord_less_eq_set_nat @ B2 @ X4 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_444_le__supI,axiom,
! [A: nat,X4: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ X4 )
=> ( ( ord_less_eq_nat @ B2 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ X4 ) ) ) ).
% le_supI
thf(fact_445_sup__ge1,axiom,
! [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y4 ) ) ).
% sup_ge1
thf(fact_446_sup__ge1,axiom,
! [X4: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y4 ) ) ).
% sup_ge1
thf(fact_447_sup__ge1,axiom,
! [X4: nat,Y4: nat] : ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ X4 @ Y4 ) ) ).
% sup_ge1
thf(fact_448_sup__ge2,axiom,
! [Y4: set_a,X4: set_a] : ( ord_less_eq_set_a @ Y4 @ ( sup_sup_set_a @ X4 @ Y4 ) ) ).
% sup_ge2
thf(fact_449_sup__ge2,axiom,
! [Y4: set_nat,X4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( sup_sup_set_nat @ X4 @ Y4 ) ) ).
% sup_ge2
thf(fact_450_sup__ge2,axiom,
! [Y4: nat,X4: nat] : ( ord_less_eq_nat @ Y4 @ ( sup_sup_nat @ X4 @ Y4 ) ) ).
% sup_ge2
thf(fact_451_le__supI1,axiom,
! [X4: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X4 @ A )
=> ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_452_le__supI1,axiom,
! [X4: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ X4 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_453_le__supI1,axiom,
! [X4: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ X4 @ A )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI1
thf(fact_454_le__supI2,axiom,
! [X4: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ X4 @ B2 )
=> ( ord_less_eq_set_a @ X4 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_455_le__supI2,axiom,
! [X4: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ B2 )
=> ( ord_less_eq_set_nat @ X4 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_456_le__supI2,axiom,
! [X4: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ X4 @ B2 )
=> ( ord_less_eq_nat @ X4 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% le_supI2
thf(fact_457_sup_Omono,axiom,
! [C2: set_a,A: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C2 @ D2 ) @ ( sup_sup_set_a @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_458_sup_Omono,axiom,
! [C2: set_nat,A: set_nat,D2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ D2 @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D2 ) @ ( sup_sup_set_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_459_sup_Omono,axiom,
! [C2: nat,A: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A @ B2 ) ) ) ) ).
% sup.mono
thf(fact_460_sup__mono,axiom,
! [A: set_a,C2: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A @ C2 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A @ B2 ) @ ( sup_sup_set_a @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_461_sup__mono,axiom,
! [A: set_nat,C2: set_nat,B2: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B2 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_462_sup__mono,axiom,
! [A: nat,C2: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B2 ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_463_sup__least,axiom,
! [Y4: set_a,X4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( ord_less_eq_set_a @ Z2 @ X4 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y4 @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_464_sup__least,axiom,
! [Y4: set_nat,X4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X4 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y4 @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_465_sup__least,axiom,
! [Y4: nat,X4: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y4 @ Z2 ) @ X4 ) ) ) ).
% sup_least
thf(fact_466_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y: set_a] :
( ( sup_sup_set_a @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_467_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X2: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_468_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ) ).
% le_iff_sup
thf(fact_469_sup_OorderE,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( A
= ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_470_sup_OorderE,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A
= ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_471_sup_OorderE,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( A
= ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.orderE
thf(fact_472_sup_OorderI,axiom,
! [A: set_a,B2: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A ) ) ).
% sup.orderI
thf(fact_473_sup_OorderI,axiom,
! [A: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ A @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_474_sup_OorderI,axiom,
! [A: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ A @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% sup.orderI
thf(fact_475_sup__unique,axiom,
! [F: set_a > set_a > set_a,X4: set_a,Y4: set_a] :
( ! [X3: set_a,Y5: set_a] : ( ord_less_eq_set_a @ X3 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: set_a,Y5: set_a] : ( ord_less_eq_set_a @ Y5 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: set_a,Y5: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y5 @ X3 )
=> ( ( ord_less_eq_set_a @ Z @ X3 )
=> ( ord_less_eq_set_a @ ( F @ Y5 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X4 @ Y4 )
= ( F @ X4 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_476_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X4: set_nat,Y4: set_nat] :
( ! [X3: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: set_nat,Y5: set_nat] : ( ord_less_eq_set_nat @ Y5 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: set_nat,Y5: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y5 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z @ X3 )
=> ( ord_less_eq_set_nat @ ( F @ Y5 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_set_nat @ X4 @ Y4 )
= ( F @ X4 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_477_sup__unique,axiom,
! [F: nat > nat > nat,X4: nat,Y4: nat] :
( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ X3 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: nat,Y5: nat,Z: nat] :
( ( ord_less_eq_nat @ Y5 @ X3 )
=> ( ( ord_less_eq_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ ( F @ Y5 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_nat @ X4 @ Y4 )
= ( F @ X4 @ Y4 ) ) ) ) ) ).
% sup_unique
thf(fact_478_sup_Oabsorb1,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( sup_sup_set_a @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_479_sup_Oabsorb1,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( sup_sup_set_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_480_sup_Oabsorb1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( sup_sup_nat @ A @ B2 )
= A ) ) ).
% sup.absorb1
thf(fact_481_sup_Oabsorb2,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( sup_sup_set_a @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_482_sup_Oabsorb2,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( sup_sup_set_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_483_sup_Oabsorb2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( sup_sup_nat @ A @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_484_sup__absorb1,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( sup_sup_set_a @ X4 @ Y4 )
= X4 ) ) ).
% sup_absorb1
thf(fact_485_sup__absorb1,axiom,
! [Y4: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X4 )
=> ( ( sup_sup_set_nat @ X4 @ Y4 )
= X4 ) ) ).
% sup_absorb1
thf(fact_486_sup__absorb1,axiom,
! [Y4: nat,X4: nat] :
( ( ord_less_eq_nat @ Y4 @ X4 )
=> ( ( sup_sup_nat @ X4 @ Y4 )
= X4 ) ) ).
% sup_absorb1
thf(fact_487_sup__absorb2,axiom,
! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( sup_sup_set_a @ X4 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_488_sup__absorb2,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ( sup_sup_set_nat @ X4 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_489_sup__absorb2,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( sup_sup_nat @ X4 @ Y4 )
= Y4 ) ) ).
% sup_absorb2
thf(fact_490_sup_OboundedE,axiom,
! [B2: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A )
=> ~ ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_491_sup_OboundedE,axiom,
! [B2: set_nat,C2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A )
=> ~ ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_492_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A )
=> ~ ( ( ord_less_eq_nat @ B2 @ A )
=> ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% sup.boundedE
thf(fact_493_sup_OboundedI,axiom,
! [B2: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_494_sup_OboundedI,axiom,
! [B2: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_495_sup_OboundedI,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A ) ) ) ).
% sup.boundedI
thf(fact_496_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( A5
= ( sup_sup_set_a @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_497_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( A5
= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_498_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_499_sup_Ocobounded1,axiom,
! [A: set_a,B2: set_a] : ( ord_less_eq_set_a @ A @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_500_sup_Ocobounded1,axiom,
! [A: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_501_sup_Ocobounded1,axiom,
! [A: nat,B2: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded1
thf(fact_502_sup_Ocobounded2,axiom,
! [B2: set_a,A: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_503_sup_Ocobounded2,axiom,
! [B2: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_504_sup_Ocobounded2,axiom,
! [B2: nat,A: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A @ B2 ) ) ).
% sup.cobounded2
thf(fact_505_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B5: set_a,A5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_506_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_507_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_508_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( sup_sup_set_a @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_509_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_510_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_511_sup_OcoboundedI1,axiom,
! [C2: set_a,A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C2 @ A )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_512_sup_OcoboundedI1,axiom,
! [C2: set_nat,A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_513_sup_OcoboundedI1,axiom,
! [C2: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_514_sup_OcoboundedI2,axiom,
! [C2: set_a,B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_515_sup_OcoboundedI2,axiom,
! [C2: set_nat,B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_516_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_517_comp__proj,axiom,
( ( ad_agr_close_set_a @ aD_Delta_psi @ ( minus_7395159227704179404_a_nat @ ( ext_tuple_set_a @ aD_psi @ ns @ ns_phi @ ( insert2950094090816004437_a_nat @ ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) @ bot_bo1033123847703346641_a_nat ) ) @ y ) )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma @ ( uminus3259373730256538813_a_nat @ s_psi ) )
& ( ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) )
@ ns_psi2 ) ) ) ).
% comp_proj
thf(fact_518__092_060open_062ext__tuple__set_AAD_092_060psi_062_Ans_Ans_092_060phi_062_H_A_123fo__nmlz_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_H_Ans_J_125_A_N_AY_A_061_Afo__nmlz_AAD_092_060psi_062_A_096_Aproj__vals_A_123_092_060sigma_062_A_092_060in_062_A_N_AS_092_060psi_062_O_Aad__agr__list_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_Ans_J_A_Imap_A_092_060sigma_062_H_Ans_J_125_Ans_092_060psi_062_092_060close_062,axiom,
( ( minus_7395159227704179404_a_nat @ ( ext_tuple_set_a @ aD_psi @ ns @ ns_phi @ ( insert2950094090816004437_a_nat @ ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) @ bot_bo1033123847703346641_a_nat ) ) @ y )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma @ ( uminus3259373730256538813_a_nat @ s_psi ) )
& ( ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) ) )
@ ns_psi2 ) ) ) ).
% \<open>ext_tuple_set AD\<psi> ns ns\<phi>' {fo_nmlz AD\<psi> (map \<sigma>' ns)} - Y = fo_nmlz AD\<psi> ` proj_vals {\<sigma> \<in> - S\<psi>. ad_agr_list AD\<psi> (map \<sigma> ns) (map \<sigma>' ns)} ns\<psi>\<close>
thf(fact_519_R__def,axiom,
( ( insert2950094090816004437_a_nat @ ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) @ bot_bo1033123847703346641_a_nat )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) )
@ ns ) ) ) ).
% R_def
thf(fact_520_option_Oinject,axiom,
! [X22: set_li6526943997496501093_a_nat,Y22: set_li6526943997496501093_a_nat] :
( ( ( some_s2697817922134046282_a_nat @ X22 )
= ( some_s2697817922134046282_a_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_521_option_Oinject,axiom,
! [X22: list_Sum_sum_a_nat,Y22: list_Sum_sum_a_nat] :
( ( ( some_l1231941606268492394_a_nat @ X22 )
= ( some_l1231941606268492394_a_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_522_X_092_060phi_062_H__def,axiom,
( x_phi
= ( ext_tuple_set_a @ ad @ ns_phi2 @ ns_phi @ ( ad_agr_close_set_a @ aD_Delta_phi @ x_phi2 ) ) ) ).
% X\<phi>'_def
thf(fact_523_order__refl,axiom,
! [X4: set_a] : ( ord_less_eq_set_a @ X4 @ X4 ) ).
% order_refl
thf(fact_524_order__refl,axiom,
! [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_525_order__refl,axiom,
! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).
% order_refl
thf(fact_526_dual__order_Orefl,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% dual_order.refl
thf(fact_527_dual__order_Orefl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% dual_order.refl
thf(fact_528_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_529_empty__iff,axiom,
! [C2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C2 @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_530_empty__iff,axiom,
! [C2: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ C2 @ bot_bo6441361344521902642_a_nat ) ).
% empty_iff
thf(fact_531_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_532_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_533_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_534_all__not__in__conv,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ! [X2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X2 @ A2 ) )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% all_not_in_conv
thf(fact_535_all__not__in__conv,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( ! [X2: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ X2 @ A2 ) )
= ( A2 = bot_bo6441361344521902642_a_nat ) ) ).
% all_not_in_conv
thf(fact_536_all__not__in__conv,axiom,
! [A2: set_nat] :
( ( ! [X2: nat] :
~ ( member_nat @ X2 @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_537_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_538_all__not__in__conv,axiom,
! [A2: set_o] :
( ( ! [X2: $o] :
~ ( member_o @ X2 @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_539_Collect__empty__eq,axiom,
! [P: list_Sum_sum_a_nat > $o] :
( ( ( collec7555443234367654128_a_nat @ P )
= bot_bo1033123847703346641_a_nat )
= ( ! [X2: list_Sum_sum_a_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_540_Collect__empty__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o] :
( ( ( collec5629555741568564177_a_nat @ P )
= bot_bo6441361344521902642_a_nat )
= ( ! [X2: nat > sum_sum_a_nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_541_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_542_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_543_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_544_empty__Collect__eq,axiom,
! [P: list_Sum_sum_a_nat > $o] :
( ( bot_bo1033123847703346641_a_nat
= ( collec7555443234367654128_a_nat @ P ) )
= ( ! [X2: list_Sum_sum_a_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_545_empty__Collect__eq,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o] :
( ( bot_bo6441361344521902642_a_nat
= ( collec5629555741568564177_a_nat @ P ) )
= ( ! [X2: nat > sum_sum_a_nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_546_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X2: nat] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_547_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_548_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X2: $o] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_549_insertCI,axiom,
! [A: $o,B: set_o,B2: $o] :
( ( ~ ( member_o @ A @ B )
=> ( A = B2 ) )
=> ( member_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% insertCI
thf(fact_550_insertCI,axiom,
! [A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat,B2: nat > sum_sum_a_nat] :
( ( ~ ( member8690443509505302927_a_nat @ A @ B )
=> ( A = B2 ) )
=> ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_551_insertCI,axiom,
! [A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ A @ B )
=> ( A = B2 ) )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_552_insertCI,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( ~ ( member_nat @ A @ B )
=> ( A = B2 ) )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertCI
thf(fact_553_insert__iff,axiom,
! [A: $o,B2: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_o @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_554_insert__iff,axiom,
! [A: nat > sum_sum_a_nat,B2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member8690443509505302927_a_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_555_insert__iff,axiom,
! [A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member408289922725080238_a_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_556_insert__iff,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_nat @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_557_insert__absorb2,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( insert2950094090816004437_a_nat @ X4 @ ( insert2950094090816004437_a_nat @ X4 @ A2 ) )
= ( insert2950094090816004437_a_nat @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_558_insert__absorb2,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( insert5265011953798106934_a_nat @ X4 @ ( insert5265011953798106934_a_nat @ X4 @ A2 ) )
= ( insert5265011953798106934_a_nat @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_559_insert__absorb2,axiom,
! [X4: $o,A2: set_o] :
( ( insert_o @ X4 @ ( insert_o @ X4 @ A2 ) )
= ( insert_o @ X4 @ A2 ) ) ).
% insert_absorb2
thf(fact_560_image__is__empty,axiom,
! [F: nat > nat,A2: set_nat] :
( ( ( image_nat_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_561_image__is__empty,axiom,
! [F: a > nat,A2: set_a] :
( ( ( image_a_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_562_image__is__empty,axiom,
! [F: $o > nat,A2: set_o] :
( ( ( image_o_nat @ F @ A2 )
= bot_bot_set_nat )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_563_image__is__empty,axiom,
! [F: nat > a,A2: set_nat] :
( ( ( image_nat_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_564_image__is__empty,axiom,
! [F: a > a,A2: set_a] :
( ( ( image_a_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_565_image__is__empty,axiom,
! [F: $o > a,A2: set_o] :
( ( ( image_o_a @ F @ A2 )
= bot_bot_set_a )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_566_image__is__empty,axiom,
! [F: nat > $o,A2: set_nat] :
( ( ( image_nat_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_567_image__is__empty,axiom,
! [F: a > $o,A2: set_a] :
( ( ( image_a_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_568_image__is__empty,axiom,
! [F: $o > $o,A2: set_o] :
( ( ( image_o_o @ F @ A2 )
= bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_569_image__is__empty,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: set_nat] :
( ( ( image_6262589752765146990_a_nat @ F @ A2 )
= bot_bo1033123847703346641_a_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_570_empty__is__image,axiom,
! [F: nat > nat,A2: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_571_empty__is__image,axiom,
! [F: a > nat,A2: set_a] :
( ( bot_bot_set_nat
= ( image_a_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_572_empty__is__image,axiom,
! [F: $o > nat,A2: set_o] :
( ( bot_bot_set_nat
= ( image_o_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_573_empty__is__image,axiom,
! [F: nat > a,A2: set_nat] :
( ( bot_bot_set_a
= ( image_nat_a @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_574_empty__is__image,axiom,
! [F: a > a,A2: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_575_empty__is__image,axiom,
! [F: $o > a,A2: set_o] :
( ( bot_bot_set_a
= ( image_o_a @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_576_empty__is__image,axiom,
! [F: nat > $o,A2: set_nat] :
( ( bot_bot_set_o
= ( image_nat_o @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_577_empty__is__image,axiom,
! [F: a > $o,A2: set_a] :
( ( bot_bot_set_o
= ( image_a_o @ F @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_578_empty__is__image,axiom,
! [F: $o > $o,A2: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_579_empty__is__image,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: set_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( image_6262589752765146990_a_nat @ F @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_580_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_581_image__empty,axiom,
! [F: nat > a] :
( ( image_nat_a @ F @ bot_bot_set_nat )
= bot_bot_set_a ) ).
% image_empty
thf(fact_582_image__empty,axiom,
! [F: nat > $o] :
( ( image_nat_o @ F @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_583_image__empty,axiom,
! [F: a > nat] :
( ( image_a_nat @ F @ bot_bot_set_a )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_584_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_585_image__empty,axiom,
! [F: a > $o] :
( ( image_a_o @ F @ bot_bot_set_a )
= bot_bot_set_o ) ).
% image_empty
thf(fact_586_image__empty,axiom,
! [F: $o > nat] :
( ( image_o_nat @ F @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_587_image__empty,axiom,
! [F: $o > a] :
( ( image_o_a @ F @ bot_bot_set_o )
= bot_bot_set_a ) ).
% image_empty
thf(fact_588_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_589_image__empty,axiom,
! [F: list_Sum_sum_a_nat > nat] :
( ( image_2535339886381165584at_nat @ F @ bot_bo1033123847703346641_a_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_590_empty__subsetI,axiom,
! [A2: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ bot_bo1033123847703346641_a_nat @ A2 ) ).
% empty_subsetI
thf(fact_591_empty__subsetI,axiom,
! [A2: set_na3699693778330250182_a_nat] : ( ord_le8108555184339247974_a_nat @ bot_bo6441361344521902642_a_nat @ A2 ) ).
% empty_subsetI
thf(fact_592_empty__subsetI,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% empty_subsetI
thf(fact_593_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_594_empty__subsetI,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% empty_subsetI
thf(fact_595_subset__empty,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ bot_bo1033123847703346641_a_nat )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% subset_empty
thf(fact_596_subset__empty,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ bot_bo6441361344521902642_a_nat )
= ( A2 = bot_bo6441361344521902642_a_nat ) ) ).
% subset_empty
thf(fact_597_subset__empty,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_598_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_599_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_600_insert__image,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( insert2950094090816004437_a_nat @ ( F @ X4 ) @ ( image_6721470456781115300_a_nat @ F @ A2 ) )
= ( image_6721470456781115300_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_601_insert__image,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( insert5265011953798106934_a_nat @ ( F @ X4 ) @ ( image_6222892899998961285_a_nat @ F @ A2 ) )
= ( image_6222892899998961285_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_602_insert__image,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( insert_o @ ( F @ X4 ) @ ( image_2376713081370839351_nat_o @ F @ A2 ) )
= ( image_2376713081370839351_nat_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_603_insert__image,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( insert2950094090816004437_a_nat @ ( F @ X4 ) @ ( image_5081948215111134021_a_nat @ F @ A2 ) )
= ( image_5081948215111134021_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_604_insert__image,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( insert5265011953798106934_a_nat @ ( F @ X4 ) @ ( image_701559317304863014_a_nat @ F @ A2 ) )
= ( image_701559317304863014_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_605_insert__image,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( insert_o @ ( F @ X4 ) @ ( image_3984946558445957976_nat_o @ F @ A2 ) )
= ( image_3984946558445957976_nat_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_606_insert__image,axiom,
! [X4: nat,A2: set_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( insert2950094090816004437_a_nat @ ( F @ X4 ) @ ( image_6262589752765146990_a_nat @ F @ A2 ) )
= ( image_6262589752765146990_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_607_insert__image,axiom,
! [X4: nat,A2: set_nat,F: nat > nat > sum_sum_a_nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( insert5265011953798106934_a_nat @ ( F @ X4 ) @ ( image_1051037728736664655_a_nat @ F @ A2 ) )
= ( image_1051037728736664655_a_nat @ F @ A2 ) ) ) ).
% insert_image
thf(fact_608_insert__image,axiom,
! [X4: nat,A2: set_nat,F: nat > $o] :
( ( member_nat @ X4 @ A2 )
=> ( ( insert_o @ ( F @ X4 ) @ ( image_nat_o @ F @ A2 ) )
= ( image_nat_o @ F @ A2 ) ) ) ).
% insert_image
thf(fact_609_image__insert,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( insert2950094090816004437_a_nat @ A @ B ) )
= ( insert2950094090816004437_a_nat @ ( F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_610_image__insert,axiom,
! [F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( image_701559317304863014_a_nat @ F @ ( insert2950094090816004437_a_nat @ A @ B ) )
= ( insert5265011953798106934_a_nat @ ( F @ A ) @ ( image_701559317304863014_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_611_image__insert,axiom,
! [F: list_Sum_sum_a_nat > $o,A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( image_3984946558445957976_nat_o @ F @ ( insert2950094090816004437_a_nat @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_3984946558445957976_nat_o @ F @ B ) ) ) ).
% image_insert
thf(fact_612_image__insert,axiom,
! [F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( image_6721470456781115300_a_nat @ F @ ( insert5265011953798106934_a_nat @ A @ B ) )
= ( insert2950094090816004437_a_nat @ ( F @ A ) @ ( image_6721470456781115300_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_613_image__insert,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( image_6222892899998961285_a_nat @ F @ ( insert5265011953798106934_a_nat @ A @ B ) )
= ( insert5265011953798106934_a_nat @ ( F @ A ) @ ( image_6222892899998961285_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_614_image__insert,axiom,
! [F: ( nat > sum_sum_a_nat ) > $o,A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( image_2376713081370839351_nat_o @ F @ ( insert5265011953798106934_a_nat @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_2376713081370839351_nat_o @ F @ B ) ) ) ).
% image_insert
thf(fact_615_image__insert,axiom,
! [F: $o > list_Sum_sum_a_nat,A: $o,B: set_o] :
( ( image_733135354807394034_a_nat @ F @ ( insert_o @ A @ B ) )
= ( insert2950094090816004437_a_nat @ ( F @ A ) @ ( image_733135354807394034_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_616_image__insert,axiom,
! [F: $o > nat > sum_sum_a_nat,A: $o,B: set_o] :
( ( image_3967413701311303635_a_nat @ F @ ( insert_o @ A @ B ) )
= ( insert5265011953798106934_a_nat @ ( F @ A ) @ ( image_3967413701311303635_a_nat @ F @ B ) ) ) ).
% image_insert
thf(fact_617_image__insert,axiom,
! [F: $o > $o,A: $o,B: set_o] :
( ( image_o_o @ F @ ( insert_o @ A @ B ) )
= ( insert_o @ ( F @ A ) @ ( image_o_o @ F @ B ) ) ) ).
% image_insert
thf(fact_618_sup__bot_Oright__neutral,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ bot_bo1033123847703346641_a_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_619_sup__bot_Oright__neutral,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ( sup_su3329769938372955546_a_nat @ A @ bot_bo6441361344521902642_a_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_620_sup__bot_Oright__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% sup_bot.right_neutral
thf(fact_621_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_622_sup__bot_Oright__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_623_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( sup_su4083067149120280889_a_nat @ A @ B2 ) )
= ( ( A = bot_bo1033123847703346641_a_nat )
& ( B2 = bot_bo1033123847703346641_a_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_624_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( bot_bo6441361344521902642_a_nat
= ( sup_su3329769938372955546_a_nat @ A @ B2 ) )
= ( ( A = bot_bo6441361344521902642_a_nat )
& ( B2 = bot_bo6441361344521902642_a_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_625_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A @ B2 ) )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_626_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B2 ) )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_627_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_o,B2: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A @ B2 ) )
= ( ( A = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_628_sup__bot_Oleft__neutral,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ bot_bo1033123847703346641_a_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_629_sup__bot_Oleft__neutral,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ( sup_su3329769938372955546_a_nat @ bot_bo6441361344521902642_a_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_630_sup__bot_Oleft__neutral,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_631_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_632_sup__bot_Oleft__neutral,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_633_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ A @ B2 )
= bot_bo1033123847703346641_a_nat )
= ( ( A = bot_bo1033123847703346641_a_nat )
& ( B2 = bot_bo1033123847703346641_a_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_634_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_na3699693778330250182_a_nat,B2: set_na3699693778330250182_a_nat] :
( ( ( sup_su3329769938372955546_a_nat @ A @ B2 )
= bot_bo6441361344521902642_a_nat )
= ( ( A = bot_bo6441361344521902642_a_nat )
& ( B2 = bot_bo6441361344521902642_a_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_635_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A @ B2 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_636_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A @ B2 )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_637_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_o,B2: set_o] :
( ( ( sup_sup_set_o @ A @ B2 )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_638_sup__eq__bot__iff,axiom,
! [X4: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ X4 @ Y4 )
= bot_bo1033123847703346641_a_nat )
= ( ( X4 = bot_bo1033123847703346641_a_nat )
& ( Y4 = bot_bo1033123847703346641_a_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_639_sup__eq__bot__iff,axiom,
! [X4: set_na3699693778330250182_a_nat,Y4: set_na3699693778330250182_a_nat] :
( ( ( sup_su3329769938372955546_a_nat @ X4 @ Y4 )
= bot_bo6441361344521902642_a_nat )
= ( ( X4 = bot_bo6441361344521902642_a_nat )
& ( Y4 = bot_bo6441361344521902642_a_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_640_sup__eq__bot__iff,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( ( sup_sup_set_nat @ X4 @ Y4 )
= bot_bot_set_nat )
= ( ( X4 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_641_sup__eq__bot__iff,axiom,
! [X4: set_a,Y4: set_a] :
( ( ( sup_sup_set_a @ X4 @ Y4 )
= bot_bot_set_a )
= ( ( X4 = bot_bot_set_a )
& ( Y4 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_642_sup__eq__bot__iff,axiom,
! [X4: set_o,Y4: set_o] :
( ( ( sup_sup_set_o @ X4 @ Y4 )
= bot_bot_set_o )
= ( ( X4 = bot_bot_set_o )
& ( Y4 = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_643_bot__eq__sup__iff,axiom,
! [X4: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( sup_su4083067149120280889_a_nat @ X4 @ Y4 ) )
= ( ( X4 = bot_bo1033123847703346641_a_nat )
& ( Y4 = bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_644_bot__eq__sup__iff,axiom,
! [X4: set_na3699693778330250182_a_nat,Y4: set_na3699693778330250182_a_nat] :
( ( bot_bo6441361344521902642_a_nat
= ( sup_su3329769938372955546_a_nat @ X4 @ Y4 ) )
= ( ( X4 = bot_bo6441361344521902642_a_nat )
& ( Y4 = bot_bo6441361344521902642_a_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_645_bot__eq__sup__iff,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X4 @ Y4 ) )
= ( ( X4 = bot_bot_set_nat )
& ( Y4 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_646_bot__eq__sup__iff,axiom,
! [X4: set_a,Y4: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X4 @ Y4 ) )
= ( ( X4 = bot_bot_set_a )
& ( Y4 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_647_bot__eq__sup__iff,axiom,
! [X4: set_o,Y4: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X4 @ Y4 ) )
= ( ( X4 = bot_bot_set_o )
& ( Y4 = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_648_sup__bot__right,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ bot_bo1033123847703346641_a_nat )
= X4 ) ).
% sup_bot_right
thf(fact_649_sup__bot__right,axiom,
! [X4: set_na3699693778330250182_a_nat] :
( ( sup_su3329769938372955546_a_nat @ X4 @ bot_bo6441361344521902642_a_nat )
= X4 ) ).
% sup_bot_right
thf(fact_650_sup__bot__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% sup_bot_right
thf(fact_651_sup__bot__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ bot_bot_set_a )
= X4 ) ).
% sup_bot_right
thf(fact_652_sup__bot__right,axiom,
! [X4: set_o] :
( ( sup_sup_set_o @ X4 @ bot_bot_set_o )
= X4 ) ).
% sup_bot_right
thf(fact_653_sup__bot__left,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ bot_bo1033123847703346641_a_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_654_sup__bot__left,axiom,
! [X4: set_na3699693778330250182_a_nat] :
( ( sup_su3329769938372955546_a_nat @ bot_bo6441361344521902642_a_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_655_sup__bot__left,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_656_sup__bot__left,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_657_sup__bot__left,axiom,
! [X4: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X4 )
= X4 ) ).
% sup_bot_left
thf(fact_658_singletonI,axiom,
! [A: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% singletonI
thf(fact_659_singletonI,axiom,
! [A: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) ).
% singletonI
thf(fact_660_singletonI,axiom,
! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_661_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_662_singletonI,axiom,
! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).
% singletonI
thf(fact_663_insert__subset,axiom,
! [X4: $o,A2: set_o,B: set_o] :
( ( ord_less_eq_set_o @ ( insert_o @ X4 @ A2 ) @ B )
= ( ( member_o @ X4 @ B )
& ( ord_less_eq_set_o @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_664_insert__subset,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ ( insert5265011953798106934_a_nat @ X4 @ A2 ) @ B )
= ( ( member8690443509505302927_a_nat @ X4 @ B )
& ( ord_le8108555184339247974_a_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_665_insert__subset,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A2 ) @ B )
= ( ( member408289922725080238_a_nat @ X4 @ B )
& ( ord_le1147066620699065093_a_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_666_insert__subset,axiom,
! [X4: a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X4 @ A2 ) @ B )
= ( ( member_a @ X4 @ B )
& ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_667_insert__subset,axiom,
! [X4: nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ A2 ) @ B )
= ( ( member_nat @ X4 @ B )
& ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% insert_subset
thf(fact_668_Un__empty,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ A2 @ B )
= bot_bo1033123847703346641_a_nat )
= ( ( A2 = bot_bo1033123847703346641_a_nat )
& ( B = bot_bo1033123847703346641_a_nat ) ) ) ).
% Un_empty
thf(fact_669_Un__empty,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ( sup_su3329769938372955546_a_nat @ A2 @ B )
= bot_bo6441361344521902642_a_nat )
= ( ( A2 = bot_bo6441361344521902642_a_nat )
& ( B = bot_bo6441361344521902642_a_nat ) ) ) ).
% Un_empty
thf(fact_670_Un__empty,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_671_Un__empty,axiom,
! [A2: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_672_Un__empty,axiom,
! [A2: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ( A2 = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_673_Diff__cancel,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( minus_5517490076408937517_a_nat @ A2 @ A2 )
= bot_bo6441361344521902642_a_nat ) ).
% Diff_cancel
thf(fact_674_Diff__cancel,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ A2 )
= bot_bot_set_nat ) ).
% Diff_cancel
thf(fact_675_Diff__cancel,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ A2 )
= bot_bot_set_o ) ).
% Diff_cancel
thf(fact_676_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_677_Diff__cancel,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A2 @ A2 )
= bot_bo1033123847703346641_a_nat ) ).
% Diff_cancel
thf(fact_678_empty__Diff,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( minus_5517490076408937517_a_nat @ bot_bo6441361344521902642_a_nat @ A2 )
= bot_bo6441361344521902642_a_nat ) ).
% empty_Diff
thf(fact_679_empty__Diff,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
= bot_bot_set_nat ) ).
% empty_Diff
thf(fact_680_empty__Diff,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ bot_bot_set_o @ A2 )
= bot_bot_set_o ) ).
% empty_Diff
thf(fact_681_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_682_empty__Diff,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ bot_bo1033123847703346641_a_nat @ A2 )
= bot_bo1033123847703346641_a_nat ) ).
% empty_Diff
thf(fact_683_Diff__empty,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( minus_5517490076408937517_a_nat @ A2 @ bot_bo6441361344521902642_a_nat )
= A2 ) ).
% Diff_empty
thf(fact_684_Diff__empty,axiom,
! [A2: set_nat] :
( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Diff_empty
thf(fact_685_Diff__empty,axiom,
! [A2: set_o] :
( ( minus_minus_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Diff_empty
thf(fact_686_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_687_Diff__empty,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A2 @ bot_bo1033123847703346641_a_nat )
= A2 ) ).
% Diff_empty
thf(fact_688_Un__insert__right,axiom,
! [A2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B ) )
= ( insert2950094090816004437_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_689_Un__insert__right,axiom,
! [A2: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( sup_su3329769938372955546_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ B ) )
= ( insert5265011953798106934_a_nat @ A @ ( sup_su3329769938372955546_a_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_690_Un__insert__right,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( sup_sup_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( insert_o @ A @ ( sup_sup_set_o @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_691_Un__insert__right,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( sup_sup_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_692_Un__insert__right,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( insert_nat @ A @ ( sup_sup_set_nat @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_693_Un__insert__left,axiom,
! [A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ ( insert2950094090816004437_a_nat @ A @ B ) @ C )
= ( insert2950094090816004437_a_nat @ A @ ( sup_su4083067149120280889_a_nat @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_694_Un__insert__left,axiom,
! [A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat,C: set_na3699693778330250182_a_nat] :
( ( sup_su3329769938372955546_a_nat @ ( insert5265011953798106934_a_nat @ A @ B ) @ C )
= ( insert5265011953798106934_a_nat @ A @ ( sup_su3329769938372955546_a_nat @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_695_Un__insert__left,axiom,
! [A: $o,B: set_o,C: set_o] :
( ( sup_sup_set_o @ ( insert_o @ A @ B ) @ C )
= ( insert_o @ A @ ( sup_sup_set_o @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_696_Un__insert__left,axiom,
! [A: a,B: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B ) @ C )
= ( insert_a @ A @ ( sup_sup_set_a @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_697_Un__insert__left,axiom,
! [A: nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( insert_nat @ A @ B ) @ C )
= ( insert_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_698_insert__Diff1,axiom,
! [X4: $o,B: set_o,A2: set_o] :
( ( member_o @ X4 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X4 @ A2 ) @ B )
= ( minus_minus_set_o @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_699_insert__Diff1,axiom,
! [X4: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ B )
=> ( ( minus_5517490076408937517_a_nat @ ( insert5265011953798106934_a_nat @ X4 @ A2 ) @ B )
= ( minus_5517490076408937517_a_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_700_insert__Diff1,axiom,
! [X4: nat,B: set_nat,A2: set_nat] :
( ( member_nat @ X4 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_701_insert__Diff1,axiom,
! [X4: a,B: set_a,A2: set_a] :
( ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_702_insert__Diff1,axiom,
! [X4: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A2 ) @ B )
= ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ).
% insert_Diff1
thf(fact_703_Diff__insert0,axiom,
! [X4: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X4 @ A2 )
=> ( ( minus_minus_set_o @ A2 @ ( insert_o @ X4 @ B ) )
= ( minus_minus_set_o @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_704_Diff__insert0,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X4 @ B ) )
= ( minus_5517490076408937517_a_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_705_Diff__insert0,axiom,
! [X4: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X4 @ A2 )
=> ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ B ) )
= ( minus_minus_set_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_706_Diff__insert0,axiom,
! [X4: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X4 @ B ) )
= ( minus_minus_set_a @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_707_Diff__insert0,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X4 @ B ) )
= ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ).
% Diff_insert0
thf(fact_708_singleton__conv2,axiom,
! [A: list_Sum_sum_a_nat] :
( ( collec7555443234367654128_a_nat
@ ( ^ [Y3: list_Sum_sum_a_nat,Z3: list_Sum_sum_a_nat] : ( Y3 = Z3 )
@ A ) )
= ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% singleton_conv2
thf(fact_709_singleton__conv2,axiom,
! [A: nat > sum_sum_a_nat] :
( ( collec5629555741568564177_a_nat
@ ( ^ [Y3: nat > sum_sum_a_nat,Z3: nat > sum_sum_a_nat] : ( Y3 = Z3 )
@ A ) )
= ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) ).
% singleton_conv2
thf(fact_710_singleton__conv2,axiom,
! [A: nat] :
( ( collect_nat
@ ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 )
@ A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv2
thf(fact_711_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 )
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_712_singleton__conv2,axiom,
! [A: $o] :
( ( collect_o
@ ( ^ [Y3: $o,Z3: $o] : ( Y3 = Z3 )
@ A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv2
thf(fact_713_singleton__conv,axiom,
! [A: list_Sum_sum_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] : ( X2 = A ) )
= ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% singleton_conv
thf(fact_714_singleton__conv,axiom,
! [A: nat > sum_sum_a_nat] :
( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] : ( X2 = A ) )
= ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) ).
% singleton_conv
thf(fact_715_singleton__conv,axiom,
! [A: nat] :
( ( collect_nat
@ ^ [X2: nat] : ( X2 = A ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% singleton_conv
thf(fact_716_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X2: a] : ( X2 = A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_717_singleton__conv,axiom,
! [A: $o] :
( ( collect_o
@ ^ [X2: $o] : ( X2 = A ) )
= ( insert_o @ A @ bot_bot_set_o ) ) ).
% singleton_conv
thf(fact_718_singleton__insert__inj__eq_H,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( ( insert2950094090816004437_a_nat @ A @ A2 )
= ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) )
= ( ( A = B2 )
& ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_719_singleton__insert__inj__eq_H,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: nat > sum_sum_a_nat] :
( ( ( insert5265011953798106934_a_nat @ A @ A2 )
= ( insert5265011953798106934_a_nat @ B2 @ bot_bo6441361344521902642_a_nat ) )
= ( ( A = B2 )
& ( ord_le8108555184339247974_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ B2 @ bot_bo6441361344521902642_a_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_720_singleton__insert__inj__eq_H,axiom,
! [A: $o,A2: set_o,B2: $o] :
( ( ( insert_o @ A @ A2 )
= ( insert_o @ B2 @ bot_bot_set_o ) )
= ( ( A = B2 )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_721_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B2: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_722_singleton__insert__inj__eq_H,axiom,
! [A: nat,A2: set_nat,B2: nat] :
( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_723_singleton__insert__inj__eq,axiom,
! [B2: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat )
= ( insert2950094090816004437_a_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_724_singleton__insert__inj__eq,axiom,
! [B2: nat > sum_sum_a_nat,A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( ( insert5265011953798106934_a_nat @ B2 @ bot_bo6441361344521902642_a_nat )
= ( insert5265011953798106934_a_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_le8108555184339247974_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ B2 @ bot_bo6441361344521902642_a_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_725_singleton__insert__inj__eq,axiom,
! [B2: $o,A: $o,A2: set_o] :
( ( ( insert_o @ B2 @ bot_bot_set_o )
= ( insert_o @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ bot_bot_set_o ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_726_singleton__insert__inj__eq,axiom,
! [B2: a,A: a,A2: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_727_singleton__insert__inj__eq,axiom,
! [B2: nat,A: nat,A2: set_nat] :
( ( ( insert_nat @ B2 @ bot_bot_set_nat )
= ( insert_nat @ A @ A2 ) )
= ( ( A = B2 )
& ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_728_Diff__eq__empty__iff,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ( minus_5517490076408937517_a_nat @ A2 @ B )
= bot_bo6441361344521902642_a_nat )
= ( ord_le8108555184339247974_a_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_729_Diff__eq__empty__iff,axiom,
! [A2: set_o,B: set_o] :
( ( ( minus_minus_set_o @ A2 @ B )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_730_Diff__eq__empty__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( minus_7395159227704179404_a_nat @ A2 @ B )
= bot_bo1033123847703346641_a_nat )
= ( ord_le1147066620699065093_a_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_731_Diff__eq__empty__iff,axiom,
! [A2: set_a,B: set_a] :
( ( ( minus_minus_set_a @ A2 @ B )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_732_Diff__eq__empty__iff,axiom,
! [A2: set_nat,B: set_nat] :
( ( ( minus_minus_set_nat @ A2 @ B )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ).
% Diff_eq_empty_iff
thf(fact_733_insert__Diff__single,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( insert5265011953798106934_a_nat @ A @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) )
= ( insert5265011953798106934_a_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_734_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_735_insert__Diff__single,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= ( insert_o @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_736_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_737_insert__Diff__single,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( insert2950094090816004437_a_nat @ A @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) )
= ( insert2950094090816004437_a_nat @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_738_subset__Compl__singleton,axiom,
! [A2: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( uminus2192744996606729052_a_nat @ ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) ) )
= ( ~ ( member408289922725080238_a_nat @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_739_subset__Compl__singleton,axiom,
! [A2: set_o,B2: $o] :
( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ ( insert_o @ B2 @ bot_bot_set_o ) ) )
= ( ~ ( member_o @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_740_subset__Compl__singleton,axiom,
! [A2: set_na3699693778330250182_a_nat,B2: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ ( uminus3259373730256538813_a_nat @ ( insert5265011953798106934_a_nat @ B2 @ bot_bo6441361344521902642_a_nat ) ) )
= ( ~ ( member8690443509505302927_a_nat @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_741_subset__Compl__singleton,axiom,
! [A2: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ ( insert_a @ B2 @ bot_bot_set_a ) ) )
= ( ~ ( member_a @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_742_subset__Compl__singleton,axiom,
! [A2: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B2 @ A2 ) ) ) ).
% subset_Compl_singleton
thf(fact_743__092_060open_062ext__tuple__set_AAD_092_060psi_062_Ans_Ans_092_060phi_062_H_A_123fo__nmlz_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_H_Ans_J_125_A_061_Afo__nmlz_AAD_092_060psi_062_A_096_Aproj__vals_A_123_092_060sigma_062_O_Aad__agr__list_AAD_092_060psi_062_A_Imap_A_092_060sigma_062_Ans_J_A_Imap_A_092_060sigma_062_H_Ans_J_125_Ans_092_060psi_062_092_060close_062,axiom,
( ( ext_tuple_set_a @ aD_psi @ ns @ ns_phi @ ( insert2950094090816004437_a_nat @ ( fo_nmlz_a @ aD_psi @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) @ bot_bo1033123847703346641_a_nat ) )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi )
@ ( proj_v3643391342904276326_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ Sigma @ ns ) @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ) )
@ ns_psi2 ) ) ) ).
% \<open>ext_tuple_set AD\<psi> ns ns\<phi>' {fo_nmlz AD\<psi> (map \<sigma>' ns)} = fo_nmlz AD\<psi> ` proj_vals {\<sigma>. ad_agr_list AD\<psi> (map \<sigma> ns) (map \<sigma>' ns)} ns\<psi>\<close>
thf(fact_744_ext__tuple__set__mono,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,AD: set_a,Ns: list_nat,Ns3: list_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B )
=> ( ord_le1147066620699065093_a_nat @ ( ext_tuple_set_a @ AD @ Ns @ Ns3 @ A2 ) @ ( ext_tuple_set_a @ AD @ Ns @ Ns3 @ B ) ) ) ).
% ext_tuple_set_mono
thf(fact_745_subset__singleton__iff,axiom,
! [X: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ X @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
= ( ( X = bot_bo1033123847703346641_a_nat )
| ( X
= ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_746_subset__singleton__iff,axiom,
! [X: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ X @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) )
= ( ( X = bot_bo6441361344521902642_a_nat )
| ( X
= ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_747_subset__singleton__iff,axiom,
! [X: set_o,A: $o] :
( ( ord_less_eq_set_o @ X @ ( insert_o @ A @ bot_bot_set_o ) )
= ( ( X = bot_bot_set_o )
| ( X
= ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).
% subset_singleton_iff
thf(fact_748_subset__singleton__iff,axiom,
! [X: set_a,A: a] :
( ( ord_less_eq_set_a @ X @ ( insert_a @ A @ bot_bot_set_a ) )
= ( ( X = bot_bot_set_a )
| ( X
= ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_749_subset__singleton__iff,axiom,
! [X: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ X @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( ( X = bot_bot_set_nat )
| ( X
= ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% subset_singleton_iff
thf(fact_750_subset__singletonD,axiom,
! [A2: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) )
=> ( ( A2 = bot_bo1033123847703346641_a_nat )
| ( A2
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_751_subset__singletonD,axiom,
! [A2: set_na3699693778330250182_a_nat,X4: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) )
=> ( ( A2 = bot_bo6441361344521902642_a_nat )
| ( A2
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) ) ) ) ).
% subset_singletonD
thf(fact_752_subset__singletonD,axiom,
! [A2: set_o,X4: $o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X4 @ bot_bot_set_o ) )
=> ( ( A2 = bot_bot_set_o )
| ( A2
= ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).
% subset_singletonD
thf(fact_753_subset__singletonD,axiom,
! [A2: set_a,X4: a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X4 @ bot_bot_set_a ) )
=> ( ( A2 = bot_bot_set_a )
| ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_754_subset__singletonD,axiom,
! [A2: set_nat,X4: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
=> ( ( A2 = bot_bot_set_nat )
| ( A2
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).
% subset_singletonD
thf(fact_755_bot_Oextremum,axiom,
! [A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ bot_bo1033123847703346641_a_nat @ A ) ).
% bot.extremum
thf(fact_756_bot_Oextremum,axiom,
! [A: set_na3699693778330250182_a_nat] : ( ord_le8108555184339247974_a_nat @ bot_bo6441361344521902642_a_nat @ A ) ).
% bot.extremum
thf(fact_757_bot_Oextremum,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% bot.extremum
thf(fact_758_bot_Oextremum,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A ) ).
% bot.extremum
thf(fact_759_bot_Oextremum,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% bot.extremum
thf(fact_760_bot_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% bot.extremum
thf(fact_761_bot_Oextremum__unique,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ bot_bo1033123847703346641_a_nat )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% bot.extremum_unique
thf(fact_762_bot_Oextremum__unique,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A @ bot_bo6441361344521902642_a_nat )
= ( A = bot_bo6441361344521902642_a_nat ) ) ).
% bot.extremum_unique
thf(fact_763_bot_Oextremum__unique,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_764_bot_Oextremum__unique,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
= ( A = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_765_bot_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_766_bot_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
= ( A = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_767_bot_Oextremum__uniqueI,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ bot_bo1033123847703346641_a_nat )
=> ( A = bot_bo1033123847703346641_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_768_bot_Oextremum__uniqueI,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A @ bot_bo6441361344521902642_a_nat )
=> ( A = bot_bo6441361344521902642_a_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_769_bot_Oextremum__uniqueI,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
=> ( A = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_770_bot_Oextremum__uniqueI,axiom,
! [A: set_a] :
( ( ord_less_eq_set_a @ A @ bot_bot_set_a )
=> ( A = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_771_bot_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
=> ( A = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_772_bot_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ bot_bot_nat )
=> ( A = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_773_singleton__Un__iff,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat )
= ( sup_su4083067149120280889_a_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bo1033123847703346641_a_nat )
& ( B
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) )
| ( ( A2
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) )
& ( B = bot_bo1033123847703346641_a_nat ) )
| ( ( A2
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) )
& ( B
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_774_singleton__Un__iff,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat )
= ( sup_su3329769938372955546_a_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bo6441361344521902642_a_nat )
& ( B
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) ) )
| ( ( A2
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) )
& ( B = bot_bo6441361344521902642_a_nat ) )
| ( ( A2
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) )
& ( B
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_775_singleton__Un__iff,axiom,
! [X4: nat,A2: set_nat,B: set_nat] :
( ( ( insert_nat @ X4 @ bot_bot_set_nat )
= ( sup_sup_set_nat @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_776_singleton__Un__iff,axiom,
! [X4: a,A2: set_a,B: set_a] :
( ( ( insert_a @ X4 @ bot_bot_set_a )
= ( sup_sup_set_a @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_777_singleton__Un__iff,axiom,
! [X4: $o,A2: set_o,B: set_o] :
( ( ( insert_o @ X4 @ bot_bot_set_o )
= ( sup_sup_set_o @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B
= ( insert_o @ X4 @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X4 @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X4 @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_778_Un__singleton__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat] :
( ( ( sup_su4083067149120280889_a_nat @ A2 @ B )
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) )
= ( ( ( A2 = bot_bo1033123847703346641_a_nat )
& ( B
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) )
| ( ( A2
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) )
& ( B = bot_bo1033123847703346641_a_nat ) )
| ( ( A2
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) )
& ( B
= ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_779_Un__singleton__iff,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,X4: nat > sum_sum_a_nat] :
( ( ( sup_su3329769938372955546_a_nat @ A2 @ B )
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) )
= ( ( ( A2 = bot_bo6441361344521902642_a_nat )
& ( B
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) ) )
| ( ( A2
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) )
& ( B = bot_bo6441361344521902642_a_nat ) )
| ( ( A2
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) )
& ( B
= ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_780_Un__singleton__iff,axiom,
! [A2: set_nat,B: set_nat,X4: nat] :
( ( ( sup_sup_set_nat @ A2 @ B )
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
= ( ( ( A2 = bot_bot_set_nat )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
| ( ( A2
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B = bot_bot_set_nat ) )
| ( ( A2
= ( insert_nat @ X4 @ bot_bot_set_nat ) )
& ( B
= ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_781_Un__singleton__iff,axiom,
! [A2: set_a,B: set_a,X4: a] :
( ( ( sup_sup_set_a @ A2 @ B )
= ( insert_a @ X4 @ bot_bot_set_a ) )
= ( ( ( A2 = bot_bot_set_a )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) )
| ( ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B = bot_bot_set_a ) )
| ( ( A2
= ( insert_a @ X4 @ bot_bot_set_a ) )
& ( B
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_782_Un__singleton__iff,axiom,
! [A2: set_o,B: set_o,X4: $o] :
( ( ( sup_sup_set_o @ A2 @ B )
= ( insert_o @ X4 @ bot_bot_set_o ) )
= ( ( ( A2 = bot_bot_set_o )
& ( B
= ( insert_o @ X4 @ bot_bot_set_o ) ) )
| ( ( A2
= ( insert_o @ X4 @ bot_bot_set_o ) )
& ( B = bot_bot_set_o ) )
| ( ( A2
= ( insert_o @ X4 @ bot_bot_set_o ) )
& ( B
= ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_783_insert__is__Un,axiom,
( insert2950094090816004437_a_nat
= ( ^ [A5: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ ( insert2950094090816004437_a_nat @ A5 @ bot_bo1033123847703346641_a_nat ) ) ) ) ).
% insert_is_Un
thf(fact_784_insert__is__Un,axiom,
( insert5265011953798106934_a_nat
= ( ^ [A5: nat > sum_sum_a_nat] : ( sup_su3329769938372955546_a_nat @ ( insert5265011953798106934_a_nat @ A5 @ bot_bo6441361344521902642_a_nat ) ) ) ) ).
% insert_is_Un
thf(fact_785_insert__is__Un,axiom,
( insert_nat
= ( ^ [A5: nat] : ( sup_sup_set_nat @ ( insert_nat @ A5 @ bot_bot_set_nat ) ) ) ) ).
% insert_is_Un
thf(fact_786_insert__is__Un,axiom,
( insert_a
= ( ^ [A5: a] : ( sup_sup_set_a @ ( insert_a @ A5 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_787_insert__is__Un,axiom,
( insert_o
= ( ^ [A5: $o] : ( sup_sup_set_o @ ( insert_o @ A5 @ bot_bot_set_o ) ) ) ) ).
% insert_is_Un
thf(fact_788_in__image__insert__iff,axiom,
! [B: set_se5822283258546872870_a_nat,X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ! [C3: set_na3699693778330250182_a_nat] :
( ( member3060896489619847151_a_nat @ C3 @ B )
=> ~ ( member8690443509505302927_a_nat @ X4 @ C3 ) )
=> ( ( member3060896489619847151_a_nat @ A2 @ ( image_4398635103182451333_a_nat @ ( insert5265011953798106934_a_nat @ X4 ) @ B ) )
= ( ( member8690443509505302927_a_nat @ X4 @ A2 )
& ( member3060896489619847151_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_789_in__image__insert__iff,axiom,
! [B: set_set_nat,X4: nat,A2: set_nat] :
( ! [C3: set_nat] :
( ( member_set_nat @ C3 @ B )
=> ~ ( member_nat @ X4 @ C3 ) )
=> ( ( member_set_nat @ A2 @ ( image_7916887816326733075et_nat @ ( insert_nat @ X4 ) @ B ) )
= ( ( member_nat @ X4 @ A2 )
& ( member_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_790_in__image__insert__iff,axiom,
! [B: set_set_o,X4: $o,A2: set_o] :
( ! [C3: set_o] :
( ( member_set_o @ C3 @ B )
=> ~ ( member_o @ X4 @ C3 ) )
=> ( ( member_set_o @ A2 @ ( image_set_o_set_o @ ( insert_o @ X4 ) @ B ) )
= ( ( member_o @ X4 @ A2 )
& ( member_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_791_in__image__insert__iff,axiom,
! [B: set_set_a,X4: a,A2: set_a] :
( ! [C3: set_a] :
( ( member_set_a @ C3 @ B )
=> ~ ( member_a @ X4 @ C3 ) )
=> ( ( member_set_a @ A2 @ ( image_set_a_set_a @ ( insert_a @ X4 ) @ B ) )
= ( ( member_a @ X4 @ A2 )
& ( member_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_792_in__image__insert__iff,axiom,
! [B: set_se4330304633200676677_a_nat,X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ! [C3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ C3 @ B )
=> ~ ( member408289922725080238_a_nat @ X4 @ C3 ) )
=> ( ( member5553968465346197646_a_nat @ A2 @ ( image_3472601871771700037_a_nat @ ( insert2950094090816004437_a_nat @ X4 ) @ B ) )
= ( ( member408289922725080238_a_nat @ X4 @ A2 )
& ( member5553968465346197646_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) @ B ) ) ) ) ).
% in_image_insert_iff
thf(fact_793_Diff__insert__absorb,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( minus_5517490076408937517_a_nat @ ( insert5265011953798106934_a_nat @ X4 @ A2 ) @ ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_794_Diff__insert__absorb,axiom,
! [X4: nat,A2: set_nat] :
( ~ ( member_nat @ X4 @ A2 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A2 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_795_Diff__insert__absorb,axiom,
! [X4: $o,A2: set_o] :
( ~ ( member_o @ X4 @ A2 )
=> ( ( minus_minus_set_o @ ( insert_o @ X4 @ A2 ) @ ( insert_o @ X4 @ bot_bot_set_o ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_796_Diff__insert__absorb,axiom,
! [X4: a,A2: set_a] :
( ~ ( member_a @ X4 @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A2 ) @ ( insert_a @ X4 @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_797_Diff__insert__absorb,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A2 ) @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_798_Diff__insert2,axiom,
! [A2: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ B ) )
= ( minus_5517490076408937517_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_799_Diff__insert2,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_800_Diff__insert2,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B ) ) ).
% Diff_insert2
thf(fact_801_Diff__insert2,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B ) ) ).
% Diff_insert2
thf(fact_802_Diff__insert2,axiom,
! [A2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B ) )
= ( minus_7395159227704179404_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) @ B ) ) ).
% Diff_insert2
thf(fact_803_insert__Diff,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( insert5265011953798106934_a_nat @ A @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_804_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_805_insert__Diff,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ ( minus_minus_set_o @ A2 @ ( insert_o @ A @ bot_bot_set_o ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_806_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_807_insert__Diff,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( insert2950094090816004437_a_nat @ A @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_808_Diff__insert,axiom,
! [A2: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ A @ B ) )
= ( minus_5517490076408937517_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ B ) @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) ) ).
% Diff_insert
thf(fact_809_Diff__insert,axiom,
! [A2: set_nat,A: nat,B: set_nat] :
( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B ) )
= ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% Diff_insert
thf(fact_810_Diff__insert,axiom,
! [A2: set_o,A: $o,B: set_o] :
( ( minus_minus_set_o @ A2 @ ( insert_o @ A @ B ) )
= ( minus_minus_set_o @ ( minus_minus_set_o @ A2 @ B ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).
% Diff_insert
thf(fact_811_Diff__insert,axiom,
! [A2: set_a,A: a,B: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_812_Diff__insert,axiom,
! [A2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ A @ B ) )
= ( minus_7395159227704179404_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ B ) @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ) ).
% Diff_insert
thf(fact_813_insert__Collect,axiom,
! [A: $o,P: $o > $o] :
( ( insert_o @ A @ ( collect_o @ P ) )
= ( collect_o
@ ^ [U: $o] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_814_insert__Collect,axiom,
! [A: nat > sum_sum_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( insert5265011953798106934_a_nat @ A @ ( collec5629555741568564177_a_nat @ P ) )
= ( collec5629555741568564177_a_nat
@ ^ [U: nat > sum_sum_a_nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_815_insert__Collect,axiom,
! [A: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( insert2950094090816004437_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) )
= ( collec7555443234367654128_a_nat
@ ^ [U: list_Sum_sum_a_nat] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_816_insert__compr,axiom,
( insert_o
= ( ^ [A5: $o,B3: set_o] :
( collect_o
@ ^ [X2: $o] :
( ( X2 = A5 )
| ( member_o @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_817_insert__compr,axiom,
( insert_nat
= ( ^ [A5: nat,B3: set_nat] :
( collect_nat
@ ^ [X2: nat] :
( ( X2 = A5 )
| ( member_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_818_insert__compr,axiom,
( insert5265011953798106934_a_nat
= ( ^ [A5: nat > sum_sum_a_nat,B3: set_na3699693778330250182_a_nat] :
( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( X2 = A5 )
| ( member8690443509505302927_a_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_819_insert__compr,axiom,
( insert2950094090816004437_a_nat
= ( ^ [A5: list_Sum_sum_a_nat,B3: set_li6526943997496501093_a_nat] :
( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( X2 = A5 )
| ( member408289922725080238_a_nat @ X2 @ B3 ) ) ) ) ) ).
% insert_compr
thf(fact_820_Set_Oempty__def,axiom,
( bot_bo1033123847703346641_a_nat
= ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] : $false ) ) ).
% Set.empty_def
thf(fact_821_Set_Oempty__def,axiom,
( bot_bo6441361344521902642_a_nat
= ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] : $false ) ) ).
% Set.empty_def
thf(fact_822_Set_Oempty__def,axiom,
( bot_bot_set_nat
= ( collect_nat
@ ^ [X2: nat] : $false ) ) ).
% Set.empty_def
thf(fact_823_Set_Oempty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% Set.empty_def
thf(fact_824_Set_Oempty__def,axiom,
( bot_bot_set_o
= ( collect_o
@ ^ [X2: $o] : $false ) ) ).
% Set.empty_def
thf(fact_825_emptyE,axiom,
! [A: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_826_emptyE,axiom,
! [A: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ).
% emptyE
thf(fact_827_emptyE,axiom,
! [A: nat] :
~ ( member_nat @ A @ bot_bot_set_nat ) ).
% emptyE
thf(fact_828_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_829_emptyE,axiom,
! [A: $o] :
~ ( member_o @ A @ bot_bot_set_o ) ).
% emptyE
thf(fact_830_insertE,axiom,
! [A: $o,B2: $o,A2: set_o] :
( ( member_o @ A @ ( insert_o @ B2 @ A2 ) )
=> ( ( A = ~ B2 )
=> ( member_o @ A @ A2 ) ) ) ).
% insertE
thf(fact_831_insertE,axiom,
! [A: nat > sum_sum_a_nat,B2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member8690443509505302927_a_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_832_insertE,axiom,
! [A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member408289922725080238_a_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_833_insertE,axiom,
! [A: nat,B2: nat,A2: set_nat] :
( ( member_nat @ A @ ( insert_nat @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_nat @ A @ A2 ) ) ) ).
% insertE
thf(fact_834_equals0D,axiom,
! [A2: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat] :
( ( A2 = bot_bo1033123847703346641_a_nat )
=> ~ ( member408289922725080238_a_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_835_equals0D,axiom,
! [A2: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat] :
( ( A2 = bot_bo6441361344521902642_a_nat )
=> ~ ( member8690443509505302927_a_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_836_equals0D,axiom,
! [A2: set_nat,A: nat] :
( ( A2 = bot_bot_set_nat )
=> ~ ( member_nat @ A @ A2 ) ) ).
% equals0D
thf(fact_837_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_838_equals0D,axiom,
! [A2: set_o,A: $o] :
( ( A2 = bot_bot_set_o )
=> ~ ( member_o @ A @ A2 ) ) ).
% equals0D
thf(fact_839_equals0I,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ! [Y5: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y5 @ A2 )
=> ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_840_equals0I,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ! [Y5: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ Y5 @ A2 )
=> ( A2 = bot_bo6441361344521902642_a_nat ) ) ).
% equals0I
thf(fact_841_equals0I,axiom,
! [A2: set_nat] :
( ! [Y5: nat] :
~ ( member_nat @ Y5 @ A2 )
=> ( A2 = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_842_equals0I,axiom,
! [A2: set_a] :
( ! [Y5: a] :
~ ( member_a @ Y5 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_843_equals0I,axiom,
! [A2: set_o] :
( ! [Y5: $o] :
~ ( member_o @ Y5 @ A2 )
=> ( A2 = bot_bot_set_o ) ) ).
% equals0I
thf(fact_844_insertI1,axiom,
! [A: $o,B: set_o] : ( member_o @ A @ ( insert_o @ A @ B ) ) ).
% insertI1
thf(fact_845_insertI1,axiom,
! [A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] : ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ A @ B ) ) ).
% insertI1
thf(fact_846_insertI1,axiom,
! [A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] : ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ A @ B ) ) ).
% insertI1
thf(fact_847_insertI1,axiom,
! [A: nat,B: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B ) ) ).
% insertI1
thf(fact_848_insertI2,axiom,
! [A: $o,B: set_o,B2: $o] :
( ( member_o @ A @ B )
=> ( member_o @ A @ ( insert_o @ B2 @ B ) ) ) ).
% insertI2
thf(fact_849_insertI2,axiom,
! [A: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat,B2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ A @ B )
=> ( member8690443509505302927_a_nat @ A @ ( insert5265011953798106934_a_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_850_insertI2,axiom,
! [A: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ A @ B )
=> ( member408289922725080238_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_851_insertI2,axiom,
! [A: nat,B: set_nat,B2: nat] :
( ( member_nat @ A @ B )
=> ( member_nat @ A @ ( insert_nat @ B2 @ B ) ) ) ).
% insertI2
thf(fact_852_ex__in__conv,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ? [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ A2 ) )
= ( A2 != bot_bo1033123847703346641_a_nat ) ) ).
% ex_in_conv
thf(fact_853_ex__in__conv,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( ? [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ A2 ) )
= ( A2 != bot_bo6441361344521902642_a_nat ) ) ).
% ex_in_conv
thf(fact_854_ex__in__conv,axiom,
! [A2: set_nat] :
( ( ? [X2: nat] : ( member_nat @ X2 @ A2 ) )
= ( A2 != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_855_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_856_ex__in__conv,axiom,
! [A2: set_o] :
( ( ? [X2: $o] : ( member_o @ X2 @ A2 ) )
= ( A2 != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_857_Set_Oset__insert,axiom,
! [X4: $o,A2: set_o] :
( ( member_o @ X4 @ A2 )
=> ~ ! [B6: set_o] :
( ( A2
= ( insert_o @ X4 @ B6 ) )
=> ( member_o @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_858_Set_Oset__insert,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ~ ! [B6: set_na3699693778330250182_a_nat] :
( ( A2
= ( insert5265011953798106934_a_nat @ X4 @ B6 ) )
=> ( member8690443509505302927_a_nat @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_859_Set_Oset__insert,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ~ ! [B6: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ X4 @ B6 ) )
=> ( member408289922725080238_a_nat @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_860_Set_Oset__insert,axiom,
! [X4: nat,A2: set_nat] :
( ( member_nat @ X4 @ A2 )
=> ~ ! [B6: set_nat] :
( ( A2
= ( insert_nat @ X4 @ B6 ) )
=> ( member_nat @ X4 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_861_singletonD,axiom,
! [B2: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_862_singletonD,axiom,
! [B2: nat > sum_sum_a_nat,A: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ B2 @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_863_singletonD,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_864_singletonD,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_865_singletonD,axiom,
! [B2: $o,A: $o] :
( ( member_o @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_866_insert__ident,axiom,
! [X4: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X4 @ A2 )
=> ( ~ ( member_o @ X4 @ B )
=> ( ( ( insert_o @ X4 @ A2 )
= ( insert_o @ X4 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_867_insert__ident,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ~ ( member8690443509505302927_a_nat @ X4 @ B )
=> ( ( ( insert5265011953798106934_a_nat @ X4 @ A2 )
= ( insert5265011953798106934_a_nat @ X4 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_868_insert__ident,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( ( insert2950094090816004437_a_nat @ X4 @ A2 )
= ( insert2950094090816004437_a_nat @ X4 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_869_insert__ident,axiom,
! [X4: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X4 @ A2 )
=> ( ~ ( member_nat @ X4 @ B )
=> ( ( ( insert_nat @ X4 @ A2 )
= ( insert_nat @ X4 @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_870_insert__absorb,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ( ( insert_o @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_871_insert__absorb,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ( insert5265011953798106934_a_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_872_insert__absorb,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ( ( insert2950094090816004437_a_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_873_insert__absorb,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ( ( insert_nat @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_874_insert__eq__iff,axiom,
! [A: $o,A2: set_o,B2: $o,B: set_o] :
( ~ ( member_o @ A @ A2 )
=> ( ~ ( member_o @ B2 @ B )
=> ( ( ( insert_o @ A @ A2 )
= ( insert_o @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A = ~ B2 )
=> ? [C4: set_o] :
( ( A2
= ( insert_o @ B2 @ C4 ) )
& ~ ( member_o @ B2 @ C4 )
& ( B
= ( insert_o @ A @ C4 ) )
& ~ ( member_o @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_875_insert__eq__iff,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B2: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ A @ A2 )
=> ( ~ ( member8690443509505302927_a_nat @ B2 @ B )
=> ( ( ( insert5265011953798106934_a_nat @ A @ A2 )
= ( insert5265011953798106934_a_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C4: set_na3699693778330250182_a_nat] :
( ( A2
= ( insert5265011953798106934_a_nat @ B2 @ C4 ) )
& ~ ( member8690443509505302927_a_nat @ B2 @ C4 )
& ( B
= ( insert5265011953798106934_a_nat @ A @ C4 ) )
& ~ ( member8690443509505302927_a_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_876_insert__eq__iff,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ A @ A2 )
=> ( ~ ( member408289922725080238_a_nat @ B2 @ B )
=> ( ( ( insert2950094090816004437_a_nat @ A @ A2 )
= ( insert2950094090816004437_a_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C4: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ B2 @ C4 ) )
& ~ ( member408289922725080238_a_nat @ B2 @ C4 )
& ( B
= ( insert2950094090816004437_a_nat @ A @ C4 ) )
& ~ ( member408289922725080238_a_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_877_insert__eq__iff,axiom,
! [A: nat,A2: set_nat,B2: nat,B: set_nat] :
( ~ ( member_nat @ A @ A2 )
=> ( ~ ( member_nat @ B2 @ B )
=> ( ( ( insert_nat @ A @ A2 )
= ( insert_nat @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C4: set_nat] :
( ( A2
= ( insert_nat @ B2 @ C4 ) )
& ~ ( member_nat @ B2 @ C4 )
& ( B
= ( insert_nat @ A @ C4 ) )
& ~ ( member_nat @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_878_singleton__iff,axiom,
! [B2: list_Sum_sum_a_nat,A: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_879_singleton__iff,axiom,
! [B2: nat > sum_sum_a_nat,A: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ B2 @ ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_880_singleton__iff,axiom,
! [B2: nat,A: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A @ bot_bot_set_nat ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_881_singleton__iff,axiom,
! [B2: a,A: a] :
( ( member_a @ B2 @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_882_singleton__iff,axiom,
! [B2: $o,A: $o] :
( ( member_o @ B2 @ ( insert_o @ A @ bot_bot_set_o ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_883_insert__commute,axiom,
! [X4: list_Sum_sum_a_nat,Y4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( insert2950094090816004437_a_nat @ X4 @ ( insert2950094090816004437_a_nat @ Y4 @ A2 ) )
= ( insert2950094090816004437_a_nat @ Y4 @ ( insert2950094090816004437_a_nat @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_884_insert__commute,axiom,
! [X4: nat > sum_sum_a_nat,Y4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( insert5265011953798106934_a_nat @ X4 @ ( insert5265011953798106934_a_nat @ Y4 @ A2 ) )
= ( insert5265011953798106934_a_nat @ Y4 @ ( insert5265011953798106934_a_nat @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_885_insert__commute,axiom,
! [X4: $o,Y4: $o,A2: set_o] :
( ( insert_o @ X4 @ ( insert_o @ Y4 @ A2 ) )
= ( insert_o @ Y4 @ ( insert_o @ X4 @ A2 ) ) ) ).
% insert_commute
thf(fact_886_doubleton__eq__iff,axiom,
! [A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat,C2: list_Sum_sum_a_nat,D2: list_Sum_sum_a_nat] :
( ( ( insert2950094090816004437_a_nat @ A @ ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) )
= ( insert2950094090816004437_a_nat @ C2 @ ( insert2950094090816004437_a_nat @ D2 @ bot_bo1033123847703346641_a_nat ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_887_doubleton__eq__iff,axiom,
! [A: nat > sum_sum_a_nat,B2: nat > sum_sum_a_nat,C2: nat > sum_sum_a_nat,D2: nat > sum_sum_a_nat] :
( ( ( insert5265011953798106934_a_nat @ A @ ( insert5265011953798106934_a_nat @ B2 @ bot_bo6441361344521902642_a_nat ) )
= ( insert5265011953798106934_a_nat @ C2 @ ( insert5265011953798106934_a_nat @ D2 @ bot_bo6441361344521902642_a_nat ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_888_doubleton__eq__iff,axiom,
! [A: nat,B2: nat,C2: nat,D2: nat] :
( ( ( insert_nat @ A @ ( insert_nat @ B2 @ bot_bot_set_nat ) )
= ( insert_nat @ C2 @ ( insert_nat @ D2 @ bot_bot_set_nat ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_889_doubleton__eq__iff,axiom,
! [A: a,B2: a,C2: a,D2: a] :
( ( ( insert_a @ A @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_890_doubleton__eq__iff,axiom,
! [A: $o,B2: $o,C2: $o,D2: $o] :
( ( ( insert_o @ A @ ( insert_o @ B2 @ bot_bot_set_o ) )
= ( insert_o @ C2 @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
= ( ( ( A = C2 )
& ( B2 = D2 ) )
| ( ( A = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_891_insert__not__empty,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( insert2950094090816004437_a_nat @ A @ A2 )
!= bot_bo1033123847703346641_a_nat ) ).
% insert_not_empty
thf(fact_892_insert__not__empty,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( insert5265011953798106934_a_nat @ A @ A2 )
!= bot_bo6441361344521902642_a_nat ) ).
% insert_not_empty
thf(fact_893_insert__not__empty,axiom,
! [A: nat,A2: set_nat] :
( ( insert_nat @ A @ A2 )
!= bot_bot_set_nat ) ).
% insert_not_empty
thf(fact_894_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_895_insert__not__empty,axiom,
! [A: $o,A2: set_o] :
( ( insert_o @ A @ A2 )
!= bot_bot_set_o ) ).
% insert_not_empty
thf(fact_896_singleton__inject,axiom,
! [A: list_Sum_sum_a_nat,B2: list_Sum_sum_a_nat] :
( ( ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat )
= ( insert2950094090816004437_a_nat @ B2 @ bot_bo1033123847703346641_a_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_897_singleton__inject,axiom,
! [A: nat > sum_sum_a_nat,B2: nat > sum_sum_a_nat] :
( ( ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat )
= ( insert5265011953798106934_a_nat @ B2 @ bot_bo6441361344521902642_a_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_898_singleton__inject,axiom,
! [A: nat,B2: nat] :
( ( ( insert_nat @ A @ bot_bot_set_nat )
= ( insert_nat @ B2 @ bot_bot_set_nat ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_899_singleton__inject,axiom,
! [A: a,B2: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_900_singleton__inject,axiom,
! [A: $o,B2: $o] :
( ( ( insert_o @ A @ bot_bot_set_o )
= ( insert_o @ B2 @ bot_bot_set_o ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_901_mk__disjoint__insert,axiom,
! [A: $o,A2: set_o] :
( ( member_o @ A @ A2 )
=> ? [B6: set_o] :
( ( A2
= ( insert_o @ A @ B6 ) )
& ~ ( member_o @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_902_mk__disjoint__insert,axiom,
! [A: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ A2 )
=> ? [B6: set_na3699693778330250182_a_nat] :
( ( A2
= ( insert5265011953798106934_a_nat @ A @ B6 ) )
& ~ ( member8690443509505302927_a_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_903_mk__disjoint__insert,axiom,
! [A: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ A2 )
=> ? [B6: set_li6526943997496501093_a_nat] :
( ( A2
= ( insert2950094090816004437_a_nat @ A @ B6 ) )
& ~ ( member408289922725080238_a_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_904_mk__disjoint__insert,axiom,
! [A: nat,A2: set_nat] :
( ( member_nat @ A @ A2 )
=> ? [B6: set_nat] :
( ( A2
= ( insert_nat @ A @ B6 ) )
& ~ ( member_nat @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_905_Collect__conv__if2,axiom,
! [P: list_Sum_sum_a_nat > $o,A: list_Sum_sum_a_nat] :
( ( ( P @ A )
=> ( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% Collect_conv_if2
thf(fact_906_Collect__conv__if2,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,A: nat > sum_sum_a_nat] :
( ( ( P @ A )
=> ( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bo6441361344521902642_a_nat ) ) ) ).
% Collect_conv_if2
thf(fact_907_Collect__conv__if2,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if2
thf(fact_908_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_909_Collect__conv__if2,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if2
thf(fact_910_Collect__conv__if,axiom,
! [P: list_Sum_sum_a_nat > $o,A: list_Sum_sum_a_nat] :
( ( ( P @ A )
=> ( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert2950094090816004437_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bo1033123847703346641_a_nat ) ) ) ).
% Collect_conv_if
thf(fact_911_Collect__conv__if,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,A: nat > sum_sum_a_nat] :
( ( ( P @ A )
=> ( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert5265011953798106934_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bo6441361344521902642_a_nat ) ) ) ).
% Collect_conv_if
thf(fact_912_Collect__conv__if,axiom,
! [P: nat > $o,A: nat] :
( ( ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_nat @ A @ bot_bot_set_nat ) ) )
& ( ~ ( P @ A )
=> ( ( collect_nat
@ ^ [X2: nat] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_nat ) ) ) ).
% Collect_conv_if
thf(fact_913_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_914_Collect__conv__if,axiom,
! [P: $o > $o,A: $o] :
( ( ( P @ A )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_o @ A @ bot_bot_set_o ) ) )
& ( ~ ( P @ A )
=> ( ( collect_o
@ ^ [X2: $o] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_o ) ) ) ).
% Collect_conv_if
thf(fact_915_image__constant__conv,axiom,
! [A2: set_nat,C2: nat] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X2: nat] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_nat
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_916_image__constant__conv,axiom,
! [A2: set_nat,C2: a] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_a
@ ^ [X2: nat] : C2
@ A2 )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_a
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_917_image__constant__conv,axiom,
! [A2: set_nat,C2: $o] :
( ( ( A2 = bot_bot_set_nat )
=> ( ( image_nat_o
@ ^ [X2: nat] : C2
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_nat )
=> ( ( image_nat_o
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_918_image__constant__conv,axiom,
! [A2: set_a,C2: nat] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_nat
@ ^ [X2: a] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_nat
@ ^ [X2: a] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_919_image__constant__conv,axiom,
! [A2: set_a,C2: a] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X2: a] : C2
@ A2 )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_a
@ ^ [X2: a] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_920_image__constant__conv,axiom,
! [A2: set_a,C2: $o] :
( ( ( A2 = bot_bot_set_a )
=> ( ( image_a_o
@ ^ [X2: a] : C2
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_a )
=> ( ( image_a_o
@ ^ [X2: a] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_921_image__constant__conv,axiom,
! [A2: set_o,C2: nat] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_nat
@ ^ [X2: $o] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_nat
@ ^ [X2: $o] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_922_image__constant__conv,axiom,
! [A2: set_o,C2: a] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_a
@ ^ [X2: $o] : C2
@ A2 )
= bot_bot_set_a ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_a
@ ^ [X2: $o] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ) ).
% image_constant_conv
thf(fact_923_image__constant__conv,axiom,
! [A2: set_o,C2: $o] :
( ( ( A2 = bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X2: $o] : C2
@ A2 )
= bot_bot_set_o ) )
& ( ( A2 != bot_bot_set_o )
=> ( ( image_o_o
@ ^ [X2: $o] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ) ).
% image_constant_conv
thf(fact_924_image__constant__conv,axiom,
! [A2: set_li6526943997496501093_a_nat,C2: nat] :
( ( ( A2 = bot_bo1033123847703346641_a_nat )
=> ( ( image_2535339886381165584at_nat
@ ^ [X2: list_Sum_sum_a_nat] : C2
@ A2 )
= bot_bot_set_nat ) )
& ( ( A2 != bot_bo1033123847703346641_a_nat )
=> ( ( image_2535339886381165584at_nat
@ ^ [X2: list_Sum_sum_a_nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ) ).
% image_constant_conv
thf(fact_925_image__constant,axiom,
! [X4: nat,A2: set_nat,C2: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( image_nat_nat
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_926_image__constant,axiom,
! [X4: nat,A2: set_nat,C2: a] :
( ( member_nat @ X4 @ A2 )
=> ( ( image_nat_a
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_927_image__constant,axiom,
! [X4: nat,A2: set_nat,C2: $o] :
( ( member_nat @ X4 @ A2 )
=> ( ( image_nat_o
@ ^ [X2: nat] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_928_image__constant,axiom,
! [X4: nat,A2: set_nat,C2: list_Sum_sum_a_nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( image_6262589752765146990_a_nat
@ ^ [X2: nat] : C2
@ A2 )
= ( insert2950094090816004437_a_nat @ C2 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% image_constant
thf(fact_929_image__constant,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,C2: nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( image_2535339886381165584at_nat
@ ^ [X2: list_Sum_sum_a_nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_930_image__constant,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,C2: a] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( image_4685209397977471678_nat_a
@ ^ [X2: list_Sum_sum_a_nat] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_931_image__constant,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,C2: $o] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( image_3984946558445957976_nat_o
@ ^ [X2: list_Sum_sum_a_nat] : C2
@ A2 )
= ( insert_o @ C2 @ bot_bot_set_o ) ) ) ).
% image_constant
thf(fact_932_image__constant,axiom,
! [X4: nat,A2: set_nat,C2: nat > sum_sum_a_nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( image_1051037728736664655_a_nat
@ ^ [X2: nat] : C2
@ A2 )
= ( insert5265011953798106934_a_nat @ C2 @ bot_bo6441361344521902642_a_nat ) ) ) ).
% image_constant
thf(fact_933_image__constant,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,C2: nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( image_5786201776793816049at_nat
@ ^ [X2: nat > sum_sum_a_nat] : C2
@ A2 )
= ( insert_nat @ C2 @ bot_bot_set_nat ) ) ) ).
% image_constant
thf(fact_934_image__constant,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,C2: a] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( image_7126633039417159965_nat_a
@ ^ [X2: nat > sum_sum_a_nat] : C2
@ A2 )
= ( insert_a @ C2 @ bot_bot_set_a ) ) ) ).
% image_constant
thf(fact_935_Diff__single__insert,axiom,
! [A2: set_na3699693778330250182_a_nat,X4: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) ) @ B )
=> ( ord_le8108555184339247974_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_936_Diff__single__insert,axiom,
! [A2: set_o,X4: $o,B: set_o] :
( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ B )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_937_Diff__single__insert,axiom,
! [A2: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) @ B )
=> ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_938_Diff__single__insert,axiom,
! [A2: set_a,X4: a,B: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_939_Diff__single__insert,axiom,
! [A2: set_nat,X4: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X4 @ B ) ) ) ).
% Diff_single_insert
thf(fact_940_subset__insert__iff,axiom,
! [A2: set_na3699693778330250182_a_nat,X4: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X4 @ B ) )
= ( ( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ord_le8108555184339247974_a_nat @ ( minus_5517490076408937517_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) ) @ B ) )
& ( ~ ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ord_le8108555184339247974_a_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_941_subset__insert__iff,axiom,
! [A2: set_o,X4: $o,B: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X4 @ B ) )
= ( ( ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ ( minus_minus_set_o @ A2 @ ( insert_o @ X4 @ bot_bot_set_o ) ) @ B ) )
& ( ~ ( member_o @ X4 @ A2 )
=> ( ord_less_eq_set_o @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_942_subset__insert__iff,axiom,
! [A2: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X4 @ B ) )
= ( ( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( minus_7395159227704179404_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) @ B ) )
& ( ~ ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_943_subset__insert__iff,axiom,
! [A2: set_a,X4: a,B: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X4 @ B ) )
= ( ( ( member_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X4 @ bot_bot_set_a ) ) @ B ) )
& ( ~ ( member_a @ X4 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_944_subset__insert__iff,axiom,
! [A2: set_nat,X4: nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X4 @ B ) )
= ( ( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) @ B ) )
& ( ~ ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ B ) ) ) ) ).
% subset_insert_iff
thf(fact_945_Compl__insert,axiom,
! [X4: nat,A2: set_nat] :
( ( uminus5710092332889474511et_nat @ ( insert_nat @ X4 @ A2 ) )
= ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ).
% Compl_insert
thf(fact_946_Compl__insert,axiom,
! [X4: $o,A2: set_o] :
( ( uminus_uminus_set_o @ ( insert_o @ X4 @ A2 ) )
= ( minus_minus_set_o @ ( uminus_uminus_set_o @ A2 ) @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ).
% Compl_insert
thf(fact_947_Compl__insert,axiom,
! [X4: a,A2: set_a] :
( ( uminus_uminus_set_a @ ( insert_a @ X4 @ A2 ) )
= ( minus_minus_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% Compl_insert
thf(fact_948_Compl__insert,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( uminus2192744996606729052_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A2 ) )
= ( minus_7395159227704179404_a_nat @ ( uminus2192744996606729052_a_nat @ A2 ) @ ( insert2950094090816004437_a_nat @ X4 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% Compl_insert
thf(fact_949_Compl__insert,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( uminus3259373730256538813_a_nat @ ( insert5265011953798106934_a_nat @ X4 @ A2 ) )
= ( minus_5517490076408937517_a_nat @ ( uminus3259373730256538813_a_nat @ A2 ) @ ( insert5265011953798106934_a_nat @ X4 @ bot_bo6441361344521902642_a_nat ) ) ) ).
% Compl_insert
thf(fact_950_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ X4 @ bot_bo1033123847703346641_a_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_951_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_na3699693778330250182_a_nat] :
( ( sup_su3329769938372955546_a_nat @ X4 @ bot_bo6441361344521902642_a_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_952_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_nat] :
( ( sup_sup_set_nat @ X4 @ bot_bot_set_nat )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_953_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_a] :
( ( sup_sup_set_a @ X4 @ bot_bot_set_a )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_954_boolean__algebra_Odisj__zero__right,axiom,
! [X4: set_o] :
( ( sup_sup_set_o @ X4 @ bot_bot_set_o )
= X4 ) ).
% boolean_algebra.disj_zero_right
thf(fact_955_subset__insertI2,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,B2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ B )
=> ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_956_subset__insertI2,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,B2: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ B )
=> ( ord_le8108555184339247974_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_957_subset__insertI2,axiom,
! [A2: set_o,B: set_o,B2: $o] :
( ( ord_less_eq_set_o @ A2 @ B )
=> ( ord_less_eq_set_o @ A2 @ ( insert_o @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_958_subset__insertI2,axiom,
! [A2: set_a,B: set_a,B2: a] :
( ( ord_less_eq_set_a @ A2 @ B )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_959_subset__insertI2,axiom,
! [A2: set_nat,B: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B )
=> ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B2 @ B ) ) ) ).
% subset_insertI2
thf(fact_960_subset__insertI,axiom,
! [B: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat] : ( ord_le1147066620699065093_a_nat @ B @ ( insert2950094090816004437_a_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_961_subset__insertI,axiom,
! [B: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat] : ( ord_le8108555184339247974_a_nat @ B @ ( insert5265011953798106934_a_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_962_subset__insertI,axiom,
! [B: set_o,A: $o] : ( ord_less_eq_set_o @ B @ ( insert_o @ A @ B ) ) ).
% subset_insertI
thf(fact_963_subset__insertI,axiom,
! [B: set_a,A: a] : ( ord_less_eq_set_a @ B @ ( insert_a @ A @ B ) ) ).
% subset_insertI
thf(fact_964_subset__insertI,axiom,
! [B: set_nat,A: nat] : ( ord_less_eq_set_nat @ B @ ( insert_nat @ A @ B ) ) ).
% subset_insertI
thf(fact_965_subset__insert,axiom,
! [X4: $o,A2: set_o,B: set_o] :
( ~ ( member_o @ X4 @ A2 )
=> ( ( ord_less_eq_set_o @ A2 @ ( insert_o @ X4 @ B ) )
= ( ord_less_eq_set_o @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_966_subset__insert,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ~ ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( ord_le8108555184339247974_a_nat @ A2 @ ( insert5265011953798106934_a_nat @ X4 @ B ) )
= ( ord_le8108555184339247974_a_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_967_subset__insert,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ~ ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ A2 @ ( insert2950094090816004437_a_nat @ X4 @ B ) )
= ( ord_le1147066620699065093_a_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_968_subset__insert,axiom,
! [X4: a,A2: set_a,B: set_a] :
( ~ ( member_a @ X4 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X4 @ B ) )
= ( ord_less_eq_set_a @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_969_subset__insert,axiom,
! [X4: nat,A2: set_nat,B: set_nat] :
( ~ ( member_nat @ X4 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X4 @ B ) )
= ( ord_less_eq_set_nat @ A2 @ B ) ) ) ).
% subset_insert
thf(fact_970_insert__mono,axiom,
! [C: set_li6526943997496501093_a_nat,D: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C @ D )
=> ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ A @ C ) @ ( insert2950094090816004437_a_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_971_insert__mono,axiom,
! [C: set_na3699693778330250182_a_nat,D: set_na3699693778330250182_a_nat,A: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ C @ D )
=> ( ord_le8108555184339247974_a_nat @ ( insert5265011953798106934_a_nat @ A @ C ) @ ( insert5265011953798106934_a_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_972_insert__mono,axiom,
! [C: set_o,D: set_o,A: $o] :
( ( ord_less_eq_set_o @ C @ D )
=> ( ord_less_eq_set_o @ ( insert_o @ A @ C ) @ ( insert_o @ A @ D ) ) ) ).
% insert_mono
thf(fact_973_insert__mono,axiom,
! [C: set_a,D: set_a,A: a] :
( ( ord_less_eq_set_a @ C @ D )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C ) @ ( insert_a @ A @ D ) ) ) ).
% insert_mono
thf(fact_974_insert__mono,axiom,
! [C: set_nat,D: set_nat,A: nat] :
( ( ord_less_eq_set_nat @ C @ D )
=> ( ord_less_eq_set_nat @ ( insert_nat @ A @ C ) @ ( insert_nat @ A @ D ) ) ) ).
% insert_mono
thf(fact_975_insert__Diff__if,axiom,
! [X4: $o,B: set_o,A2: set_o] :
( ( ( member_o @ X4 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X4 @ A2 ) @ B )
= ( minus_minus_set_o @ A2 @ B ) ) )
& ( ~ ( member_o @ X4 @ B )
=> ( ( minus_minus_set_o @ ( insert_o @ X4 @ A2 ) @ B )
= ( insert_o @ X4 @ ( minus_minus_set_o @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_976_insert__Diff__if,axiom,
! [X4: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat,A2: set_na3699693778330250182_a_nat] :
( ( ( member8690443509505302927_a_nat @ X4 @ B )
=> ( ( minus_5517490076408937517_a_nat @ ( insert5265011953798106934_a_nat @ X4 @ A2 ) @ B )
= ( minus_5517490076408937517_a_nat @ A2 @ B ) ) )
& ( ~ ( member8690443509505302927_a_nat @ X4 @ B )
=> ( ( minus_5517490076408937517_a_nat @ ( insert5265011953798106934_a_nat @ X4 @ A2 ) @ B )
= ( insert5265011953798106934_a_nat @ X4 @ ( minus_5517490076408937517_a_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_977_insert__Diff__if,axiom,
! [X4: nat,B: set_nat,A2: set_nat] :
( ( ( member_nat @ X4 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A2 ) @ B )
= ( minus_minus_set_nat @ A2 @ B ) ) )
& ( ~ ( member_nat @ X4 @ B )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X4 @ A2 ) @ B )
= ( insert_nat @ X4 @ ( minus_minus_set_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_978_insert__Diff__if,axiom,
! [X4: a,B: set_a,A2: set_a] :
( ( ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A2 ) @ B )
= ( minus_minus_set_a @ A2 @ B ) ) )
& ( ~ ( member_a @ X4 @ B )
=> ( ( minus_minus_set_a @ ( insert_a @ X4 @ A2 ) @ B )
= ( insert_a @ X4 @ ( minus_minus_set_a @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_979_insert__Diff__if,axiom,
! [X4: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A2 ) @ B )
= ( minus_7395159227704179404_a_nat @ A2 @ B ) ) )
& ( ~ ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( minus_7395159227704179404_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ A2 ) @ B )
= ( insert2950094090816004437_a_nat @ X4 @ ( minus_7395159227704179404_a_nat @ A2 @ B ) ) ) ) ) ).
% insert_Diff_if
thf(fact_980_Un__empty__right,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A2 @ bot_bo1033123847703346641_a_nat )
= A2 ) ).
% Un_empty_right
thf(fact_981_Un__empty__right,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( sup_su3329769938372955546_a_nat @ A2 @ bot_bo6441361344521902642_a_nat )
= A2 ) ).
% Un_empty_right
thf(fact_982_Un__empty__right,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% Un_empty_right
thf(fact_983_Un__empty__right,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Un_empty_right
thf(fact_984_Un__empty__right,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% Un_empty_right
thf(fact_985_Un__empty__left,axiom,
! [B: set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ bot_bo1033123847703346641_a_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_986_Un__empty__left,axiom,
! [B: set_na3699693778330250182_a_nat] :
( ( sup_su3329769938372955546_a_nat @ bot_bo6441361344521902642_a_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_987_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_988_Un__empty__left,axiom,
! [B: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B )
= B ) ).
% Un_empty_left
thf(fact_989_Un__empty__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B )
= B ) ).
% Un_empty_left
thf(fact_990_Set_Oinsert__def,axiom,
( insert_o
= ( ^ [A5: $o] :
( sup_sup_set_o
@ ( collect_o
@ ^ [X2: $o] : ( X2 = A5 ) ) ) ) ) ).
% Set.insert_def
thf(fact_991_Set_Oinsert__def,axiom,
( insert5265011953798106934_a_nat
= ( ^ [A5: nat > sum_sum_a_nat] :
( sup_su3329769938372955546_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] : ( X2 = A5 ) ) ) ) ) ).
% Set.insert_def
thf(fact_992_Set_Oinsert__def,axiom,
( insert2950094090816004437_a_nat
= ( ^ [A5: list_Sum_sum_a_nat] :
( sup_su4083067149120280889_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] : ( X2 = A5 ) ) ) ) ) ).
% Set.insert_def
thf(fact_993_Set_Oinsert__def,axiom,
( insert_a
= ( ^ [A5: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X2: a] : ( X2 = A5 ) ) ) ) ) ).
% Set.insert_def
thf(fact_994_Set_Oinsert__def,axiom,
( insert_nat
= ( ^ [A5: nat] :
( sup_sup_set_nat
@ ( collect_nat
@ ^ [X2: nat] : ( X2 = A5 ) ) ) ) ) ).
% Set.insert_def
thf(fact_995_set__minus__filter__out,axiom,
! [Xs: list_n989787106983797996_a_nat,Y4: nat > sum_sum_a_nat] :
( ( minus_5517490076408937517_a_nat @ ( set_na645604395003041787_a_nat @ Xs ) @ ( insert5265011953798106934_a_nat @ Y4 @ bot_bo6441361344521902642_a_nat ) )
= ( set_na645604395003041787_a_nat
@ ( filter7153797121941418979_a_nat
@ ^ [X2: nat > sum_sum_a_nat] : ( X2 != Y4 )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_996_set__minus__filter__out,axiom,
! [Xs: list_nat,Y4: nat] :
( ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ Y4 @ bot_bot_set_nat ) )
= ( set_nat2
@ ( filter_nat
@ ^ [X2: nat] : ( X2 != Y4 )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_997_set__minus__filter__out,axiom,
! [Xs: list_o,Y4: $o] :
( ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ Y4 @ bot_bot_set_o ) )
= ( set_o2
@ ( filter_o
@ ^ [X2: $o] : ( X2 != Y4 )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_998_set__minus__filter__out,axiom,
! [Xs: list_a,Y4: a] :
( ( minus_minus_set_a @ ( set_a2 @ Xs ) @ ( insert_a @ Y4 @ bot_bot_set_a ) )
= ( set_a2
@ ( filter_a
@ ^ [X2: a] : ( X2 != Y4 )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_999_set__minus__filter__out,axiom,
! [Xs: list_l4703314356710769291_a_nat,Y4: list_Sum_sum_a_nat] :
( ( minus_7395159227704179404_a_nat @ ( set_li2392974972034027290_a_nat @ Xs ) @ ( insert2950094090816004437_a_nat @ Y4 @ bot_bo1033123847703346641_a_nat ) )
= ( set_li2392974972034027290_a_nat
@ ( filter5373755100216644354_a_nat
@ ^ [X2: list_Sum_sum_a_nat] : ( X2 != Y4 )
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_1000_diff__shunt__var,axiom,
! [X4: set_na3699693778330250182_a_nat,Y4: set_na3699693778330250182_a_nat] :
( ( ( minus_5517490076408937517_a_nat @ X4 @ Y4 )
= bot_bo6441361344521902642_a_nat )
= ( ord_le8108555184339247974_a_nat @ X4 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1001_diff__shunt__var,axiom,
! [X4: set_o,Y4: set_o] :
( ( ( minus_minus_set_o @ X4 @ Y4 )
= bot_bot_set_o )
= ( ord_less_eq_set_o @ X4 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1002_diff__shunt__var,axiom,
! [X4: set_li6526943997496501093_a_nat,Y4: set_li6526943997496501093_a_nat] :
( ( ( minus_7395159227704179404_a_nat @ X4 @ Y4 )
= bot_bo1033123847703346641_a_nat )
= ( ord_le1147066620699065093_a_nat @ X4 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1003_diff__shunt__var,axiom,
! [X4: set_a,Y4: set_a] :
( ( ( minus_minus_set_a @ X4 @ Y4 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ X4 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1004_diff__shunt__var,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( ( minus_minus_set_nat @ X4 @ Y4 )
= bot_bot_set_nat )
= ( ord_less_eq_set_nat @ X4 @ Y4 ) ) ).
% diff_shunt_var
thf(fact_1005_subset__Diff__insert,axiom,
! [A2: set_o,B: set_o,X4: $o,C: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B @ ( insert_o @ X4 @ C ) ) )
= ( ( ord_less_eq_set_o @ A2 @ ( minus_minus_set_o @ B @ C ) )
& ~ ( member_o @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1006_subset__Diff__insert,axiom,
! [A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,X4: nat > sum_sum_a_nat,C: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ ( minus_5517490076408937517_a_nat @ B @ ( insert5265011953798106934_a_nat @ X4 @ C ) ) )
= ( ( ord_le8108555184339247974_a_nat @ A2 @ ( minus_5517490076408937517_a_nat @ B @ C ) )
& ~ ( member8690443509505302927_a_nat @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1007_subset__Diff__insert,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,X4: list_Sum_sum_a_nat,C: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( minus_7395159227704179404_a_nat @ B @ ( insert2950094090816004437_a_nat @ X4 @ C ) ) )
= ( ( ord_le1147066620699065093_a_nat @ A2 @ ( minus_7395159227704179404_a_nat @ B @ C ) )
& ~ ( member408289922725080238_a_nat @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1008_subset__Diff__insert,axiom,
! [A2: set_a,B: set_a,X4: a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ ( insert_a @ X4 @ C ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B @ C ) )
& ~ ( member_a @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1009_subset__Diff__insert,axiom,
! [A2: set_nat,B: set_nat,X4: nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ ( insert_nat @ X4 @ C ) ) )
= ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B @ C ) )
& ~ ( member_nat @ X4 @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_1010_subset__Compl__self__eq,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A2 @ ( uminus2192744996606729052_a_nat @ A2 ) )
= ( A2 = bot_bo1033123847703346641_a_nat ) ) ).
% subset_Compl_self_eq
thf(fact_1011_subset__Compl__self__eq,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ ( uminus_uminus_set_o @ A2 ) )
= ( A2 = bot_bot_set_o ) ) ).
% subset_Compl_self_eq
thf(fact_1012_subset__Compl__self__eq,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A2 @ ( uminus3259373730256538813_a_nat @ A2 ) )
= ( A2 = bot_bo6441361344521902642_a_nat ) ) ).
% subset_Compl_self_eq
thf(fact_1013_subset__Compl__self__eq,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( uminus_uminus_set_a @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% subset_Compl_self_eq
thf(fact_1014_subset__Compl__self__eq,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
= ( A2 = bot_bot_set_nat ) ) ).
% subset_Compl_self_eq
thf(fact_1015_order__antisym__conv,axiom,
! [Y4: set_a,X4: set_a] :
( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( ( ord_less_eq_set_a @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_1016_order__antisym__conv,axiom,
! [Y4: set_nat,X4: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_set_nat @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_1017_order__antisym__conv,axiom,
! [Y4: nat,X4: nat] :
( ( ord_less_eq_nat @ Y4 @ X4 )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
= ( X4 = Y4 ) ) ) ).
% order_antisym_conv
thf(fact_1018_linorder__le__cases,axiom,
! [X4: nat,Y4: nat] :
( ~ ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ).
% linorder_le_cases
thf(fact_1019_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1020_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > set_nat,C2: set_nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1021_ord__le__eq__subst,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1022_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_a,C2: set_a] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1023_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1024_ord__le__eq__subst,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1025_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1026_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1027_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1028_ord__eq__le__subst,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1029_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_a > set_nat,B2: set_a,C2: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1030_ord__eq__le__subst,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1031_ord__eq__le__subst,axiom,
! [A: set_a,F: set_nat > set_a,B2: set_nat,C2: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1032_ord__eq__le__subst,axiom,
! [A: set_nat,F: set_nat > set_nat,B2: set_nat,C2: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1033_ord__eq__le__subst,axiom,
! [A: nat,F: set_nat > nat,B2: set_nat,C2: set_nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1034_ord__eq__le__subst,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1035_ord__eq__le__subst,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1036_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1037_linorder__linear,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
| ( ord_less_eq_nat @ Y4 @ X4 ) ) ).
% linorder_linear
thf(fact_1038_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_1039_order__eq__refl,axiom,
! [X4: set_a,Y4: set_a] :
( ( X4 = Y4 )
=> ( ord_less_eq_set_a @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_1040_order__eq__refl,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( X4 = Y4 )
=> ( ord_less_eq_set_nat @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_1041_order__eq__refl,axiom,
! [X4: nat,Y4: nat] :
( ( X4 = Y4 )
=> ( ord_less_eq_nat @ X4 @ Y4 ) ) ).
% order_eq_refl
thf(fact_1042_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1043_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > set_nat,C2: set_nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1044_order__subst2,axiom,
! [A: set_a,B2: set_a,F: set_a > nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1045_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_a,C2: set_a] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1046_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1047_order__subst2,axiom,
! [A: set_nat,B2: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1048_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1049_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1050_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_1051_order__subst1,axiom,
! [A: set_a,F: set_a > set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1052_order__subst1,axiom,
! [A: set_a,F: set_nat > set_a,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1053_order__subst1,axiom,
! [A: set_a,F: nat > set_a,B2: nat,C2: nat] :
( ( ord_less_eq_set_a @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_a @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1054_order__subst1,axiom,
! [A: set_nat,F: set_a > set_nat,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1055_order__subst1,axiom,
! [A: set_nat,F: set_nat > set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1056_order__subst1,axiom,
! [A: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_set_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1057_order__subst1,axiom,
! [A: nat,F: set_a > nat,B2: set_a,C2: set_a] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ! [X3: set_a,Y5: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1058_order__subst1,axiom,
! [A: nat,F: set_nat > nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X3: set_nat,Y5: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1059_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_1060_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z3: set_a] : ( Y3 = Z3 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1061_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ B5 )
& ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1062_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1063_antisym,axiom,
! [A: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_1064_antisym,axiom,
! [A: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_1065_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_1066_dual__order_Otrans,axiom,
! [B2: set_a,A: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ C2 @ B2 )
=> ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_1067_dual__order_Otrans,axiom,
! [B2: set_nat,A: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_1068_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_1069_dual__order_Oantisym,axiom,
! [B2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B2 @ A )
=> ( ( ord_less_eq_set_a @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_1070_dual__order_Oantisym,axiom,
! [B2: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A )
=> ( ( ord_less_eq_set_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_1071_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_1072_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z3: set_a] : ( Y3 = Z3 ) )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ B5 @ A5 )
& ( ord_less_eq_set_a @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1073_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [A5: set_nat,B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ A5 )
& ( ord_less_eq_set_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1074_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1075_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A6: nat,B7: nat] :
( ( ord_less_eq_nat @ A6 @ B7 )
=> ( P @ A6 @ B7 ) )
=> ( ! [A6: nat,B7: nat] :
( ( P @ B7 @ A6 )
=> ( P @ A6 @ B7 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_1076_order__trans,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ Z2 )
=> ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_1077_order__trans,axiom,
! [X4: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ Z2 )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_1078_order__trans,axiom,
! [X4: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% order_trans
thf(fact_1079_order_Otrans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_1080_order_Otrans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_1081_order_Otrans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_1082_order__antisym,axiom,
! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ( ord_less_eq_set_a @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_1083_order__antisym,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_set_nat @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_1084_order__antisym,axiom,
! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X4 )
=> ( X4 = Y4 ) ) ) ).
% order_antisym
thf(fact_1085_ord__le__eq__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_1086_ord__le__eq__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_1087_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_1088_ord__eq__le__trans,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( A = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C2 )
=> ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1089_ord__eq__le__trans,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( A = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1090_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_1091_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_a,Z3: set_a] : ( Y3 = Z3 ) )
= ( ^ [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
& ( ord_less_eq_set_a @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1092_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: set_nat,Z3: set_nat] : ( Y3 = Z3 ) )
= ( ^ [X2: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y )
& ( ord_less_eq_set_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1093_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 ) )
= ( ^ [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
& ( ord_less_eq_nat @ Y @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1094_le__cases3,axiom,
! [X4: nat,Y4: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X4 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1095_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_1096_verit__comp__simplify1_I2_J,axiom,
! [A: set_a] : ( ord_less_eq_set_a @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1097_verit__comp__simplify1_I2_J,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1098_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1099_x__proj__singleton,axiom,
( ( insert2950094090816004437_a_nat @ x2 @ bot_bo1033123847703346641_a_nat )
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( proj_v3643391342904276326_a_nat @ ( insert5265011953798106934_a_nat @ sigma3 @ bot_bo6441361344521902642_a_nat ) @ ns_phi2 ) ) ) ).
% x_proj_singleton
thf(fact_1100_sup__Un__eq,axiom,
! [R2: set_na3699693778330250182_a_nat,S: set_na3699693778330250182_a_nat] :
( ( sup_su3226716170639437251_nat_o
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ R2 )
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ S ) )
= ( ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ ( sup_su3329769938372955546_a_nat @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1101_sup__Un__eq,axiom,
! [R2: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( sup_su1334248866174809316_nat_o
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ R2 )
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ S ) )
= ( ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ ( sup_su4083067149120280889_a_nat @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1102_sup__Un__eq,axiom,
! [R2: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R2 )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( sup_sup_set_a @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1103_sup__Un__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( sup_sup_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R2 )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ^ [X2: nat] : ( member_nat @ X2 @ ( sup_sup_set_nat @ R2 @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_1104_res__def,axiom,
( res
= ( map_va3267824910732782952_a_nat
@ ^ [Xs2: list_Sum_sum_a_nat,X6: set_li6526943997496501093_a_nat] :
( case_o5757545902052400075_a_nat @ ( ext_tuple_set_a @ ad @ ns_phi2 @ ns_phi @ X6 )
@ ^ [Y6: set_li6526943997496501093_a_nat] : ( eval_conj_set_a @ ad @ ns_phi2 @ X6 @ ns_psi2 @ ( ad_agr_close_set_a @ aD_Delta_psi @ ( minus_7395159227704179404_a_nat @ ( ext_tuple_set_a @ aD_psi @ ns @ ns_phi @ ( insert2950094090816004437_a_nat @ Xs2 @ bot_bo1033123847703346641_a_nat ) ) @ Y6 ) ) )
@ ( lookup8968130772531298012_a_nat @ idx_psi @ Xs2 ) )
@ idx_phi ) ) ).
% res_def
thf(fact_1105_res__eq,axiom,
( res
= ( map_va3267824910732782952_a_nat
@ ^ [Xs2: list_Sum_sum_a_nat,X6: set_li6526943997496501093_a_nat] :
( case_o5757545902052400075_a_nat @ ( ext_tuple_set_a @ ad @ ns_phi2 @ ns_phi @ X6 )
@ ^ [Y6: set_li6526943997496501093_a_nat] : ( idx_join_a @ ad @ ns @ ns_phi2 @ X6 @ ns_psi2 @ ( ad_agr_close_set_a @ aD_Delta_psi @ ( minus_7395159227704179404_a_nat @ ( ext_tuple_set_a @ aD_psi @ ns @ ns_phi @ ( insert2950094090816004437_a_nat @ Xs2 @ bot_bo1033123847703346641_a_nat ) ) @ Y6 ) ) )
@ ( lookup8968130772531298012_a_nat @ idx_psi @ Xs2 ) )
@ idx_phi ) ) ).
% res_eq
thf(fact_1106_idx_092_060phi_062__def,axiom,
( idx_phi
= ( cluste8110478419151344300_a_nat
@ ( comp_l8558361960953920476_a_nat @ some_l1231941606268492394_a_nat
@ ^ [Xs2: list_Sum_sum_a_nat] : ( fo_nmlz_a @ aD_psi @ ( proj_tuple_a @ ns @ ( zip_na2013496608136855606_a_nat @ ns_phi2 @ Xs2 ) ) ) )
@ ( ad_agr_close_set_a @ aD_Delta_phi @ x_phi2 ) ) ) ).
% idx\<phi>_def
thf(fact_1107_pred__subset__eq,axiom,
! [R2: set_na3699693778330250182_a_nat,S: set_na3699693778330250182_a_nat] :
( ( ord_le6982925868732858103_nat_o
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ R2 )
@ ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ S ) )
= ( ord_le8108555184339247974_a_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1108_pred__subset__eq,axiom,
! [R2: set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( ord_le8737610411969296920_nat_o
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ R2 )
@ ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ S ) )
= ( ord_le1147066620699065093_a_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1109_pred__subset__eq,axiom,
! [R2: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R2 )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ord_less_eq_set_a @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1110_pred__subset__eq,axiom,
! [R2: set_nat,S: set_nat] :
( ( ord_less_eq_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ R2 )
@ ^ [X2: nat] : ( member_nat @ X2 @ S ) )
= ( ord_less_eq_set_nat @ R2 @ S ) ) ).
% pred_subset_eq
thf(fact_1111_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
=> ( member8690443509505302927_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le8108555184339247974_a_nat @ ( image_6222892899998961285_a_nat @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1112_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6721470456781115300_a_nat @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1113_image__Collect__subsetI,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
=> ( member8690443509505302927_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le8108555184339247974_a_nat @ ( image_701559317304863014_a_nat @ F @ ( collec7555443234367654128_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1114_image__Collect__subsetI,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( collec7555443234367654128_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1115_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > a,B: set_a] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_7126633039417159965_nat_a @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1116_image__Collect__subsetI,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > a,B: set_a] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
=> ( member_a @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_a @ ( image_4685209397977471678_nat_a @ F @ ( collec7555443234367654128_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1117_image__Collect__subsetI,axiom,
! [P: ( nat > sum_sum_a_nat ) > $o,F: ( nat > sum_sum_a_nat ) > nat,B: set_nat] :
( ! [X3: nat > sum_sum_a_nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_5786201776793816049at_nat @ F @ ( collec5629555741568564177_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1118_image__Collect__subsetI,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > nat,B: set_nat] :
( ! [X3: list_Sum_sum_a_nat] :
( ( P @ X3 )
=> ( member_nat @ ( F @ X3 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_2535339886381165584at_nat @ F @ ( collec7555443234367654128_a_nat @ P ) ) @ B ) ) ).
% image_Collect_subsetI
thf(fact_1119_comp__apply,axiom,
( comp_l8558361960953920476_a_nat
= ( ^ [F2: list_Sum_sum_a_nat > option2159618704653257803_a_nat,G2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X2: list_Sum_sum_a_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_1120_map__comp__map,axiom,
! [F: list_Sum_sum_a_nat > option2159618704653257803_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( comp_l6861429754633178466_a_nat @ ( map_li5411300359766877642_a_nat @ F ) @ ( map_li6507455427659069316_a_nat @ G ) )
= ( map_li5411300359766877642_a_nat @ ( comp_l8558361960953920476_a_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_1121_map__comp__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( comp_l2590854360456707st_nat @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_na823391071729141993_a_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_1122_map__comp__map,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat] :
( ( comp_l4895937012117054562st_nat @ ( map_na823391071729141993_a_nat @ F ) @ ( map_nat_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_1123_List_Omap_Ocomp,axiom,
! [F: list_Sum_sum_a_nat > option2159618704653257803_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( comp_l6861429754633178466_a_nat @ ( map_li5411300359766877642_a_nat @ F ) @ ( map_li6507455427659069316_a_nat @ G ) )
= ( map_li5411300359766877642_a_nat @ ( comp_l8558361960953920476_a_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_1124_List_Omap_Ocomp,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( comp_l2590854360456707st_nat @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_na823391071729141993_a_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_1125_List_Omap_Ocomp,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat] :
( ( comp_l4895937012117054562st_nat @ ( map_na823391071729141993_a_nat @ F ) @ ( map_nat_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_1126_map__map,axiom,
! [F: list_Sum_sum_a_nat > option2159618704653257803_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( map_li5411300359766877642_a_nat @ F @ ( map_li6507455427659069316_a_nat @ G @ Xs ) )
= ( map_li5411300359766877642_a_nat @ ( comp_l8558361960953920476_a_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_1127_map__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,Xs: list_nat] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_na823391071729141993_a_nat @ G @ Xs ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_1128_map__map,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,Xs: list_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( map_nat_nat @ G @ Xs ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_1129_List_Omap_Ocompositionality,axiom,
! [F: list_Sum_sum_a_nat > option2159618704653257803_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,List: list_l4703314356710769291_a_nat] :
( ( map_li5411300359766877642_a_nat @ F @ ( map_li6507455427659069316_a_nat @ G @ List ) )
= ( map_li5411300359766877642_a_nat @ ( comp_l8558361960953920476_a_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_1130_List_Omap_Ocompositionality,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,List: list_nat] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_na823391071729141993_a_nat @ G @ List ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_1131_List_Omap_Ocompositionality,axiom,
! [F: nat > sum_sum_a_nat,G: nat > nat,List: list_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( map_nat_nat @ G @ List ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_1132_list_Omap__comp,axiom,
! [G: list_Sum_sum_a_nat > option2159618704653257803_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: list_l4703314356710769291_a_nat] :
( ( map_li5411300359766877642_a_nat @ G @ ( map_li6507455427659069316_a_nat @ F @ V ) )
= ( map_li5411300359766877642_a_nat @ ( comp_l8558361960953920476_a_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_1133_list_Omap__comp,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: nat > sum_sum_a_nat,V: list_nat] :
( ( map_Su2790769393171190532_a_nat @ G @ ( map_na823391071729141993_a_nat @ F @ V ) )
= ( map_na823391071729141993_a_nat @ ( comp_S2395038772458240499at_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_1134_list_Omap__comp,axiom,
! [G: nat > sum_sum_a_nat,F: nat > nat,V: list_nat] :
( ( map_na823391071729141993_a_nat @ G @ ( map_nat_nat @ F @ V ) )
= ( map_na823391071729141993_a_nat @ ( comp_n1522318729830440540at_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_1135_idx_092_060psi_062__def,axiom,
( idx_psi
= ( cluste8110478419151344300_a_nat
@ ( comp_l8558361960953920476_a_nat @ some_l1231941606268492394_a_nat
@ ^ [Ys2: list_Sum_sum_a_nat] : ( fo_nmlz_a @ aD_psi @ ( proj_tuple_a @ ns @ ( zip_na2013496608136855606_a_nat @ ns_psi2 @ Ys2 ) ) ) )
@ x_psi ) ) ).
% idx\<psi>_def
thf(fact_1136_bot__empty__eq,axiom,
( bot_bo9042073657639083596_nat_o
= ( ^ [X2: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X2 @ bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_1137_bot__empty__eq,axiom,
( bot_bo3382309974966529835_nat_o
= ( ^ [X2: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X2 @ bot_bo6441361344521902642_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_1138_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_1139_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_1140_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X2: $o] : ( member_o @ X2 @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_1141_option_Ocase__distrib,axiom,
! [H: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,F1: set_li6526943997496501093_a_nat,F22: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,Option: option4559388950256476203_a_nat] :
( ( H @ ( case_o5757545902052400075_a_nat @ F1 @ F22 @ Option ) )
= ( case_o5757545902052400075_a_nat @ ( H @ F1 )
@ ^ [X2: set_li6526943997496501093_a_nat] : ( H @ ( F22 @ X2 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_1142_comp__eq__dest__lhs,axiom,
! [A: list_Sum_sum_a_nat > option2159618704653257803_a_nat,B2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,C2: list_Sum_sum_a_nat > option2159618704653257803_a_nat,V: list_Sum_sum_a_nat] :
( ( ( comp_l8558361960953920476_a_nat @ A @ B2 )
= C2 )
=> ( ( A @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_1143_comp__eq__elim,axiom,
! [A: list_Sum_sum_a_nat > option2159618704653257803_a_nat,B2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,C2: list_Sum_sum_a_nat > option2159618704653257803_a_nat,D2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ( comp_l8558361960953920476_a_nat @ A @ B2 )
= ( comp_l8558361960953920476_a_nat @ C2 @ D2 ) )
=> ! [V2: list_Sum_sum_a_nat] :
( ( A @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_1144_comp__eq__dest,axiom,
! [A: list_Sum_sum_a_nat > option2159618704653257803_a_nat,B2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,C2: list_Sum_sum_a_nat > option2159618704653257803_a_nat,D2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,V: list_Sum_sum_a_nat] :
( ( ( comp_l8558361960953920476_a_nat @ A @ B2 )
= ( comp_l8558361960953920476_a_nat @ C2 @ D2 ) )
=> ( ( A @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_1145_comp__assoc,axiom,
! [F: option2159618704653257803_a_nat > option2159618704653257803_a_nat,G: list_Sum_sum_a_nat > option2159618704653257803_a_nat,H: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( comp_l8558361960953920476_a_nat @ ( comp_o2481550590722179362_a_nat @ F @ G ) @ H )
= ( comp_o2481550590722179362_a_nat @ F @ ( comp_l8558361960953920476_a_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_1146_comp__assoc,axiom,
! [F: list_Sum_sum_a_nat > option2159618704653257803_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,H: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( comp_l8558361960953920476_a_nat @ ( comp_l8558361960953920476_a_nat @ F @ G ) @ H )
= ( comp_l8558361960953920476_a_nat @ F @ ( comp_l3689981812903912354_a_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_1147_comp__def,axiom,
( comp_l8558361960953920476_a_nat
= ( ^ [F2: list_Sum_sum_a_nat > option2159618704653257803_a_nat,G2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,X2: list_Sum_sum_a_nat] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_1148_bot__set__def,axiom,
( bot_bo1033123847703346641_a_nat
= ( collec7555443234367654128_a_nat @ bot_bo9042073657639083596_nat_o ) ) ).
% bot_set_def
thf(fact_1149_bot__set__def,axiom,
( bot_bo6441361344521902642_a_nat
= ( collec5629555741568564177_a_nat @ bot_bo3382309974966529835_nat_o ) ) ).
% bot_set_def
thf(fact_1150_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_1151_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_1152_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_1153_image__comp,axiom,
! [F: list_Sum_sum_a_nat > option2159618704653257803_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,R3: set_li6526943997496501093_a_nat] :
( ( image_6628469031518839819_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ R3 ) )
= ( image_6628469031518839819_a_nat @ ( comp_l8558361960953920476_a_nat @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_1154_image__comp,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,R3: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ R3 ) )
= ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ F @ G ) @ R3 ) ) ).
% image_comp
thf(fact_1155_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,H: list_Sum_sum_a_nat > option2159618704653257803_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A2 )
= ( image_5081948215111134021_a_nat @ G @ B ) )
=> ( ( image_6628469031518839819_a_nat @ ( comp_l8558361960953920476_a_nat @ H @ F ) @ A2 )
= ( image_6628469031518839819_a_nat @ ( comp_l8558361960953920476_a_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1156_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,H: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A2 )
= ( image_5081948215111134021_a_nat @ G @ B ) )
=> ( ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ H @ F ) @ A2 )
= ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_1157_option_Osimps_I5_J,axiom,
! [F1: set_li6526943997496501093_a_nat,F22: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,X22: set_li6526943997496501093_a_nat] :
( ( case_o5757545902052400075_a_nat @ F1 @ F22 @ ( some_s2697817922134046282_a_nat @ X22 ) )
= ( F22 @ X22 ) ) ).
% option.simps(5)
thf(fact_1158_map__values__cong,axiom,
! [T: mappin5678568201568998833_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,F3: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat,Y5: set_li6526943997496501093_a_nat] :
( ( ( lookup8968130772531298012_a_nat @ T @ X3 )
= ( some_s2697817922134046282_a_nat @ Y5 ) )
=> ( ( F @ X3 @ Y5 )
= ( F3 @ X3 @ Y5 ) ) )
=> ( ( map_va3267824910732782952_a_nat @ F @ T )
= ( map_va3267824910732782952_a_nat @ F3 @ T ) ) ) ).
% map_values_cong
thf(fact_1159_prop__restrict,axiom,
! [X4: nat > sum_sum_a_nat,Z4: set_na3699693778330250182_a_nat,X: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ X4 @ Z4 )
=> ( ( ord_le8108555184339247974_a_nat @ Z4
@ ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ X )
& ( P @ X2 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_1160_prop__restrict,axiom,
! [X4: list_Sum_sum_a_nat,Z4: set_li6526943997496501093_a_nat,X: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X4 @ Z4 )
=> ( ( ord_le1147066620699065093_a_nat @ Z4
@ ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ X )
& ( P @ X2 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_1161_prop__restrict,axiom,
! [X4: a,Z4: set_a,X: set_a,P: a > $o] :
( ( member_a @ X4 @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X )
& ( P @ X2 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_1162_prop__restrict,axiom,
! [X4: nat,Z4: set_nat,X: set_nat,P: nat > $o] :
( ( member_nat @ X4 @ Z4 )
=> ( ( ord_less_eq_set_nat @ Z4
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X )
& ( P @ X2 ) ) ) )
=> ( P @ X4 ) ) ) ).
% prop_restrict
thf(fact_1163_Collect__restrict,axiom,
! [X: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ord_le8108555184339247974_a_nat
@ ( collec5629555741568564177_a_nat
@ ^ [X2: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X2 @ X )
& ( P @ X2 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_1164_Collect__restrict,axiom,
! [X: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ord_le1147066620699065093_a_nat
@ ( collec7555443234367654128_a_nat
@ ^ [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ X )
& ( P @ X2 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_1165_Collect__restrict,axiom,
! [X: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ X )
& ( P @ X2 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_1166_Collect__restrict,axiom,
! [X: set_nat,P: nat > $o] :
( ord_less_eq_set_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ X )
& ( P @ X2 ) ) )
@ X ) ).
% Collect_restrict
thf(fact_1167_subset__emptyI,axiom,
! [A2: set_li6526943997496501093_a_nat] :
( ! [X3: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X3 @ A2 )
=> ( ord_le1147066620699065093_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ) ).
% subset_emptyI
thf(fact_1168_subset__emptyI,axiom,
! [A2: set_na3699693778330250182_a_nat] :
( ! [X3: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ X3 @ A2 )
=> ( ord_le8108555184339247974_a_nat @ A2 @ bot_bo6441361344521902642_a_nat ) ) ).
% subset_emptyI
thf(fact_1169_subset__emptyI,axiom,
! [A2: set_o] :
( ! [X3: $o] :
~ ( member_o @ X3 @ A2 )
=> ( ord_less_eq_set_o @ A2 @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_1170_subset__emptyI,axiom,
! [A2: set_a] :
( ! [X3: a] :
~ ( member_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ A2 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_1171_subset__emptyI,axiom,
! [A2: set_nat] :
( ! [X3: nat] :
~ ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1172_insert__subsetI,axiom,
! [X4: $o,A2: set_o,X: set_o] :
( ( member_o @ X4 @ A2 )
=> ( ( ord_less_eq_set_o @ X @ A2 )
=> ( ord_less_eq_set_o @ ( insert_o @ X4 @ X ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1173_insert__subsetI,axiom,
! [X4: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,X: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A2 )
=> ( ( ord_le8108555184339247974_a_nat @ X @ A2 )
=> ( ord_le8108555184339247974_a_nat @ ( insert5265011953798106934_a_nat @ X4 @ X ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1174_insert__subsetI,axiom,
! [X4: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,X: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A2 )
=> ( ( ord_le1147066620699065093_a_nat @ X @ A2 )
=> ( ord_le1147066620699065093_a_nat @ ( insert2950094090816004437_a_nat @ X4 @ X ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1175_insert__subsetI,axiom,
! [X4: a,A2: set_a,X: set_a] :
( ( member_a @ X4 @ A2 )
=> ( ( ord_less_eq_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( insert_a @ X4 @ X ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1176_insert__subsetI,axiom,
! [X4: nat,A2: set_nat,X: set_nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ ( insert_nat @ X4 @ X ) @ A2 ) ) ) ).
% insert_subsetI
thf(fact_1177_aux6_I1_J,axiom,
( ( inf_inf_set_nat @ ( set_nat2 @ ns_psi2 ) @ ( set_nat2 @ ns_psi ) )
= bot_bot_set_nat ) ).
% aux6(1)
thf(fact_1178_aux_I4_J,axiom,
( ( inf_inf_set_nat @ ( set_nat2 @ ns_phi2 ) @ ( set_nat2 @ ns_phi ) )
= bot_bot_set_nat ) ).
% aux(4)
thf(fact_1179_aux4_I1_J,axiom,
( ( inf_inf_set_nat @ ( set_nat2 @ ns ) @ ( set_nat2 @ ns_phi ) )
= bot_bot_set_nat ) ).
% aux4(1)
thf(fact_1180_aux3_I1_J,axiom,
( ( inf_inf_set_nat @ ( set_nat2 @ ns_phi ) @ ( set_nat2 @ ns ) )
= bot_bot_set_nat ) ).
% aux3(1)
thf(fact_1181_AD__disj_I1_J,axiom,
( ( inf_inf_set_a @ aD_phi @ aD_Delta_phi )
= bot_bot_set_a ) ).
% AD_disj(1)
thf(fact_1182_inf_Oidem,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% inf.idem
thf(fact_1183_inf_Oidem,axiom,
! [A: set_a] :
( ( inf_inf_set_a @ A @ A )
= A ) ).
% inf.idem
thf(fact_1184_inf__idem,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_1185_inf__idem,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ X4 )
= X4 ) ).
% inf_idem
thf(fact_1186_inf_Oleft__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B2 ) )
= ( inf_inf_set_nat @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_1187_inf_Oleft__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ A @ ( inf_inf_set_a @ A @ B2 ) )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_1188_inf__left__idem,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( inf_inf_set_nat @ X4 @ Y4 ) )
= ( inf_inf_set_nat @ X4 @ Y4 ) ) ).
% inf_left_idem
thf(fact_1189_inf__left__idem,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y4 ) )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_left_idem
thf(fact_1190_inf_Oright__idem,axiom,
! [A: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B2 ) @ B2 )
= ( inf_inf_set_nat @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_1191_inf_Oright__idem,axiom,
! [A: set_a,B2: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ A @ B2 ) @ B2 )
= ( inf_inf_set_a @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_1192_inf__right__idem,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X4 @ Y4 ) @ Y4 )
= ( inf_inf_set_nat @ X4 @ Y4 ) ) ).
% inf_right_idem
thf(fact_1193_inf__right__idem,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ ( inf_inf_set_a @ X4 @ Y4 ) @ Y4 )
= ( inf_inf_set_a @ X4 @ Y4 ) ) ).
% inf_right_idem
thf(fact_1194_IntI,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ A2 )
=> ( ( member8690443509505302927_a_nat @ C2 @ B )
=> ( member8690443509505302927_a_nat @ C2 @ ( inf_in8399021836546144180_a_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_1195_IntI,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ A2 )
=> ( ( member408289922725080238_a_nat @ C2 @ B )
=> ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_1196_IntI,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A2 )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_1197_IntI,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ A2 )
=> ( ( member_a @ C2 @ B )
=> ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_1198_Int__iff,axiom,
! [C2: nat > sum_sum_a_nat,A2: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( inf_in8399021836546144180_a_nat @ A2 @ B ) )
= ( ( member8690443509505302927_a_nat @ C2 @ A2 )
& ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_1199_Int__iff,axiom,
! [C2: list_Sum_sum_a_nat,A2: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A2 @ B ) )
= ( ( member408289922725080238_a_nat @ C2 @ A2 )
& ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_1200_Int__iff,axiom,
! [C2: nat,A2: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( member_nat @ C2 @ A2 )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_1201_Int__iff,axiom,
! [C2: a,A2: set_a,B: set_a] :
( ( member_a @ C2 @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( member_a @ C2 @ A2 )
& ( member_a @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_1202_inf_Obounded__iff,axiom,
! [A: set_a,B2: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( inf_inf_set_a @ B2 @ C2 ) )
= ( ( ord_less_eq_set_a @ A @ B2 )
& ( ord_less_eq_set_a @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_1203_inf_Obounded__iff,axiom,
! [A: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_set_nat @ A @ B2 )
& ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_1204_inf_Obounded__iff,axiom,
! [A: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A @ B2 )
& ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_1205_le__inf__iff,axiom,
! [X4: set_a,Y4: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X4 @ ( inf_inf_set_a @ Y4 @ Z2 ) )
= ( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1206_le__inf__iff,axiom,
! [X4: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ ( inf_inf_set_nat @ Y4 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X4 @ Y4 )
& ( ord_less_eq_set_nat @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1207_le__inf__iff,axiom,
! [X4: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X4 @ ( inf_inf_nat @ Y4 @ Z2 ) )
= ( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ X4 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_1208_inf__bot__left,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ bot_bo1033123847703346641_a_nat @ X4 )
= bot_bo1033123847703346641_a_nat ) ).
% inf_bot_left
thf(fact_1209_inf__bot__left,axiom,
! [X4: set_na3699693778330250182_a_nat] :
( ( inf_in8399021836546144180_a_nat @ bot_bo6441361344521902642_a_nat @ X4 )
= bot_bo6441361344521902642_a_nat ) ).
% inf_bot_left
thf(fact_1210_inf__bot__left,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X4 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_1211_inf__bot__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% inf_bot_left
thf(fact_1212_inf__bot__left,axiom,
! [X4: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X4 )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_1213_inf__bot__right,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ bot_bo1033123847703346641_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% inf_bot_right
thf(fact_1214_inf__bot__right,axiom,
! [X4: set_na3699693778330250182_a_nat] :
( ( inf_in8399021836546144180_a_nat @ X4 @ bot_bo6441361344521902642_a_nat )
= bot_bo6441361344521902642_a_nat ) ).
% inf_bot_right
thf(fact_1215_inf__bot__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_1216_inf__bot__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% inf_bot_right
thf(fact_1217_inf__bot__right,axiom,
! [X4: set_o] :
( ( inf_inf_set_o @ X4 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_1218_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ bot_bo1033123847703346641_a_nat @ X4 )
= bot_bo1033123847703346641_a_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1219_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_na3699693778330250182_a_nat] :
( ( inf_in8399021836546144180_a_nat @ bot_bo6441361344521902642_a_nat @ X4 )
= bot_bo6441361344521902642_a_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1220_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X4 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_1221_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ bot_bot_set_a @ X4 )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_left
thf(fact_1222_boolean__algebra_Oconj__zero__left,axiom,
! [X4: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X4 )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_1223_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_li6526943997496501093_a_nat] :
( ( inf_in3249246906714053971_a_nat @ X4 @ bot_bo1033123847703346641_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1224_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_na3699693778330250182_a_nat] :
( ( inf_in8399021836546144180_a_nat @ X4 @ bot_bo6441361344521902642_a_nat )
= bot_bo6441361344521902642_a_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1225_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_nat] :
( ( inf_inf_set_nat @ X4 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_1226_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_a] :
( ( inf_inf_set_a @ X4 @ bot_bot_set_a )
= bot_bot_set_a ) ).
% boolean_algebra.conj_zero_right
thf(fact_1227_boolean__algebra_Oconj__zero__right,axiom,
! [X4: set_o] :
( ( inf_inf_set_o @ X4 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_1228_sup__inf__absorb,axiom,
! [X4: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X4 @ ( inf_inf_set_a @ X4 @ Y4 ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_1229_sup__inf__absorb,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( inf_inf_set_nat @ X4 @ Y4 ) )
= X4 ) ).
% sup_inf_absorb
thf(fact_1230_inf__sup__absorb,axiom,
! [X4: set_a,Y4: set_a] :
( ( inf_inf_set_a @ X4 @ ( sup_sup_set_a @ X4 @ Y4 ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_1231_inf__sup__absorb,axiom,
! [X4: set_nat,Y4: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( sup_sup_set_nat @ X4 @ Y4 ) )
= X4 ) ).
% inf_sup_absorb
thf(fact_1232_Int__subset__iff,axiom,
! [C: set_a,A2: set_a,B: set_a] :
( ( ord_less_eq_set_a @ C @ ( inf_inf_set_a @ A2 @ B ) )
= ( ( ord_less_eq_set_a @ C @ A2 )
& ( ord_less_eq_set_a @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_1233_Int__subset__iff,axiom,
! [C: set_nat,A2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A2 @ B ) )
= ( ( ord_less_eq_set_nat @ C @ A2 )
& ( ord_less_eq_set_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_1234_AD__disj_I2_J,axiom,
( ( inf_inf_set_a @ aD_psi @ aD_Delta_psi )
= bot_bot_set_a ) ).
% AD_disj(2)
thf(fact_1235_ts__def_I2_J,axiom,
( t_psi
= ( produc1626021218680649873_a_nat @ aD_psi @ ( produc4622690550020908860_a_nat @ n_psi @ x_psi ) ) ) ).
% ts_def(2)
thf(fact_1236_ts__def_I1_J,axiom,
( t_phi
= ( produc1626021218680649873_a_nat @ aD_phi @ ( produc4622690550020908860_a_nat @ n_phi @ x_phi2 ) ) ) ).
% ts_def(1)
thf(fact_1237_diff__diff__cancel,axiom,
! [I: nat,N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( minus_minus_nat @ N3 @ ( minus_minus_nat @ N3 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1238_eq__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N3 @ K ) )
= ( M2 = N3 ) ) ) ) ).
% eq_diff_iff
thf(fact_1239_le__diff__iff,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ) ) ).
% le_diff_iff
thf(fact_1240_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N3 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N3 @ K ) )
= ( minus_minus_nat @ M2 @ N3 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1241_diff__le__mono,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N3 @ L ) ) ) ).
% diff_le_mono
thf(fact_1242_diff__le__self,axiom,
! [M2: nat,N3: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N3 ) @ M2 ) ).
% diff_le_self
thf(fact_1243_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1244_diff__le__mono2,axiom,
! [M2: nat,N3: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N3 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1245_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1246_nat__le__linear,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
| ( ord_less_eq_nat @ N3 @ M2 ) ) ).
% nat_le_linear
thf(fact_1247_le__antisym,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( ord_less_eq_nat @ N3 @ M2 )
=> ( M2 = N3 ) ) ) ).
% le_antisym
thf(fact_1248_eq__imp__le,axiom,
! [M2: nat,N3: nat] :
( ( M2 = N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% eq_imp_le
thf(fact_1249_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1250_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_1251_X_092_060phi_062__def,axiom,
( x_phi2
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_phi ) @ ( proj_v3643391342904276326_a_nat @ s_phi @ ( fv_fo_fmla_list_a_b @ phi ) ) ) ) ).
% X\<phi>_def
thf(fact_1252_ns_092_060phi_062__def,axiom,
( ns_phi2
= ( fv_fo_fmla_list_a_b @ phi ) ) ).
% ns\<phi>_def
thf(fact_1253_ad__agr___092_060phi_062_I1_J,axiom,
! [Sigma2: nat > sum_sum_a_nat,Tau3: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ ( fv_fo_fmla_list_a_b @ phi ) ) @ ( set_nat2 @ ( fv_fo_fmla_list_a_b @ phi ) ) @ aD_phi @ Sigma2 @ Tau3 )
=> ( ( member8690443509505302927_a_nat @ Sigma2 @ s_phi )
= ( member8690443509505302927_a_nat @ Tau3 @ s_phi ) ) ) ).
% ad_agr_\<phi>(1)
thf(fact_1254_ad__agr___092_060phi_062_I2_J,axiom,
! [Sigma2: nat > sum_sum_a_nat,Tau3: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ ( fv_fo_fmla_list_a_b @ phi ) ) @ ( set_nat2 @ ( fv_fo_fmla_list_a_b @ phi ) ) @ ad @ Sigma2 @ Tau3 )
=> ( ( member8690443509505302927_a_nat @ Sigma2 @ s_phi )
= ( member8690443509505302927_a_nat @ Tau3 @ s_phi ) ) ) ).
% ad_agr_\<phi>(2)
thf(fact_1255_AD__X__def,axiom,
( ( eval_ajoin_a @ ( fv_fo_fmla_list_a_b @ phi ) @ t_phi @ ( fv_fo_fmla_list_a_b @ psi ) @ t_psi )
= ( produc1626021218680649873_a_nat @ ad @ ( produc4622690550020908860_a_nat @ n @ x ) ) ) ).
% AD_X_def
thf(fact_1256_X_092_060psi_062__def,axiom,
( x_psi
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ aD_psi ) @ ( proj_v3643391342904276326_a_nat @ s_psi @ ( fv_fo_fmla_list_a_b @ psi ) ) ) ) ).
% X\<psi>_def
thf(fact_1257_ns_092_060psi_062__def,axiom,
( ns_psi2
= ( fv_fo_fmla_list_a_b @ psi ) ) ).
% ns\<psi>_def
thf(fact_1258_ad__agr___092_060psi_062,axiom,
! [Sigma2: nat > sum_sum_a_nat,Tau3: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ ( fv_fo_fmla_list_a_b @ psi ) ) @ ( set_nat2 @ ( fv_fo_fmla_list_a_b @ psi ) ) @ aD_psi @ Sigma2 @ Tau3 )
=> ( ( member8690443509505302927_a_nat @ Sigma2 @ s_psi )
= ( member8690443509505302927_a_nat @ Tau3 @ s_psi ) ) ) ).
% ad_agr_\<psi>
thf(fact_1259__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062AD_An_AX_O_Aeval__ajoin_A_Ifv__fo__fmla__list_A_092_060phi_062_J_At_092_060phi_062_A_Ifv__fo__fmla__list_A_092_060psi_062_J_At_092_060psi_062_A_061_A_IAD_M_An_M_AX_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [AD2: set_a,N4: nat,X7: set_li6526943997496501093_a_nat] :
( ( eval_ajoin_a @ ( fv_fo_fmla_list_a_b @ phi ) @ t_phi @ ( fv_fo_fmla_list_a_b @ psi ) @ t_psi )
!= ( produc1626021218680649873_a_nat @ AD2 @ ( produc4622690550020908860_a_nat @ N4 @ X7 ) ) ) ).
% \<open>\<And>thesis. (\<And>AD n X. eval_ajoin (fv_fo_fmla_list \<phi>) t\<phi> (fv_fo_fmla_list \<psi>) t\<psi> = (AD, n, X) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1260_ns_092_060psi_062_H__def,axiom,
( ns_psi
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( fv_fo_fmla_a_b @ psi ) )
@ ( fv_fo_fmla_list_a_b @ phi ) ) ) ).
% ns\<psi>'_def
thf(fact_1261_ns_092_060phi_062_H__def,axiom,
( ns_phi
= ( filter_nat
@ ^ [N2: nat] :
~ ( member_nat @ N2 @ ( fv_fo_fmla_a_b @ phi ) )
@ ( fv_fo_fmla_list_a_b @ psi ) ) ) ).
% ns\<phi>'_def
thf(fact_1262_ns,axiom,
( ns
= ( filter_nat
@ ^ [N2: nat] : ( member_nat @ N2 @ ( fv_fo_fmla_a_b @ phi ) )
@ ( fv_fo_fmla_list_a_b @ psi ) ) ) ).
% ns
thf(fact_1263__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062AD_092_060phi_062_An_092_060phi_062_AX_092_060phi_062_AAD_092_060psi_062_An_092_060psi_062_AX_092_060psi_062_O_A_092_060lbrakk_062t_092_060phi_062_A_061_A_IAD_092_060phi_062_M_An_092_060phi_062_M_AX_092_060phi_062_J_059_At_092_060psi_062_A_061_A_IAD_092_060psi_062_M_An_092_060psi_062_M_AX_092_060psi_062_J_059_AAD_092_060phi_062_A_061_Aact__edom_A_092_060phi_062_AI_059_AAD_092_060psi_062_A_061_Aact__edom_A_092_060psi_062_AI_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [AD_phi: set_a] :
( ? [N_phi: nat,X_phi: set_li6526943997496501093_a_nat] :
( t_phi
= ( produc1626021218680649873_a_nat @ AD_phi @ ( produc4622690550020908860_a_nat @ N_phi @ X_phi ) ) )
=> ! [AD_psi: set_a] :
( ? [N_psi: nat,X_psi: set_li6526943997496501093_a_nat] :
( t_psi
= ( produc1626021218680649873_a_nat @ AD_psi @ ( produc4622690550020908860_a_nat @ N_psi @ X_psi ) ) )
=> ( ( AD_phi
= ( act_edom_a_b @ phi @ i ) )
=> ( AD_psi
!= ( act_edom_a_b @ psi @ i ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>AD\<phi> n\<phi> X\<phi> AD\<psi> n\<psi> X\<psi>. \<lbrakk>t\<phi> = (AD\<phi>, n\<phi>, X\<phi>); t\<psi> = (AD\<psi>, n\<psi>, X\<psi>); AD\<phi> = act_edom \<phi> I; AD\<psi> = act_edom \<psi> I\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1264_ts__def_I4_J,axiom,
( aD_psi
= ( act_edom_a_b @ psi @ i ) ) ).
% ts_def(4)
thf(fact_1265_ts__def_I3_J,axiom,
( aD_phi
= ( act_edom_a_b @ phi @ i ) ) ).
% ts_def(3)
thf(fact_1266_AD__sub_I2_J,axiom,
ord_less_eq_set_a @ ( act_edom_a_b @ psi @ i ) @ aD_psi ).
% AD_sub(2)
thf(fact_1267_AD__sub_I1_J,axiom,
ord_less_eq_set_a @ ( act_edom_a_b @ phi @ i ) @ aD_phi ).
% AD_sub(1)
thf(fact_1268_local_Owf_I1_J,axiom,
fo_wf_a_b @ phi @ i @ t_phi ).
% local.wf(1)
thf(fact_1269_local_Owf_I2_J,axiom,
fo_wf_a_b @ psi @ i @ t_psi ).
% local.wf(2)
thf(fact_1270_S_092_060phi_062__def,axiom,
( s_phi
= ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ phi @ i @ Sigma @ top_to795618464972521135_a_nat ) ) ) ).
% S\<phi>_def
thf(fact_1271_S_092_060psi_062__def,axiom,
( s_psi
= ( collec5629555741568564177_a_nat
@ ^ [Sigma: nat > sum_sum_a_nat] : ( esat_a_b @ psi @ i @ Sigma @ top_to795618464972521135_a_nat ) ) ) ).
% S\<psi>_def
thf(fact_1272_len__ns_092_060psi_062,axiom,
( ( plus_plus_nat @ ( size_size_list_nat @ ns ) @ ( size_size_list_nat @ ns_phi ) )
= ( size_size_list_nat @ ns_psi2 ) ) ).
% len_ns\<psi>
thf(fact_1273_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N3 ) )
= ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% nat_add_left_cancel_le
thf(fact_1274_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1275_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1276_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1277_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
% Conjectures (5)
thf(conj_0,hypothesis,
ad_agr_list_a_nat @ ad @ ( map_na823391071729141993_a_nat @ sigma @ ns_psi2 ) @ ( map_na823391071729141993_a_nat @ tau @ ns_psi2 ) ).
thf(conj_1,hypothesis,
~ ( member8690443509505302927_a_nat @ sigma @ s_psi ) ).
thf(conj_2,hypothesis,
ad_agr_list_a_nat @ aD_psi @ ( map_na823391071729141993_a_nat @ sigma @ ns ) @ ( map_na823391071729141993_a_nat @ sigma2 @ ns ) ).
thf(conj_3,hypothesis,
member8690443509505302927_a_nat @ tau @ s_psi ).
thf(conj_4,conjecture,
$false ).
%------------------------------------------------------------------------------