TPTP Problem File: SLH0389^1.p
View Solutions
- 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_01975_073670__15776940_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1664 ( 620 unt; 377 typ; 0 def)
% Number of atoms : 3434 (1494 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10774 ( 212 ~; 12 |; 147 &;9010 @)
% ( 0 <=>;1393 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 47 ( 46 usr)
% Number of type conns : 1795 (1795 >; 0 *; 0 +; 0 <<)
% Number of symbols : 334 ( 331 usr; 25 con; 0-5 aty)
% Number of variables : 3749 ( 226 ^;3437 !; 86 ?;3749 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:06:52.940
%------------------------------------------------------------------------------
% Could-be-implicit typings (46)
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_P8364476329498976096_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_P397628741534574662_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_P8619360983253792231_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_P1195027771636113901_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
list_P2991316961754610805at_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J_J,type,
list_P604317786108825364at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
set_li5408732054060274282_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
set_se1776965611924809360_a_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc7017002724195966439_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
set_se2794976184135595813_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
list_P6164600145584960654at_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__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_Pr1021281732701644634_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__List__Olist_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
list_s602657755630655083_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_P2388504626358630548_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
set_na8448764090737828173_nat_o: $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__Set__Oset_I_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
set_li4526012430949197550_nat_o: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
produc7258583773236448510at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
produc5454855657571582244_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
list_o6767316136206324043_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_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_op253850770698000421_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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
list_P6011104703257516679at_nat: $tType ).
thf(ty_n_t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
option_Sum_sum_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
list_P6285523579766656935_o_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_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Nat__Onat_M_Eo_J_J,type,
set_nat_o: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
set_set_o: $tType ).
thf(ty_n_t__Option__Ooption_It__Nat__Onat_J,type,
option_nat: $tType ).
thf(ty_n_t__Set__Oset_I_062_I_Eo_M_Eo_J_J,type,
set_o_o: $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__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 ).
% Explicit typings (331)
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_Oad__equiv__list_001tf__a_001t__Nat__Onat,type,
ad_equiv_list_a_nat: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Oext__tuple_001tf__a,type,
ext_tuple_a: set_a > list_nat > list_nat > list_Sum_sum_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__nmlzd_001tf__a,type,
fo_nmlzd_a: set_a > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Omerge_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
merge_Sum_sum_a_nat: list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_Ailamazyan_Onall__tuples__rec_001tf__a,type,
nall_tuples_rec_a: set_a > nat > 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_Ailamazyan_Osp__equiv_001tf__a_001t__Nat__Onat,type,
sp_equiv_a_nat: ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > set_nat > $o ).
thf(sy_c_Ailamazyan_Osp__equiv__list_001tf__a_001t__Nat__Onat,type,
sp_equiv_list_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_Complete__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,
comple8250499579560596074_nat_o: set_na8448764090737828173_nat_o > ( nat > sum_sum_a_nat ) > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_I_Eo_M_Eo_J,type,
complete_Sup_Sup_o_o: set_o_o > $o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
comple1334631452232999051_nat_o: set_li4526012430949197550_nat_o > list_Sum_sum_a_nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__Nat__Onat_M_Eo_J,type,
comple8317665133742190828_nat_o: set_nat_o > nat > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_Eo,type,
complete_Sup_Sup_o: set_o > $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
comple8643769897048643123_a_nat: set_se5822283258546872870_a_nat > set_na3699693778330250182_a_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_Eo_J,type,
comple90263536869209701_set_o: set_set_o > set_o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
comple1686717674086456018_a_nat: set_se4330304633200676677_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
comple3314747135086582061_a_nat: set_se1776965611924809360_a_nat > set_Pr1021281732701644634_a_nat ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
comple1778571073001984690_a_nat: set_se2794976184135595813_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Finite__Set_OFpow_001_Eo,type,
finite_Fpow_o: set_o > set_set_o ).
thf(sy_c_Finite__Set_OFpow_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
finite3225531020566593003_a_nat: set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
finite_Fpow_nat: set_nat > set_set_nat ).
thf(sy_c_Finite__Set_OFpow_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
finite9077140459278341579_a_nat: set_se4330304633200676677_a_nat > set_se2794976184135595813_a_nat ).
thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
finite9161971191270313901_a_nat: set_li6526943997496501093_a_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
finite152669285360869261_a_nat: set_se4330304633200676677_a_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
finite6080979521523705895_a_nat: set_Sum_sum_a_nat > nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
comp_l7223822213492037765st_nat: ( list_nat > list_nat ) > ( list_nat > list_nat ) > list_nat > list_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__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
comp_l7562833077401626166_a_nat: ( list_nat > list_Sum_sum_a_nat ) > ( list_P5056861408695629236_a_nat > list_nat ) > list_P5056861408695629236_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Option__Ooption_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_001t__List__Olist_It__Nat__Onat_J,type,
comp_l5575366868326257725st_nat: ( list_o6767316136206324043_a_nat > list_Sum_sum_a_nat ) > ( list_nat > list_o6767316136206324043_a_nat ) > list_nat > list_Sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__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_001t__List__Olist_It__Nat__Onat_J,type,
comp_l6946783421279048718st_nat: ( list_P5056861408695629236_a_nat > list_Sum_sum_a_nat ) > ( list_nat > list_P5056861408695629236_a_nat ) > list_nat > list_Sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__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_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
comp_l1389448023631161314_a_nat: ( list_P5056861408695629236_a_nat > list_Sum_sum_a_nat ) > ( list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat ) > list_P5056861408695629236_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_001_Eo_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_l7931591902798318543_a_nat: ( list_Sum_sum_a_nat > $o ) > ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat > $o ).
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__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
comp_l6851066656763219287_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > ( list_P5056861408695629236_a_nat > list_Sum_sum_a_nat ) > list_P5056861408695629236_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__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__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_l6309227251058912209_a_nat: ( list_Sum_sum_a_nat > nat ) > ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat > nat ).
thf(sy_c_Fun_Ocomp_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_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_l6800670225517141954_a_nat: ( list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ) > ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Nat__Onat_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
comp_n848383228264675746at_nat: ( nat > option_Sum_sum_a_nat ) > ( nat > nat ) > nat > option_Sum_sum_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__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_n6276394432912108976_a_nat: ( nat > sum_sum_a_nat ) > ( produc5454855657571582244_a_nat > nat ) > produc5454855657571582244_a_nat > sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat_001t__Nat__Onat,type,
comp_o4663963155070737957at_nat: ( option_nat > nat ) > ( nat > option_nat ) > nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Nat__Onat_J_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
comp_o571110771943018602_a_nat: ( option_nat > nat ) > ( sum_sum_a_nat > option_nat ) > sum_sum_a_nat > nat ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
comp_o5487264423576322163at_nat: ( option_Sum_sum_a_nat > option_Sum_sum_a_nat ) > ( nat > option_Sum_sum_a_nat ) > nat > option_Sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
comp_o2721059924169077421at_nat: ( option_Sum_sum_a_nat > sum_sum_a_nat ) > ( nat > option_Sum_sum_a_nat ) > nat > sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_o8686311273808465576_a_nat: ( option_Sum_sum_a_nat > sum_sum_a_nat ) > ( option_Sum_sum_a_nat > option_Sum_sum_a_nat ) > option_Sum_sum_a_nat > sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_o6804431283847804801_a_nat: ( option_Sum_sum_a_nat > sum_sum_a_nat ) > ( produc5454855657571582244_a_nat > option_Sum_sum_a_nat ) > produc5454855657571582244_a_nat > sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
comp_o6940958636155918818_a_nat: ( option_Sum_sum_a_nat > sum_sum_a_nat ) > ( sum_sum_a_nat > option_Sum_sum_a_nat ) > sum_sum_a_nat > sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
comp_P2433158451856970632at_nat: ( produc5454855657571582244_a_nat > sum_sum_a_nat ) > ( nat > produc5454855657571582244_a_nat ) > nat > sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_P9187809017694532828_a_nat: ( produc5454855657571582244_a_nat > sum_sum_a_nat ) > ( produc5454855657571582244_a_nat > produc5454855657571582244_a_nat ) > produc5454855657571582244_a_nat > sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001_Eo_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_s7989947936688375727_a_nat: ( set_li6526943997496501093_a_nat > $o ) > ( list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ) > list_Sum_sum_a_nat > $o ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_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_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_s2318311659520683906_a_nat: ( set_li6526943997496501093_a_nat > list_Sum_sum_a_nat ) > ( list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_s8210616430925637617_a_nat: ( set_li6526943997496501093_a_nat > nat ) > ( list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ) > list_Sum_sum_a_nat > nat ).
thf(sy_c_Fun_Ocomp_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_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_s3377174507952090018_a_nat: ( set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ) > ( list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ) > list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
comp_s8964582002068861047et_nat: ( set_nat > set_nat ) > ( set_nat > set_nat ) > set_nat > set_nat ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_s5818590256416363566_a_nat: ( set_nat > set_nat ) > ( set_Sum_sum_a_nat > set_nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_s3847575663057944277_a_nat: ( set_op253850770698000421_a_nat > set_nat ) > ( set_Sum_sum_a_nat > set_op253850770698000421_a_nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Fun_Ocomp_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_s1691789666922741659_a_nat: ( set_Sum_sum_a_nat > set_nat ) > ( set_Sum_sum_a_nat > set_Sum_sum_a_nat ) > set_Sum_sum_a_nat > set_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_Fun_Ocomp_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_S2748962947160188642_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > ( option_Sum_sum_a_nat > sum_sum_a_nat ) > option_Sum_sum_a_nat > sum_sum_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__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
comp_S8059809979792035783_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > ( produc5454855657571582244_a_nat > sum_sum_a_nat ) > produc5454855657571582244_a_nat > sum_sum_a_nat ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_Eo,type,
inj_on1633075686209370787_nat_o: ( ( nat > sum_sum_a_nat ) > $o ) > set_na3699693778330250182_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
inj_on3141676652335697285at_nat: ( ( nat > sum_sum_a_nat ) > nat ) > set_na3699693778330250182_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001_Eo,type,
inj_on_o_o: ( $o > $o ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
inj_on8234360183494294622_a_nat: ( $o > list_Sum_sum_a_nat ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001_Eo_001t__Nat__Onat,type,
inj_on_o_nat: ( $o > nat ) > set_o > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
inj_on1175180479830800114_a_nat: ( list_nat > list_Sum_sum_a_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__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,
inj_on543452247319097886_a_nat: ( list_P5056861408695629236_a_nat > list_Sum_sum_a_nat ) > set_li5408732054060274282_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_Eo,type,
inj_on2262799350278082756_nat_o: ( list_Sum_sum_a_nat > $o ) > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_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,
inj_on6609798167860701873_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
inj_on6397736275557941668at_nat: ( list_Sum_sum_a_nat > nat ) > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
inj_on7795941589172611729_a_nat: ( list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ) > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001_Eo,type,
inj_on_nat_o: ( nat > $o ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
inj_on901614105087147266_a_nat: ( nat > list_Sum_sum_a_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
inj_on8649581803800938946_a_nat: ( nat > option_Sum_sum_a_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
inj_on4348161877322679292_a_nat: ( nat > sum_sum_a_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
inj_on5841394699700873579_a_nat: ( option_Sum_sum_a_nat > sum_sum_a_nat ) > set_op253850770698000421_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
inj_on663950023194574504_a_nat: ( produc5454855657571582244_a_nat > sum_sum_a_nat ) > set_Pr1021281732701644634_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_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,
inj_on1831002258565622481_a_nat: ( set_li6526943997496501093_a_nat > list_Sum_sum_a_nat ) > set_se4330304633200676677_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_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,
inj_on561899399213738673_a_nat: ( set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ) > set_se4330304633200676677_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
inj_on4382358243947751057_a_nat: ( set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ) > set_se4330304633200676677_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
inj_on4604407203859583615et_nat: ( set_nat > set_nat ) > set_set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
inj_on8752143810983750942at_nat: ( sum_sum_a_nat > nat ) > set_Sum_sum_a_nat > $o ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
if_lis4685338526944683083_a_nat: $o > list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
if_set7709265119413304363_a_nat: $o > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
if_Sum_sum_a_nat: $o > sum_sum_a_nat > sum_sum_a_nat > sum_sum_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
inf_inf_o: $o > $o > $o ).
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_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
inf_in5367731912061063475_a_nat: set_se4330304633200676677_a_nat > set_se4330304633200676677_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
inf_in7084830621192376909_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
sup_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
sup_su3329769938372955546_a_nat: set_na3699693778330250182_a_nat > set_na3699693778330250182_a_nat > set_na3699693778330250182_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
sup_su4083067149120280889_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
sup_su4565425741012426374_a_nat: set_Pr1021281732701644634_a_nat > set_Pr1021281732701644634_a_nat > set_Pr1021281732701644634_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
sup_su499249268922660121_a_nat: set_se4330304633200676677_a_nat > set_se4330304633200676677_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
sup_su6804446743777130803_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_List_Oappend_001_Eo,type,
append_o: list_o > list_o > list_o ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
append338925788367110473_a_nat: list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
append2142653904031976739at_nat: list_P6164600145584960654at_nat > list_P6164600145584960654at_nat > list_P6164600145584960654at_nat ).
thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
append1996214168388709506_a_nat: list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat > list_P1195027771636113901_a_nat ).
thf(sy_c_List_Oappend_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
append_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Odistinct_001_Eo,type,
distinct_o: list_o > $o ).
thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
distinct_nat: list_nat > $o ).
thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
distin5548549501492119275_a_nat: list_P5056861408695629236_a_nat > $o ).
thf(sy_c_List_Odistinct_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
distin2701893636801681144_a_nat: list_Sum_sum_a_nat > $o ).
thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Oenumerate_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
enumer3164015132978342500_a_nat: nat > list_Sum_sum_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001_Eo_001_Eo,type,
linord6472470733373143810ey_o_o: ( $o > $o ) > $o > list_o > list_o ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Nat__Onat_001_Eo,type,
linord2970189811765474244_nat_o: ( nat > $o ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord1921536354676448932at_nat: ( nat > nat ) > nat > list_nat > list_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001_Eo,type,
linord8391790818113774811_nat_o: ( sum_sum_a_nat > $o ) > sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
linord458367490959630669at_nat: ( sum_sum_a_nat > nat ) > sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olinorder__class_Osort__key_001t__Nat__Onat_001t__Nat__Onat,type,
linord738340561235409698at_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
map_na722425985163074756_a_nat: ( ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat ) > list_n989787106983797996_a_nat > list_n989787106983797996_a_nat ).
thf(sy_c_List_Olist_Omap_001_Eo_001_Eo,type,
map_o_o: ( $o > $o ) > list_o > list_o ).
thf(sy_c_List_Olist_Omap_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,
map_li6507455427659069316_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > list_l4703314356710769291_a_nat > list_l4703314356710769291_a_nat ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
map_li4181295100893080420_a_nat: ( list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ) > list_l4703314356710769291_a_nat > list_s602657755630655083_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
map_na4778969939102885679_a_nat: ( nat > option_Sum_sum_a_nat ) > list_nat > list_o6767316136206324043_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
map_na4983825868277236666_a_nat: ( nat > produc5454855657571582244_a_nat ) > list_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_na823391071729141993_a_nat: ( nat > sum_sum_a_nat ) > list_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_op8672217432457595070_a_nat: ( option_Sum_sum_a_nat > sum_sum_a_nat ) > list_o6767316136206324043_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_Pr3938374229010428429at_nat: ( product_prod_nat_nat > nat ) > list_P6011104703257516679at_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
map_Pr2058079554187054650at_nat: ( produc5454855657571582244_a_nat > nat ) > list_P5056861408695629236_a_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
map_Pr4115244106412896910_a_nat: ( produc5454855657571582244_a_nat > produc5454855657571582244_a_nat ) > list_P5056861408695629236_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_Pr2515564165894599445_a_nat: ( produc5454855657571582244_a_nat > sum_sum_a_nat ) > list_P5056861408695629236_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_Pr8434518024483680788at_nat: ( produc7258583773236448510at_nat > nat ) > list_P6164600145584960654at_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_Pr6349035483858946235_a_nat: ( produc7258583773236448510at_nat > sum_sum_a_nat ) > list_P6164600145584960654at_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_Pr3527087583225239196_a_nat: ( produc7017002724195966439_a_nat > sum_sum_a_nat ) > list_P1195027771636113901_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_Su5227373005390213643at_nat: ( sum_sum_a_nat > nat ) > list_Sum_sum_a_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_Su2790769393171190532_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Oset_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_na645604395003041787_a_nat: list_n989787106983797996_a_nat > set_na3699693778330250182_a_nat ).
thf(sy_c_List_Olist_Oset_001_Eo,type,
set_o2: list_o > set_o ).
thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_li2392974972034027290_a_nat: list_l4703314356710769291_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Pr2196415342531872975_a_nat: list_P5056861408695629236_a_nat > set_Pr1021281732701644634_a_nat ).
thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
set_se268393680088689402_a_nat: list_s602657755630655083_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_List_Olist_Oset_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
set_Sum_sum_a_nat2: list_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_List_Osorted__wrt_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
sorted5637561102903432531_a_nat: ( ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o ) > list_n989787106983797996_a_nat > $o ).
thf(sy_c_List_Osorted__wrt_001_Eo,type,
sorted_wrt_o: ( $o > $o > $o ) > list_o > $o ).
thf(sy_c_List_Osorted__wrt_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
sorted1521004414301201522_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ) > list_l4703314356710769291_a_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(sy_c_List_Osorted__wrt_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sorted6245805940552704876_a_nat: ( sum_sum_a_nat > sum_sum_a_nat > $o ) > list_Sum_sum_a_nat > $o ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
union_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Ozip_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
zip_na6430312473843624639at_nat: list_n989787106983797996_a_nat > list_nat > list_P2991316961754610805at_nat ).
thf(sy_c_List_Ozip_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
zip_na5841697299243076944_a_nat: list_n989787106983797996_a_nat > list_Sum_sum_a_nat > list_P397628741534574662_a_nat ).
thf(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
thf(sy_c_List_Ozip_001_Eo_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
zip_o_Sum_sum_a_nat: list_o > list_Sum_sum_a_nat > list_P2388504626358630548_a_nat ).
thf(sy_c_List_Ozip_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
zip_li3415139562815969502at_nat: list_l4703314356710769291_a_nat > list_nat > list_P604317786108825364at_nat ).
thf(sy_c_List_Ozip_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
zip_li3824709458655879025_a_nat: list_l4703314356710769291_a_nat > list_Sum_sum_a_nat > list_P8619360983253792231_a_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
zip_na2013496608136855606_a_nat: list_nat > list_Sum_sum_a_nat > list_P5056861408695629236_a_nat ).
thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
zip_Pr8089753336904788450_a_nat: list_P5056861408695629236_a_nat > list_Sum_sum_a_nat > list_P8364476329498976096_a_nat ).
thf(sy_c_List_Ozip_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
zip_Su6417478541797927256at_nat: list_Sum_sum_a_nat > list_nat > list_P6164600145584960654at_nat ).
thf(sy_c_List_Ozip_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
zip_Su7355543910597222519_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_P1195027771636113901_a_nat ).
thf(sy_c_Map_Omap__of_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
map_of9065542035934512787at_nat: list_P2991316961754610805at_nat > ( nat > sum_sum_a_nat ) > option_nat ).
thf(sy_c_Map_Omap__of_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_of4712045957748317308_a_nat: list_P397628741534574662_a_nat > ( nat > sum_sum_a_nat ) > option_Sum_sum_a_nat ).
thf(sy_c_Map_Omap__of_001_Eo_001t__Nat__Onat,type,
map_of_o_nat: list_P6285523579766656935_o_nat > $o > option_nat ).
thf(sy_c_Map_Omap__of_001_Eo_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_of7579214193106729930_a_nat: list_P2388504626358630548_a_nat > $o > option_Sum_sum_a_nat ).
thf(sy_c_Map_Omap__of_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
map_of3602546406765676722at_nat: list_P604317786108825364at_nat > list_Sum_sum_a_nat > option_nat ).
thf(sy_c_Map_Omap__of_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_of928655414695205533_a_nat: list_P8619360983253792231_a_nat > list_Sum_sum_a_nat > option_Sum_sum_a_nat ).
thf(sy_c_Map_Omap__of_001t__Nat__Onat_001t__Nat__Onat,type,
map_of_nat_nat: list_P6011104703257516679at_nat > nat > option_nat ).
thf(sy_c_Map_Omap__of_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_of1932138695521331722_a_nat: list_P5056861408695629236_a_nat > nat > option_Sum_sum_a_nat ).
thf(sy_c_Map_Omap__of_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_of3400036372718715318_a_nat: list_P8364476329498976096_a_nat > produc5454855657571582244_a_nat > option_Sum_sum_a_nat ).
thf(sy_c_Map_Omap__of_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
map_of6336120629182403372at_nat: list_P6164600145584960654at_nat > sum_sum_a_nat > option_nat ).
thf(sy_c_Map_Omap__of_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_of7357225391242338723_a_nat: list_P1195027771636113901_a_nat > sum_sum_a_nat > option_Sum_sum_a_nat ).
thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Nat__Onat,type,
ran_nat_nat: ( nat > option_nat ) > set_nat ).
thf(sy_c_Map_Oran_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
ran_na775312650358800638_a_nat: ( nat > option_Sum_sum_a_nat ) > set_Sum_sum_a_nat ).
thf(sy_c_Map_Oran_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
ran_Su5179294584019872288at_nat: ( sum_sum_a_nat > option_nat ) > set_nat ).
thf(sy_c_Map_Oran_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
ran_Su6606007654835531183_a_nat: ( sum_sum_a_nat > option_Sum_sum_a_nat ) > set_Sum_sum_a_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
size_s2119148862884149080_a_nat: list_n989787106983797996_a_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
size_size_list_o: list_o > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
size_s5212483967078203639_a_nat: list_l4703314356710769291_a_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
size_s4076174644546656840_a_nat: list_P5056861408695629236_a_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
size_s5283204784079214577_a_nat: list_Sum_sum_a_nat > nat ).
thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
none_nat: option_nat ).
thf(sy_c_Option_Ooption_ONone_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
none_Sum_sum_a_nat: option_Sum_sum_a_nat ).
thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
the_nat: option_nat > nat ).
thf(sy_c_Option_Ooption_Othe_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
the_Sum_sum_a_nat: option_Sum_sum_a_nat > sum_sum_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
bot_bo3382309974966529835_nat_o: ( nat > sum_sum_a_nat ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
bot_bot_o_o: $o > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J,type,
bot_bo9042073657639083596_nat_o: list_Sum_sum_a_nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
bot_bot_nat_o: nat > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
bot_bo6441361344521902642_a_nat: set_na3699693778330250182_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
bot_bot_set_o: set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
bot_bo1033123847703346641_a_nat: set_li6526943997496501093_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_I_Eo_J_J,type,
bot_bot_set_set_o: set_set_o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
bot_bo3237059034911209905_a_nat: set_se4330304633200676677_a_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
bot_bot_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
bot_bo3438331934148233675_a_nat: set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
ord_less_eq_o: $o > $o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
ord_le8108555184339247974_a_nat: set_na3699693778330250182_a_nat > set_na3699693778330250182_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
ord_le1147066620699065093_a_nat: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
ord_le8420040822840260538_a_nat: set_Pr1021281732701644634_a_nat > set_Pr1021281732701644634_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
ord_le8138476598237931237_a_nat: set_se4330304633200676677_a_nat > set_se4330304633200676677_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
ord_le8587131541179289285_a_nat: set_se2794976184135595813_a_nat > set_se2794976184135595813_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le1325389633284124927_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
top_top_set_o: set_o ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
produc1510075366785312874_a_nat: produc5454855657571582244_a_nat > nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc5914057300446384524at_nat: produc7258583773236448510at_nat > sum_sum_a_nat ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
produc3940427395981429571_a_nat: produc7017002724195966439_a_nat > sum_sum_a_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
produc1280857981117566888_a_nat: produc5454855657571582244_a_nat > sum_sum_a_nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
produc5684839914778638538at_nat: produc7258583773236448510at_nat > nat ).
thf(sy_c_Product__Type_Oprod_Osnd_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
produc8028804885397038213_a_nat: produc7017002724195966439_a_nat > sum_sum_a_nat ).
thf(sy_c_Set_OCollect_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
collec5629555741568564177_a_nat: ( ( nat > sum_sum_a_nat ) > $o ) > set_na3699693778330250182_a_nat ).
thf(sy_c_Set_OCollect_001_Eo,type,
collect_o: ( $o > $o ) > set_o ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
collec7555443234367654128_a_nat: ( list_Sum_sum_a_nat > $o ) > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
collec7073057861543223018_a_nat: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001_062_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J_001t__Set__Oset_I_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
image_1911308356169199528_a_nat: ( ( ( nat > sum_sum_a_nat ) > $o ) > set_na3699693778330250182_a_nat ) > set_na8448764090737828173_nat_o > set_se5822283258546872870_a_nat ).
thf(sy_c_Set_Oimage_001_062_I_Eo_M_Eo_J_001t__Set__Oset_I_Eo_J,type,
image_o_o_set_o: ( ( $o > $o ) > set_o ) > set_o_o > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_M_Eo_J_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
image_2943553480580064358_a_nat: ( ( list_Sum_sum_a_nat > $o ) > set_li6526943997496501093_a_nat ) > set_li4526012430949197550_nat_o > set_se4330304633200676677_a_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_M_Eo_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_o_set_nat: ( ( nat > $o ) > set_nat ) > set_nat_o > set_set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_Eo,type,
image_2376713081370839351_nat_o: ( ( nat > sum_sum_a_nat ) > $o ) > set_na3699693778330250182_a_nat > set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_5786201776793816049at_nat: ( ( nat > sum_sum_a_nat ) > nat ) > set_na3699693778330250182_a_nat > set_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_I_Eo_J,type,
image_9210942892871078935_set_o: ( ( nat > sum_sum_a_nat ) > set_o ) > set_na3699693778330250182_a_nat > set_set_o ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
image_494503752901444484_a_nat: ( ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat ) > set_na3699693778330250182_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set_Oimage_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_5790795286017029031et_nat: ( ( nat > sum_sum_a_nat ) > set_nat ) > set_na3699693778330250182_a_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001_Eo,type,
image_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_733135354807394034_a_nat: ( $o > list_Sum_sum_a_nat ) > set_o > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Nat__Onat,type,
image_o_nat: ( $o > nat ) > set_o > set_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_I_Eo_J,type,
image_o_set_o: ( $o > set_o ) > set_o > set_set_o ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
image_2951602039182858962_a_nat: ( $o > set_li6526943997496501093_a_nat ) > set_o > set_se4330304633200676677_a_nat ).
thf(sy_c_Set_Oimage_001_Eo_001t__Set__Oset_It__Nat__Onat_J,type,
image_o_set_nat: ( $o > set_nat ) > set_o > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_1775855109352712557et_nat: ( list_nat > set_nat ) > set_list_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
image_3068022823300244885_a_nat: ( list_P5056861408695629236_a_nat > set_Pr1021281732701644634_a_nat ) > set_li5408732054060274282_a_nat > set_se1776965611924809360_a_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_Eo,type,
image_3984946558445957976_nat_o: ( list_Sum_sum_a_nat > $o ) > set_li6526943997496501093_a_nat > set_o ).
thf(sy_c_Set_Oimage_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,
image_5081948215111134021_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_2535339886381165584at_nat: ( list_Sum_sum_a_nat > nat ) > set_li6526943997496501093_a_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_I_Eo_J,type,
image_4041951406191573560_set_o: ( list_Sum_sum_a_nat > set_o ) > set_li6526943997496501093_a_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
image_7676043921908783909_a_nat: ( list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ) > set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_2109848253374568390et_nat: ( list_Sum_sum_a_nat > set_nat ) > set_li6526943997496501093_a_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
image_7324140262315149061_a_nat: ( list_Sum_sum_a_nat > set_se4330304633200676677_a_nat ) > set_li6526943997496501093_a_nat > set_se2794976184135595813_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_1051037728736664655_a_nat: ( nat > nat > sum_sum_a_nat ) > set_nat > set_na3699693778330250182_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001_Eo,type,
image_nat_o: ( nat > $o ) > set_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_6262589752765146990_a_nat: ( nat > list_Sum_sum_a_nat ) > set_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_2070734891404281902_a_nat: ( nat > option_Sum_sum_a_nat ) > set_nat > set_op253850770698000421_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_I_Eo_J,type,
image_nat_set_o: ( nat > set_o ) > set_nat > set_set_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
image_6073155208182920014_a_nat: ( nat > set_li6526943997496501093_a_nat ) > set_nat > set_se4330304633200676677_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7293268710728258664_a_nat: ( nat > sum_sum_a_nat ) > set_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_6550528888111247359_a_nat: ( option_Sum_sum_a_nat > sum_sum_a_nat ) > set_op253850770698000421_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
image_1677410150705809403at_nat: ( produc5454855657571582244_a_nat > nat ) > set_Pr1021281732701644634_a_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_5908522764169838868_a_nat: ( produc5454855657571582244_a_nat > sum_sum_a_nat ) > set_Pr1021281732701644634_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001_Eo,type,
image_6425633437027283832_nat_o: ( set_li6526943997496501093_a_nat > $o ) > set_se4330304633200676677_a_nat > 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__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
image_1711104591301794661_a_nat: ( set_li6526943997496501093_a_nat > list_Sum_sum_a_nat ) > set_se4330304633200676677_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Nat__Onat,type,
image_4755997627985507312at_nat: ( set_li6526943997496501093_a_nat > nat ) > set_se4330304633200676677_a_nat > set_nat ).
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__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
image_7557436503106877221_a_nat: ( set_li6526943997496501093_a_nat > set_se4330304633200676677_a_nat ) > set_se4330304633200676677_a_nat > set_se2794976184135595813_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__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001_Eo,type,
image_6095136190293192542_nat_o: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat > set_o ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2473878607534554506at_nat: ( sum_sum_a_nat > nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7142520692256960453_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oinsert_001_Eo,type,
insert_o: $o > set_o > set_o ).
thf(sy_c_Set_Ovimage_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
vimage1507083042557951675at_nat: ( ( nat > sum_sum_a_nat ) > nat ) > set_nat > set_na3699693778330250182_a_nat ).
thf(sy_c_Set_Ovimage_001_Eo_001_Eo,type,
vimage_o_o: ( $o > $o ) > set_o > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage2884537194618305448_a_nat: ( $o > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_o ).
thf(sy_c_Set_Ovimage_001_Eo_001t__Nat__Onat,type,
vimage_o_nat: ( $o > nat ) > set_nat > set_o ).
thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001_Eo,type,
vimage6136348398256869390_nat_o: ( list_Sum_sum_a_nat > $o ) > set_o > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_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,
vimage7538740559230589179_a_nat: ( list_Sum_sum_a_nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
vimage4660099214859102426at_nat: ( list_Sum_sum_a_nat > nat ) > set_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
vimage5553558261632676059_a_nat: ( list_Sum_sum_a_nat > set_li6526943997496501093_a_nat ) > set_se4330304633200676677_a_nat > set_li6526943997496501093_a_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_062_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage5995291031355576089_a_nat: ( nat > nat > sum_sum_a_nat ) > set_na3699693778330250182_a_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001_Eo,type,
vimage_nat_o: ( nat > $o ) > set_o > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage8387349081243083832_a_nat: ( nat > list_Sum_sum_a_nat ) > set_li6526943997496501093_a_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
vimage7014988194023193336_a_nat: ( nat > option_Sum_sum_a_nat ) > set_op253850770698000421_a_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage3040984495076556338_a_nat: ( nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Option__Ooption_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage1635291442211470005_a_nat: ( option_Sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_op253850770698000421_a_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage6545432446589551483_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001tf__a,type,
sum_Inr_nat_a: nat > sum_sum_a_nat ).
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__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
member3650934779329710139_a_nat: produc5454855657571582244_a_nat > set_Pr1021281732701644634_a_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__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_v_AD,type,
ad: set_a ).
thf(sy_v_R,type,
r: set_na3699693778330250182_a_nat ).
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_as____,type,
as: list_Sum_sum_a_nat ).
thf(sy_v_ass,type,
ass: set_li6526943997496501093_a_nat ).
thf(sy_v_fs____,type,
fs: list_Sum_sum_a_nat ).
thf(sy_v_fv__all,type,
fv_all: list_nat ).
thf(sy_v_fv__sub,type,
fv_sub: list_nat ).
thf(sy_v_fv__sub__comp,type,
fv_sub_comp: list_nat ).
thf(sy_v_xs,type,
xs: list_Sum_sum_a_nat ).
% Relevant facts (1277)
thf(fact_0_assms_I1_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ fv_sub )
& ( distinct_nat @ fv_sub ) ) ).
% assms(1)
thf(fact_1_assms_I2_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ fv_sub_comp )
& ( distinct_nat @ fv_sub_comp ) ) ).
% assms(2)
thf(fact_2_assms_I3_J,axiom,
( ( sorted_wrt_nat @ ord_less_eq_nat @ fv_all )
& ( distinct_nat @ fv_all ) ) ).
% assms(3)
thf(fact_3_assms_I5_J,axiom,
( ( sup_sup_set_nat @ ( set_nat2 @ fv_sub ) @ ( set_nat2 @ fv_sub_comp ) )
= ( set_nat2 @ fv_all ) ) ).
% assms(5)
thf(fact_4__092_060sigma_062_H_H__def,axiom,
( xs
= ( map_na823391071729141993_a_nat @ sigma @ fv_all ) ) ).
% \<sigma>''_def
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060sigma_062_H_H_O_Axs_A_061_Amap_A_092_060sigma_062_H_H_Afv__all_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Sigma: nat > sum_sum_a_nat] :
( xs
!= ( map_na823391071729141993_a_nat @ Sigma @ fv_all ) ) ).
% \<open>\<And>thesis. (\<And>\<sigma>''. xs = map \<sigma>'' fv_all \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_proj__tuple__map,axiom,
! [Ns: list_nat,Ms: list_nat,Sigma2: nat > sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
=> ( ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Ms @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) ) )
= ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) ) ) ) ) ).
% proj_tuple_map
thf(fact_7__092_060tau_062__def_I1_J,axiom,
( as
= ( map_na823391071729141993_a_nat @ tau @ fv_sub ) ) ).
% \<tau>_def(1)
thf(fact_8_map__fv__sub,axiom,
( ( map_na823391071729141993_a_nat @ sigma2 @ fv_sub )
= ( map_na823391071729141993_a_nat @ tau @ fv_sub ) ) ).
% map_fv_sub
thf(fact_9_vs__map__fv__all,axiom,
( xs
= ( fo_nmlz_a @ ad @ ( map_na823391071729141993_a_nat @ sigma2 @ fv_all ) ) ) ).
% vs_map_fv_all
thf(fact_10_map__eq__conv,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Xs ) )
= ( ! [X: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_11_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 ) )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( ( F @ X )
= ( G @ X ) ) ) ) ) ).
% map_eq_conv
thf(fact_12_vs__norm,axiom,
fo_nmlzd_a @ ad @ xs ).
% vs_norm
thf(fact_13__092_060open_062_092_060forall_062n_092_060in_062set_Afv__sub_O_A_092_060tau_062_An_A_061_A_092_060sigma_062_H_An_092_060close_062,axiom,
! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ fv_sub ) )
=> ( ( tau @ X2 )
= ( sigma2 @ X2 ) ) ) ).
% \<open>\<forall>n\<in>set fv_sub. \<tau> n = \<sigma>' n\<close>
thf(fact_14_fo__nmlz__map,axiom,
! [AD: set_a,Sigma2: nat > sum_sum_a_nat,Ns: list_nat] :
? [Tau: nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) )
= ( map_na823391071729141993_a_nat @ Tau @ Ns ) ) ).
% fo_nmlz_map
thf(fact_15_fo__nmlz__map,axiom,
! [AD: set_a,Sigma2: produc5454855657571582244_a_nat > sum_sum_a_nat,Ns: list_P5056861408695629236_a_nat] :
? [Tau: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( fo_nmlz_a @ AD @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ns ) )
= ( map_Pr2515564165894599445_a_nat @ Tau @ Ns ) ) ).
% fo_nmlz_map
thf(fact_16_list_Omap__cong,axiom,
! [X3: list_P5056861408695629236_a_nat,Ya: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z @ ( set_Pr2196415342531872975_a_nat @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ X3 )
= ( map_Pr2515564165894599445_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_17_list_Omap__cong,axiom,
! [X3: list_nat,Ya: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( X3 = Ya )
=> ( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X3 )
= ( map_na823391071729141993_a_nat @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_18_list_Omap__cong0,axiom,
! [X3: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [Z: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z @ ( set_Pr2196415342531872975_a_nat @ X3 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ X3 )
= ( map_Pr2515564165894599445_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_19_list_Omap__cong0,axiom,
! [X3: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [Z: nat] :
( ( member_nat @ Z @ ( set_nat2 @ X3 ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ X3 )
= ( map_na823391071729141993_a_nat @ G @ X3 ) ) ) ).
% list.map_cong0
thf(fact_20_list_Oinj__map__strong,axiom,
! [X3: list_P5056861408695629236_a_nat,Xa: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,Fa: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [Z: produc5454855657571582244_a_nat,Za: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ Z @ ( set_Pr2196415342531872975_a_nat @ X3 ) )
=> ( ( member3650934779329710139_a_nat @ Za @ ( set_Pr2196415342531872975_a_nat @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_Pr2515564165894599445_a_nat @ F @ X3 )
= ( map_Pr2515564165894599445_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_21_list_Oinj__map__strong,axiom,
! [X3: 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 @ X3 ) )
=> ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map_na823391071729141993_a_nat @ F @ X3 )
= ( map_na823391071729141993_a_nat @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_22_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_23_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_24_list_Omap__ident__strong,axiom,
! [T: list_o,F: $o > $o] :
( ! [Z: $o] :
( ( member_o @ Z @ ( set_o2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_o_o @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_25_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_26_list_Omap__ident__strong,axiom,
! [T: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( set_Sum_sum_a_nat2 @ T ) )
=> ( ( F @ Z )
= Z ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ T )
= T ) ) ).
% list.map_ident_strong
thf(fact_27_map__ext,axiom,
! [Xs: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ! [X4: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X4 @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_28_map__ext,axiom,
! [Xs: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Xs ) ) ) ).
% map_ext
thf(fact_29_as__fs__def_I1_J,axiom,
member408289922725080238_a_nat @ as @ ass ).
% as_fs_def(1)
thf(fact_30__092_060sigma_062__def_I2_J,axiom,
( as
= ( fo_nmlz_a @ ad @ ( map_na823391071729141993_a_nat @ sigma3 @ fv_sub ) ) ) ).
% \<sigma>_def(2)
thf(fact_31_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 )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( member408289922725080238_a_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_32_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 )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( member8690443509505302927_a_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_33_subset__code_I1_J,axiom,
! [Xs: list_o,B: set_o] :
( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B )
= ( ! [X: $o] :
( ( member_o @ X @ ( set_o2 @ Xs ) )
=> ( member_o @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_34_subset__code_I1_J,axiom,
! [Xs: list_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ B )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( member_Sum_sum_a_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_35_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X: nat] :
( ( member_nat @ X @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X @ B ) ) ) ) ).
% subset_code(1)
thf(fact_36_sorted__distinct__set__unique,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
=> ( ( distinct_nat @ Xs )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ys )
=> ( ( distinct_nat @ Ys )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_37_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( distinct_nat @ Xs ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_38_fo__nmlz__idem,axiom,
! [AD: set_a,Ys: list_Sum_sum_a_nat] :
( ( fo_nmlzd_a @ AD @ Ys )
=> ( ( fo_nmlz_a @ AD @ Ys )
= Ys ) ) ).
% fo_nmlz_idem
thf(fact_39_fo__nmlz__sound,axiom,
! [AD: set_a,Xs: list_Sum_sum_a_nat] : ( fo_nmlzd_a @ AD @ ( fo_nmlz_a @ AD @ Xs ) ) ).
% fo_nmlz_sound
thf(fact_40_fo__nmlzd__code,axiom,
( fo_nmlzd_a
= ( ^ [AD2: set_a,Xs2: list_Sum_sum_a_nat] :
( ( fo_nmlz_a @ AD2 @ Xs2 )
= Xs2 ) ) ) ).
% fo_nmlzd_code
thf(fact_41_sorted__wrt__mono__rel,axiom,
! [Xs: list_l4703314356710769291_a_nat,P: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o] :
( ! [X4: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( member408289922725080238_a_nat @ Y @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( P @ X4 @ Y )
=> ( Q @ X4 @ Y ) ) ) )
=> ( ( sorted1521004414301201522_a_nat @ P @ Xs )
=> ( sorted1521004414301201522_a_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_42_sorted__wrt__mono__rel,axiom,
! [Xs: list_n989787106983797996_a_nat,P: ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > ( nat > sum_sum_a_nat ) > $o] :
( ! [X4: nat > sum_sum_a_nat,Y: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( member8690443509505302927_a_nat @ Y @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( P @ X4 @ Y )
=> ( Q @ X4 @ Y ) ) ) )
=> ( ( sorted5637561102903432531_a_nat @ P @ Xs )
=> ( sorted5637561102903432531_a_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_43_sorted__wrt__mono__rel,axiom,
! [Xs: list_o,P: $o > $o > $o,Q: $o > $o > $o] :
( ! [X4: $o,Y: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs ) )
=> ( ( member_o @ Y @ ( set_o2 @ Xs ) )
=> ( ( P @ X4 @ Y )
=> ( Q @ X4 @ Y ) ) ) )
=> ( ( sorted_wrt_o @ P @ Xs )
=> ( sorted_wrt_o @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_44_sorted__wrt__mono__rel,axiom,
! [Xs: list_Sum_sum_a_nat,P: sum_sum_a_nat > sum_sum_a_nat > $o,Q: sum_sum_a_nat > sum_sum_a_nat > $o] :
( ! [X4: sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( member_Sum_sum_a_nat @ Y @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( P @ X4 @ Y )
=> ( Q @ X4 @ Y ) ) ) )
=> ( ( sorted6245805940552704876_a_nat @ P @ Xs )
=> ( sorted6245805940552704876_a_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_45_sorted__wrt__mono__rel,axiom,
! [Xs: list_nat,P: nat > nat > $o,Q: nat > nat > $o] :
( ! [X4: nat,Y: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( member_nat @ Y @ ( set_nat2 @ Xs ) )
=> ( ( P @ X4 @ Y )
=> ( Q @ X4 @ Y ) ) ) )
=> ( ( sorted_wrt_nat @ P @ Xs )
=> ( sorted_wrt_nat @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_46_distinct__zipI2,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_nat] :
( ( distin2701893636801681144_a_nat @ Ys )
=> ( distin5548549501492119275_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) ) ) ).
% distinct_zipI2
thf(fact_47_distinct__zipI1,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( distinct_nat @ Xs )
=> ( distin5548549501492119275_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) ) ) ).
% distinct_zipI1
thf(fact_48_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 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y2: nat] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_49_ex__map__conv,axiom,
! [Ys: list_Sum_sum_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ? [Xs2: list_P5056861408695629236_a_nat] :
( Ys
= ( map_Pr2515564165894599445_a_nat @ F @ Xs2 ) ) )
= ( ! [X: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ? [Y2: produc5454855657571582244_a_nat] :
( X
= ( F @ Y2 ) ) ) ) ) ).
% ex_map_conv
thf(fact_50_map__cong,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X4: produc5454855657571582244_a_nat] :
( ( member3650934779329710139_a_nat @ X4 @ ( set_Pr2196415342531872975_a_nat @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_51_map__cong,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( Xs = Ys )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_52_map__idI,axiom,
! [Xs: list_l4703314356710769291_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( set_li2392974972034027290_a_nat @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_li6507455427659069316_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_53_map__idI,axiom,
! [Xs: list_n989787106983797996_a_nat,F: ( nat > sum_sum_a_nat ) > nat > sum_sum_a_nat] :
( ! [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ ( set_na645604395003041787_a_nat @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_na722425985163074756_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_54_map__idI,axiom,
! [Xs: list_o,F: $o > $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ ( set_o2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_o_o @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_55_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_56_map__idI,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X4 )
= X4 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_57_fs__map,axiom,
( fs
= ( map_na823391071729141993_a_nat @ sigma2 @ fv_sub_comp ) ) ).
% fs_map
thf(fact_58__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060sigma_062_O_A_092_060lbrakk_062_092_060sigma_062_A_092_060in_062_AR_059_Aas_A_061_Afo__nmlz_AAD_A_Imap_A_092_060sigma_062_Afv__sub_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Sigma3: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ Sigma3 @ r )
=> ( as
!= ( fo_nmlz_a @ ad @ ( map_na823391071729141993_a_nat @ Sigma3 @ fv_sub ) ) ) ) ).
% \<open>\<And>thesis. (\<And>\<sigma>. \<lbrakk>\<sigma> \<in> R; as = fo_nmlz AD (map \<sigma> fv_sub)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_59__092_060tau_062__def_I2_J,axiom,
ad_agr_list_a_nat @ ad @ ( map_na823391071729141993_a_nat @ sigma3 @ fv_sub ) @ ( map_na823391071729141993_a_nat @ tau @ fv_sub ) ).
% \<tau>_def(2)
thf(fact_60_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( ( ord_less_eq_set_nat @ A @ C )
& ( ord_less_eq_set_nat @ B @ C ) ) ) ).
% Un_subset_iff
thf(fact_61_le__sup__iff,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ Z2 )
= ( ( ord_less_eq_nat @ X3 @ Z2 )
& ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_62_le__sup__iff,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z2 )
= ( ( ord_less_eq_set_nat @ X3 @ Z2 )
& ( ord_less_eq_set_nat @ Y3 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_63_sup_Obounded__iff,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_64_sup_Obounded__iff,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_65__092_060sigma_062_H__S,axiom,
member8690443509505302927_a_nat @ sigma2 @ r ).
% \<sigma>'_S
thf(fact_66__092_060tau_062__R,axiom,
member8690443509505302927_a_nat @ tau @ r ).
% \<tau>_R
thf(fact_67_set__union,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( union_Sum_sum_a_nat @ Xs @ Ys ) )
= ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_68_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_69_as__fs__def_I3_J,axiom,
( xs
= ( fo_nmlz_a @ ad @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ fv_sub @ as ) @ ( zip_na2013496608136855606_a_nat @ fv_sub_comp @ fs ) ) ) ) ) ).
% as_fs_def(3)
thf(fact_70__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062_092_060tau_062_O_A_092_060lbrakk_062as_A_061_Amap_A_092_060tau_062_Afv__sub_059_Aad__agr__list_AAD_A_Imap_A_092_060sigma_062_Afv__sub_J_A_Imap_A_092_060tau_062_Afv__sub_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Tau: nat > sum_sum_a_nat] :
( ( as
= ( map_na823391071729141993_a_nat @ Tau @ fv_sub ) )
=> ~ ( ad_agr_list_a_nat @ ad @ ( map_na823391071729141993_a_nat @ sigma3 @ fv_sub ) @ ( map_na823391071729141993_a_nat @ Tau @ fv_sub ) ) ) ).
% \<open>\<And>thesis. (\<And>\<tau>. \<lbrakk>as = map \<tau> fv_sub; ad_agr_list AD (map \<sigma> fv_sub) (map \<tau> fv_sub)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_71_mem__Collect__eq,axiom,
! [A2: list_Sum_sum_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ A2 @ ( collec7555443234367654128_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A2: nat > sum_sum_a_nat,P: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ A2 @ ( collec5629555741568564177_a_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A2: $o,P: $o > $o] :
( ( member_o @ A2 @ ( collect_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ( collec5629555741568564177_a_nat
@ ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A: set_o] :
( ( collect_o
@ ^ [X: $o] : ( member_o @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X: nat] : ( member_nat @ X @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_79_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_80__092_060sigma_062__def_I1_J,axiom,
member8690443509505302927_a_nat @ sigma3 @ r ).
% \<sigma>_def(1)
thf(fact_81_subset__antisym,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( A = B ) ) ) ).
% subset_antisym
thf(fact_82_subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ( member408289922725080238_a_nat @ X4 @ B ) )
=> ( ord_le1147066620699065093_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_83_subsetI,axiom,
! [A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ! [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A )
=> ( member8690443509505302927_a_nat @ X4 @ B ) )
=> ( ord_le8108555184339247974_a_nat @ A @ B ) ) ).
% subsetI
thf(fact_84_subsetI,axiom,
! [A: set_o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ X4 @ B ) )
=> ( ord_less_eq_set_o @ A @ B ) ) ).
% subsetI
thf(fact_85_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ X4 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_86_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_87_sup__left__idem,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% sup_left_idem
thf(fact_88_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_89_sup__idem,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_90_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_91_Un__iff,axiom,
! [C2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
= ( ( member408289922725080238_a_nat @ C2 @ A )
| ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_92_Un__iff,axiom,
! [C2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A @ B ) )
= ( ( member8690443509505302927_a_nat @ C2 @ A )
| ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_93_Un__iff,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( sup_sup_set_o @ A @ B ) )
= ( ( member_o @ C2 @ A )
| ( member_o @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_94_Un__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
| ( member_nat @ C2 @ B ) ) ) ).
% Un_iff
thf(fact_95_UnCI,axiom,
! [C2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ~ ( member408289922725080238_a_nat @ C2 @ B )
=> ( member408289922725080238_a_nat @ C2 @ A ) )
=> ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_96_UnCI,axiom,
! [C2: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat,A: set_na3699693778330250182_a_nat] :
( ( ~ ( member8690443509505302927_a_nat @ C2 @ B )
=> ( member8690443509505302927_a_nat @ C2 @ A ) )
=> ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_97_UnCI,axiom,
! [C2: $o,B: set_o,A: set_o] :
( ( ~ ( member_o @ C2 @ B )
=> ( member_o @ C2 @ A ) )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnCI
thf(fact_98_UnCI,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ A ) )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_99_distinct__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
= ( distinct_nat @ Ys ) ) ).
% distinct_union
thf(fact_100_ad__agr__list__trans,axiom,
! [X5: 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 @ X5 @ Xs @ Ys )
=> ( ( ad_agr_list_a_nat @ X5 @ Ys @ Zs )
=> ( ad_agr_list_a_nat @ X5 @ Xs @ Zs ) ) ) ).
% ad_agr_list_trans
thf(fact_101_ad__agr__list__refl,axiom,
! [X5: set_a,Xs: list_Sum_sum_a_nat] : ( ad_agr_list_a_nat @ X5 @ Xs @ Xs ) ).
% ad_agr_list_refl
thf(fact_102_ad__agr__list__comm,axiom,
! [X5: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X5 @ Xs @ Ys )
=> ( ad_agr_list_a_nat @ X5 @ Ys @ Xs ) ) ).
% ad_agr_list_comm
thf(fact_103_ad__agr__list__mono,axiom,
! [X5: set_a,Y4: set_a,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( ad_agr_list_a_nat @ Y4 @ Ys @ Xs )
=> ( ad_agr_list_a_nat @ X5 @ Ys @ Xs ) ) ) ).
% ad_agr_list_mono
thf(fact_104_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_105_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_106_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_107_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_108_ad__agr__list__fo__nmlzd,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_nmlzd_a @ AD @ Vs )
=> ( ( fo_nmlzd_a @ AD @ Vs2 )
=> ( Vs = Vs2 ) ) ) ) ).
% ad_agr_list_fo_nmlzd
thf(fact_109_Collect__mono__iff,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
= ( ! [X: nat] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_110_set__eq__subset,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = 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_111_subset__trans,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ A @ C ) ) ) ).
% subset_trans
thf(fact_112_Collect__mono,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% Collect_mono
thf(fact_113_subset__refl,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).
% subset_refl
thf(fact_114_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_115_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_116_subset__iff,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [T2: $o] :
( ( member_o @ T2 @ A3 )
=> ( member_o @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_117_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_118_Set_OequalityD2,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ B @ A ) ) ).
% Set.equalityD2
thf(fact_119_equalityD1,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% equalityD1
thf(fact_120_subset__eq,axiom,
( ord_le1147066620699065093_a_nat
= ( ^ [A3: set_li6526943997496501093_a_nat,B3: set_li6526943997496501093_a_nat] :
! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A3 )
=> ( member408289922725080238_a_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_121_subset__eq,axiom,
( ord_le8108555184339247974_a_nat
= ( ^ [A3: set_na3699693778330250182_a_nat,B3: set_na3699693778330250182_a_nat] :
! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A3 )
=> ( member8690443509505302927_a_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_122_subset__eq,axiom,
( ord_less_eq_set_o
= ( ^ [A3: set_o,B3: set_o] :
! [X: $o] :
( ( member_o @ X @ A3 )
=> ( member_o @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_123_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ( member_nat @ X @ B3 ) ) ) ) ).
% subset_eq
thf(fact_124_equalityE,axiom,
! [A: set_nat,B: set_nat] :
( ( A = B )
=> ~ ( ( ord_less_eq_set_nat @ A @ B )
=> ~ ( ord_less_eq_set_nat @ B @ A ) ) ) ).
% equalityE
thf(fact_125_subsetD,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C2: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( member408289922725080238_a_nat @ C2 @ A )
=> ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_126_subsetD,axiom,
! [A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,C2: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A @ B )
=> ( ( member8690443509505302927_a_nat @ C2 @ A )
=> ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_127_subsetD,axiom,
! [A: set_o,B: set_o,C2: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B ) ) ) ).
% subsetD
thf(fact_128_subsetD,axiom,
! [A: set_nat,B: set_nat,C2: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% subsetD
thf(fact_129_in__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,X3: list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_130_in__mono,axiom,
! [A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,X3: nat > sum_sum_a_nat] :
( ( ord_le8108555184339247974_a_nat @ A @ B )
=> ( ( member8690443509505302927_a_nat @ X3 @ A )
=> ( member8690443509505302927_a_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_131_in__mono,axiom,
! [A: set_o,B: set_o,X3: $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ X3 @ B ) ) ) ).
% in_mono
thf(fact_132_in__mono,axiom,
! [A: set_nat,B: set_nat,X3: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ X3 @ B ) ) ) ).
% in_mono
thf(fact_133_sup__left__commute,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_134_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C2 ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_135_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X ) ) ) ).
% sup_commute
thf(fact_136_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A4 ) ) ) ).
% sup.commute
thf(fact_137_sup__assoc,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_138_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C2 )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_139_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X: set_nat,Y2: set_nat] : ( sup_sup_set_nat @ Y2 @ X ) ) ) ).
% inf_sup_aci(5)
thf(fact_140_inf__sup__aci_I6_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ Z2 )
= ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_141_inf__sup__aci_I7_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_142_inf__sup__aci_I8_J,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_aci(8)
thf(fact_143_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C ) ) ) ).
% Un_left_commute
thf(fact_144_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_145_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_146_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_147_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) ) ).
% Un_assoc
thf(fact_148_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X ) ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( P @ X ) )
& ! [X: nat] :
( ( member_nat @ X @ B )
=> ( P @ X ) ) ) ) ).
% ball_Un
thf(fact_149_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X: nat] :
( ( member_nat @ X @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X ) ) )
= ( ? [X: nat] :
( ( member_nat @ X @ A )
& ( P @ X ) )
| ? [X: nat] :
( ( member_nat @ X @ B )
& ( P @ X ) ) ) ) ).
% bex_Un
thf(fact_150_UnI2,axiom,
! [C2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ B )
=> ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_151_UnI2,axiom,
! [C2: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat,A: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ B )
=> ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_152_UnI2,axiom,
! [C2: $o,B: set_o,A: set_o] :
( ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI2
thf(fact_153_UnI2,axiom,
! [C2: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_154_UnI1,axiom,
! [C2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ A )
=> ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_155_UnI1,axiom,
! [C2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ A )
=> ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_156_UnI1,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ A )
=> ( member_o @ C2 @ ( sup_sup_set_o @ A @ B ) ) ) ).
% UnI1
thf(fact_157_UnI1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_158_UnE,axiom,
! [C2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
=> ( ~ ( member408289922725080238_a_nat @ C2 @ A )
=> ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_159_UnE,axiom,
! [C2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( sup_su3329769938372955546_a_nat @ A @ B ) )
=> ( ~ ( member8690443509505302927_a_nat @ C2 @ A )
=> ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_160_UnE,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( sup_sup_set_o @ A @ B ) )
=> ( ~ ( member_o @ C2 @ A )
=> ( member_o @ C2 @ B ) ) ) ).
% UnE
thf(fact_161_UnE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C2 @ A )
=> ( member_nat @ C2 @ B ) ) ) ).
% UnE
thf(fact_162_ad__agr__list__subset,axiom,
! [Ms: list_P5056861408695629236_a_nat,Ns: list_P5056861408695629236_a_nat,X5: set_a,Sigma2: produc5454855657571582244_a_nat > sum_sum_a_nat,Sigma4: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ord_le8420040822840260538_a_nat @ ( set_Pr2196415342531872975_a_nat @ Ms ) @ ( set_Pr2196415342531872975_a_nat @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X5 @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ns ) @ ( map_Pr2515564165894599445_a_nat @ Sigma4 @ Ns ) )
=> ( ad_agr_list_a_nat @ X5 @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ms ) @ ( map_Pr2515564165894599445_a_nat @ Sigma4 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_163_ad__agr__list__subset,axiom,
! [Ms: list_Sum_sum_a_nat,Ns: list_Sum_sum_a_nat,X5: set_a,Sigma2: sum_sum_a_nat > sum_sum_a_nat,Sigma4: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Ms ) @ ( set_Sum_sum_a_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X5 @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ns ) @ ( map_Su2790769393171190532_a_nat @ Sigma4 @ Ns ) )
=> ( ad_agr_list_a_nat @ X5 @ ( map_Su2790769393171190532_a_nat @ Sigma2 @ Ms ) @ ( map_Su2790769393171190532_a_nat @ Sigma4 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_164_ad__agr__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,X5: set_a,Sigma2: nat > sum_sum_a_nat,Sigma4: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( ad_agr_list_a_nat @ X5 @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma4 @ Ns ) )
=> ( ad_agr_list_a_nat @ X5 @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma4 @ Ms ) ) ) ) ).
% ad_agr_list_subset
thf(fact_165_sup_OcoboundedI2,axiom,
! [C2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_166_sup_OcoboundedI2,axiom,
! [C2: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_167_sup_OcoboundedI1,axiom,
! [C2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_168_sup_OcoboundedI1,axiom,
! [C2: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_169_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_170_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% sup.absorb_iff2
thf(fact_171_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_172_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( sup_sup_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_173_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_174_sup_Ocobounded2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_175_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_176_sup_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_177_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_178_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( A4
= ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).
% sup.order_iff
thf(fact_179_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_180_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_181_sup_OboundedE,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_182_sup_OboundedE,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_183_sup__absorb2,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( sup_sup_nat @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_184_sup__absorb2,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( sup_sup_set_nat @ X3 @ Y3 )
= Y3 ) ) ).
% sup_absorb2
thf(fact_185_sup__absorb1,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( sup_sup_nat @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_186_sup__absorb1,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( sup_sup_set_nat @ X3 @ Y3 )
= X3 ) ) ).
% sup_absorb1
thf(fact_187_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_188_sup_Oabsorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_189_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_190_sup_Oabsorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_191_sup__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y3: nat] :
( ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X4 )
=> ( ( ord_less_eq_nat @ Z @ X4 )
=> ( ord_less_eq_nat @ ( F @ Y @ Z ) @ X4 ) ) )
=> ( ( sup_sup_nat @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_192_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y3: set_nat] :
( ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X4 @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ Y @ ( F @ X4 @ Y ) )
=> ( ! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X4 )
=> ( ( ord_less_eq_set_nat @ Z @ X4 )
=> ( ord_less_eq_set_nat @ ( F @ Y @ Z ) @ X4 ) ) )
=> ( ( sup_sup_set_nat @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% sup_unique
thf(fact_193_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_194_sup_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_195_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_196_sup_OorderE,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_197_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y2: nat] :
( ( sup_sup_nat @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_198_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y2: set_nat] :
( ( sup_sup_set_nat @ X @ Y2 )
= Y2 ) ) ) ).
% le_iff_sup
thf(fact_199_sup__least,axiom,
! [Y3: nat,X3: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_200_sup__least,axiom,
! [Y3: set_nat,X3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z2 ) @ X3 ) ) ) ).
% sup_least
thf(fact_201_sup__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_202_sup__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).
% sup_mono
thf(fact_203_sup_Omono,axiom,
! [C2: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C2 @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_204_sup_Omono,axiom,
! [C2: set_nat,A2: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_205_le__supI2,axiom,
! [X3: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_206_le__supI2,axiom,
! [X3: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_207_le__supI1,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_208_le__supI1,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_209_sup__ge2,axiom,
! [Y3: nat,X3: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_210_sup__ge2,axiom,
! [Y3: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% sup_ge2
thf(fact_211_sup__ge1,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_212_sup__ge1,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% sup_ge1
thf(fact_213_le__supI,axiom,
! [A2: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_214_le__supI,axiom,
! [A2: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X3 ) ) ) ).
% le_supI
thf(fact_215_le__supE,axiom,
! [A2: nat,B2: nat,X3: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_216_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X3 )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ~ ( ord_less_eq_set_nat @ B2 @ X3 ) ) ) ).
% le_supE
thf(fact_217_inf__sup__ord_I3_J,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_218_inf__sup__ord_I3_J,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(3)
thf(fact_219_inf__sup__ord_I4_J,axiom,
! [Y3: nat,X3: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_220_inf__sup__ord_I4_J,axiom,
! [Y3: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_ord(4)
thf(fact_221_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_222_subset__UnE,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ A )
=> ! [B5: set_nat] :
( ( ord_less_eq_set_nat @ B5 @ B )
=> ( C
!= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ) ).
% subset_UnE
thf(fact_223_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_224_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_225_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_226_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_227_Un__least,axiom,
! [A: set_nat,C: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C ) ) ) ).
% Un_least
thf(fact_228_Un__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D2 ) ) ) ) ).
% Un_mono
thf(fact_229_assms_I7_J,axiom,
! [Sigma2: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ fv_sub ) @ ( set_nat2 @ fv_sub ) @ ad @ Sigma2 @ Tau2 )
=> ( ( member8690443509505302927_a_nat @ Sigma2 @ r )
= ( member8690443509505302927_a_nat @ Tau2 @ r ) ) ) ).
% assms(7)
thf(fact_230_assms_I6_J,axiom,
( ass
= ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( proj_v3643391342904276326_a_nat @ r @ fv_sub ) ) ) ).
% assms(6)
thf(fact_231__092_060sigma_062_H__def,axiom,
( sigma2
= ( ^ [N: nat] : ( if_Sum_sum_a_nat @ ( member_nat @ N @ ( set_nat2 @ fv_sub_comp ) ) @ ( the_Sum_sum_a_nat @ ( map_of1932138695521331722_a_nat @ ( zip_na2013496608136855606_a_nat @ fv_sub_comp @ fs ) @ N ) ) @ ( tau @ N ) ) ) ) ).
% \<sigma>'_def
thf(fact_232__092_060open_062xs_A_092_060in_062_Afo__nmlz_AAD_A_096_Aproj__vals_AR_Afv__all_092_060close_062,axiom,
member408289922725080238_a_nat @ xs @ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( proj_v3643391342904276326_a_nat @ r @ fv_all ) ) ).
% \<open>xs \<in> fo_nmlz AD ` proj_vals R fv_all\<close>
thf(fact_233_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_234_dual__order_Orefl,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_235_order__refl,axiom,
! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_236_order__refl,axiom,
! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).
% order_refl
thf(fact_237_assms_I4_J,axiom,
( ( inf_inf_set_nat @ ( set_nat2 @ fv_sub ) @ ( set_nat2 @ fv_sub_comp ) )
= bot_bot_set_nat ) ).
% assms(4)
thf(fact_238_ad__agr__list__link,axiom,
! [Ns: list_nat,AD: set_a,Sigma2: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ns ) @ AD @ Sigma2 @ Tau2 )
= ( ad_agr_list_a_nat @ AD @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau2 @ Ns ) ) ) ).
% ad_agr_list_link
thf(fact_239_map__sorted__distinct__set__unique,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( inj_on8752143810983750942at_nat @ F @ ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs ) )
=> ( ( distinct_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Su5227373005390213643at_nat @ F @ Ys ) )
=> ( ( distinct_nat @ ( map_Su5227373005390213643at_nat @ F @ Ys ) )
=> ( ( ( set_Sum_sum_a_nat2 @ Xs )
= ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_240_map__sorted__distinct__set__unique,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( ( distinct_nat @ ( map_nat_nat @ F @ Xs ) )
=> ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_nat_nat @ F @ Ys ) )
=> ( ( distinct_nat @ ( map_nat_nat @ F @ Ys ) )
=> ( ( ( set_nat2 @ Xs )
= ( set_nat2 @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_241_sp__equiv__list__subset,axiom,
! [Ms: list_P5056861408695629236_a_nat,Ns: list_P5056861408695629236_a_nat,Sigma2: produc5454855657571582244_a_nat > sum_sum_a_nat,Sigma4: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( ord_le8420040822840260538_a_nat @ ( set_Pr2196415342531872975_a_nat @ Ms ) @ ( set_Pr2196415342531872975_a_nat @ Ns ) )
=> ( ( sp_equiv_list_a_nat @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ns ) @ ( map_Pr2515564165894599445_a_nat @ Sigma4 @ Ns ) )
=> ( sp_equiv_list_a_nat @ ( map_Pr2515564165894599445_a_nat @ Sigma2 @ Ms ) @ ( map_Pr2515564165894599445_a_nat @ Sigma4 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_242_sp__equiv__list__subset,axiom,
! [Ms: list_nat,Ns: list_nat,Sigma2: nat > sum_sum_a_nat,Sigma4: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Ms ) @ ( set_nat2 @ Ns ) )
=> ( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) @ ( map_na823391071729141993_a_nat @ Sigma4 @ Ns ) )
=> ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ms ) @ ( map_na823391071729141993_a_nat @ Sigma4 @ Ms ) ) ) ) ).
% sp_equiv_list_subset
thf(fact_243_image__eqI,axiom,
! [B2: nat,F: nat > nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_244_image__eqI,axiom,
! [B2: $o,F: nat > $o,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_245_image__eqI,axiom,
! [B2: nat,F: $o > nat,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_246_image__eqI,axiom,
! [B2: $o,F: $o > $o,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_247_image__eqI,axiom,
! [B2: nat,F: list_Sum_sum_a_nat > nat,X3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_2535339886381165584at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_248_image__eqI,axiom,
! [B2: $o,F: list_Sum_sum_a_nat > $o,X3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member_o @ B2 @ ( image_3984946558445957976_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_249_image__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat,X3: nat,A: set_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_250_image__eqI,axiom,
! [B2: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat,X3: $o,A: set_o] :
( ( B2
= ( F @ X3 ) )
=> ( ( member_o @ X3 @ A )
=> ( member408289922725080238_a_nat @ B2 @ ( image_733135354807394034_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_251_image__eqI,axiom,
! [B2: nat,F: ( nat > sum_sum_a_nat ) > nat,X3: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member8690443509505302927_a_nat @ X3 @ A )
=> ( member_nat @ B2 @ ( image_5786201776793816049at_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_252_image__eqI,axiom,
! [B2: $o,F: ( nat > sum_sum_a_nat ) > $o,X3: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat] :
( ( B2
= ( F @ X3 ) )
=> ( ( member8690443509505302927_a_nat @ X3 @ A )
=> ( member_o @ B2 @ ( image_2376713081370839351_nat_o @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_253_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_254_empty__Collect__eq,axiom,
! [P: $o > $o] :
( ( bot_bot_set_o
= ( collect_o @ P ) )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% empty_Collect_eq
thf(fact_255_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X: nat] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_256_Collect__empty__eq,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( ! [X: $o] :
~ ( P @ X ) ) ) ).
% Collect_empty_eq
thf(fact_257_all__not__in__conv,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ! [X: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X @ A ) )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% all_not_in_conv
thf(fact_258_all__not__in__conv,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ( ! [X: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ X @ A ) )
= ( A = bot_bo6441361344521902642_a_nat ) ) ).
% all_not_in_conv
thf(fact_259_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X: nat] :
~ ( member_nat @ X @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_260_all__not__in__conv,axiom,
! [A: set_o] :
( ( ! [X: $o] :
~ ( member_o @ X @ A ) )
= ( A = bot_bot_set_o ) ) ).
% all_not_in_conv
thf(fact_261_empty__iff,axiom,
! [C2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ C2 @ bot_bo1033123847703346641_a_nat ) ).
% empty_iff
thf(fact_262_empty__iff,axiom,
! [C2: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ C2 @ bot_bo6441361344521902642_a_nat ) ).
% empty_iff
thf(fact_263_empty__iff,axiom,
! [C2: nat] :
~ ( member_nat @ C2 @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_264_empty__iff,axiom,
! [C2: $o] :
~ ( member_o @ C2 @ bot_bot_set_o ) ).
% empty_iff
thf(fact_265_inf__right__idem,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Y3 )
= ( inf_inf_set_nat @ X3 @ Y3 ) ) ).
% inf_right_idem
thf(fact_266_inf_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_267_inf__left__idem,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
= ( inf_inf_set_nat @ X3 @ Y3 ) ) ).
% inf_left_idem
thf(fact_268_inf_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_269_inf__idem,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ X3 )
= X3 ) ).
% inf_idem
thf(fact_270_inf_Oidem,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_271_Int__iff,axiom,
! [C2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
= ( ( member408289922725080238_a_nat @ C2 @ A )
& ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_272_Int__iff,axiom,
! [C2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( inf_in8399021836546144180_a_nat @ A @ B ) )
= ( ( member8690443509505302927_a_nat @ C2 @ A )
& ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_273_Int__iff,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
= ( ( member_o @ C2 @ A )
& ( member_o @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_274_Int__iff,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C2 @ A )
& ( member_nat @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_275_IntI,axiom,
! [C2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ A )
=> ( ( member408289922725080238_a_nat @ C2 @ B )
=> ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_276_IntI,axiom,
! [C2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ A )
=> ( ( member8690443509505302927_a_nat @ C2 @ B )
=> ( member8690443509505302927_a_nat @ C2 @ ( inf_in8399021836546144180_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_277_IntI,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ A )
=> ( ( member_o @ C2 @ B )
=> ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) ) ) ) ).
% IntI
thf(fact_278_IntI,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ A )
=> ( ( member_nat @ C2 @ B )
=> ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_279_le__inf__iff,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_nat @ X3 @ Y3 )
& ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_280_le__inf__iff,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( ( ord_less_eq_set_nat @ X3 @ Y3 )
& ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_281_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_282_inf_Obounded__iff,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
= ( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_283_image__is__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A )
= bot_bo1033123847703346641_a_nat )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% image_is_empty
thf(fact_284_image__is__empty,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( image_7676043921908783909_a_nat @ F @ A )
= bot_bo3237059034911209905_a_nat )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% image_is_empty
thf(fact_285_image__is__empty,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_286_image__is__empty,axiom,
! [F: $o > nat,A: set_o] :
( ( ( image_o_nat @ F @ A )
= bot_bot_set_nat )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_287_image__is__empty,axiom,
! [F: nat > $o,A: set_nat] :
( ( ( image_nat_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_nat ) ) ).
% image_is_empty
thf(fact_288_image__is__empty,axiom,
! [F: $o > $o,A: set_o] :
( ( ( image_o_o @ F @ A )
= bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% image_is_empty
thf(fact_289_empty__is__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( image_5081948215111134021_a_nat @ F @ A ) )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% empty_is_image
thf(fact_290_empty__is__image,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( bot_bo3237059034911209905_a_nat
= ( image_7676043921908783909_a_nat @ F @ A ) )
= ( A = bot_bo1033123847703346641_a_nat ) ) ).
% empty_is_image
thf(fact_291_empty__is__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( bot_bot_set_nat
= ( image_nat_nat @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_292_empty__is__image,axiom,
! [F: $o > nat,A: set_o] :
( ( bot_bot_set_nat
= ( image_o_nat @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_293_empty__is__image,axiom,
! [F: nat > $o,A: set_nat] :
( ( bot_bot_set_o
= ( image_nat_o @ F @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% empty_is_image
thf(fact_294_empty__is__image,axiom,
! [F: $o > $o,A: set_o] :
( ( bot_bot_set_o
= ( image_o_o @ F @ A ) )
= ( A = bot_bot_set_o ) ) ).
% empty_is_image
thf(fact_295_image__empty,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ bot_bo1033123847703346641_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% image_empty
thf(fact_296_image__empty,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ bot_bo1033123847703346641_a_nat )
= bot_bo3237059034911209905_a_nat ) ).
% image_empty
thf(fact_297_image__empty,axiom,
! [F: nat > nat] :
( ( image_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_298_image__empty,axiom,
! [F: nat > $o] :
( ( image_nat_o @ F @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% image_empty
thf(fact_299_image__empty,axiom,
! [F: $o > nat] :
( ( image_o_nat @ F @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% image_empty
thf(fact_300_image__empty,axiom,
! [F: $o > $o] :
( ( image_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% image_empty
thf(fact_301_subset__empty,axiom,
! [A: set_o] :
( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
= ( A = bot_bot_set_o ) ) ).
% subset_empty
thf(fact_302_subset__empty,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
= ( A = bot_bot_set_nat ) ) ).
% subset_empty
thf(fact_303_empty__subsetI,axiom,
! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).
% empty_subsetI
thf(fact_304_empty__subsetI,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% empty_subsetI
thf(fact_305_sup__bot__left,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_306_sup__bot__left,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_307_sup__bot__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% sup_bot_right
thf(fact_308_sup__bot__right,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ X3 @ bot_bot_set_o )
= X3 ) ).
% sup_bot_right
thf(fact_309_bot__eq__sup__iff,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X3 @ Y3 ) )
= ( ( X3 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_310_bot__eq__sup__iff,axiom,
! [X3: set_o,Y3: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ X3 @ Y3 ) )
= ( ( X3 = bot_bot_set_o )
& ( Y3 = bot_bot_set_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_311_sup__eq__bot__iff,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ( sup_sup_set_nat @ X3 @ Y3 )
= bot_bot_set_nat )
= ( ( X3 = bot_bot_set_nat )
& ( Y3 = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_312_sup__eq__bot__iff,axiom,
! [X3: set_o,Y3: set_o] :
( ( ( sup_sup_set_o @ X3 @ Y3 )
= bot_bot_set_o )
= ( ( X3 = bot_bot_set_o )
& ( Y3 = bot_bot_set_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_313_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_314_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( ( sup_sup_set_o @ A2 @ B2 )
= bot_bot_set_o )
= ( ( A2 = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_315_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_316_sup__bot_Oleft__neutral,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_317_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_318_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_o,B2: set_o] :
( ( bot_bot_set_o
= ( sup_sup_set_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_o )
& ( B2 = bot_bot_set_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_319_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_320_sup__bot_Oright__neutral,axiom,
! [A2: set_o] :
( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_321_inf__bot__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% inf_bot_right
thf(fact_322_inf__bot__right,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ X3 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% inf_bot_right
thf(fact_323_inf__bot__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% inf_bot_left
thf(fact_324_inf__bot__left,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X3 )
= bot_bot_set_o ) ).
% inf_bot_left
thf(fact_325_inf__sup__absorb,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y3 ) )
= X3 ) ).
% inf_sup_absorb
thf(fact_326_sup__inf__absorb,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
= X3 ) ).
% sup_inf_absorb
thf(fact_327_Un__empty,axiom,
! [A: set_nat,B: set_nat] :
( ( ( sup_sup_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_328_Un__empty,axiom,
! [A: set_o,B: set_o] :
( ( ( sup_sup_set_o @ A @ B )
= bot_bot_set_o )
= ( ( A = bot_bot_set_o )
& ( B = bot_bot_set_o ) ) ) ).
% Un_empty
thf(fact_329_Int__subset__iff,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C @ A )
& ( ord_less_eq_set_nat @ C @ B ) ) ) ).
% Int_subset_iff
thf(fact_330_Int__Un__eq_I4_J,axiom,
! [T3: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T3 @ ( inf_inf_set_nat @ S @ T3 ) )
= T3 ) ).
% Int_Un_eq(4)
thf(fact_331_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T3 ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_332_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Int_Un_eq(2)
thf(fact_333_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T3 ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_334_Un__Int__eq_I4_J,axiom,
! [T3: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T3 @ ( sup_sup_set_nat @ S @ T3 ) )
= T3 ) ).
% Un_Int_eq(4)
thf(fact_335_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T3 ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_336_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T3 ) @ T3 )
= T3 ) ).
% Un_Int_eq(2)
thf(fact_337_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T3 ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_338_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_339_list_Oset__map,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,V: list_l4703314356710769291_a_nat] :
( ( set_se268393680088689402_a_nat @ ( map_li4181295100893080420_a_nat @ F @ V ) )
= ( image_7676043921908783909_a_nat @ F @ ( set_li2392974972034027290_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_340_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_341_list_Oset__map,axiom,
! [F: sum_sum_a_nat > nat,V: list_Sum_sum_a_nat] :
( ( set_nat2 @ ( map_Su5227373005390213643at_nat @ F @ V ) )
= ( image_2473878607534554506at_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_342_list_Oset__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,V: list_P5056861408695629236_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ F @ V ) )
= ( image_5908522764169838868_a_nat @ F @ ( set_Pr2196415342531872975_a_nat @ V ) ) ) ).
% list.set_map
thf(fact_343_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_344_list_Oset__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,V: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ V ) )
= ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_345_disjoint__iff__not__equal,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ B )
=> ( X != Y2 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_346_disjoint__iff__not__equal,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ! [Y2: $o] :
( ( member_o @ Y2 @ B )
=> ( X = (~ Y2) ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_347_Int__left__commute,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C ) ) ) ).
% Int_left_commute
thf(fact_348_Int__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_349_Int__empty__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% Int_empty_right
thf(fact_350_Int__empty__right,axiom,
! [A: set_o] :
( ( inf_inf_set_o @ A @ bot_bot_set_o )
= bot_bot_set_o ) ).
% Int_empty_right
thf(fact_351_Int__empty__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ B )
= bot_bot_set_nat ) ).
% Int_empty_left
thf(fact_352_Int__empty__left,axiom,
! [B: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ B )
= bot_bot_set_o ) ).
% Int_empty_left
thf(fact_353_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_354_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: $o,F: nat > $o] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_355_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: nat,F: $o > nat] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_o_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_356_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: $o,F: $o > $o] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_o_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_357_rev__image__eqI,axiom,
! [X3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_2535339886381165584at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_358_rev__image__eqI,axiom,
! [X3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: $o,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_3984946558445957976_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_359_rev__image__eqI,axiom,
! [X3: nat,A: set_nat,B2: list_Sum_sum_a_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_6262589752765146990_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_360_rev__image__eqI,axiom,
! [X3: $o,A: set_o,B2: list_Sum_sum_a_nat,F: $o > list_Sum_sum_a_nat] :
( ( member_o @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( image_733135354807394034_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_361_rev__image__eqI,axiom,
! [X3: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B2: nat,F: ( nat > sum_sum_a_nat ) > nat] :
( ( member8690443509505302927_a_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_nat @ B2 @ ( image_5786201776793816049at_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_362_rev__image__eqI,axiom,
! [X3: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B2: $o,F: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ X3 @ A )
=> ( ( B2
= ( F @ X3 ) )
=> ( member_o @ B2 @ ( image_2376713081370839351_nat_o @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_363_disjoint__iff,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ~ ( member408289922725080238_a_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_364_disjoint__iff,axiom,
! [A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ( inf_in8399021836546144180_a_nat @ A @ B )
= bot_bo6441361344521902642_a_nat )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ A )
=> ~ ( member8690443509505302927_a_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_365_disjoint__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ~ ( member_nat @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_366_disjoint__iff,axiom,
! [A: set_o,B: set_o] :
( ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ~ ( member_o @ X @ B ) ) ) ) ).
% disjoint_iff
thf(fact_367_ball__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_368_ball__imageD,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ! [X4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ ( image_7676043921908783909_a_nat @ F @ A ) )
=> ( P @ X4 ) )
=> ! [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
=> ( P @ ( F @ X2 ) ) ) ) ).
% ball_imageD
thf(fact_369_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_370_image__cong,axiom,
! [M: set_li6526943997496501093_a_nat,N2: 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 = N2 )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ N2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_5081948215111134021_a_nat @ F @ M )
= ( image_5081948215111134021_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_371_image__cong,axiom,
! [M: set_li6526943997496501093_a_nat,N2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( M = N2 )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ N2 )
=> ( ( F @ X4 )
= ( G @ X4 ) ) )
=> ( ( image_7676043921908783909_a_nat @ F @ M )
= ( image_7676043921908783909_a_nat @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_372_ex__in__conv,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( ? [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ A ) )
= ( A != bot_bo1033123847703346641_a_nat ) ) ).
% ex_in_conv
thf(fact_373_ex__in__conv,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ( ? [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ A ) )
= ( A != bot_bo6441361344521902642_a_nat ) ) ).
% ex_in_conv
thf(fact_374_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X: nat] : ( member_nat @ X @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_375_ex__in__conv,axiom,
! [A: set_o] :
( ( ? [X: $o] : ( member_o @ X @ A ) )
= ( A != bot_bot_set_o ) ) ).
% ex_in_conv
thf(fact_376_bex__imageD,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ? [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ ( image_5081948215111134021_a_nat @ F @ A ) )
& ( P @ X2 ) )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_377_bex__imageD,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ? [X2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X2 @ ( image_7676043921908783909_a_nat @ F @ A ) )
& ( P @ X2 ) )
=> ? [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
& ( P @ ( F @ X4 ) ) ) ) ).
% bex_imageD
thf(fact_378_Int__emptyI,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ~ ( member408289922725080238_a_nat @ X4 @ B ) )
=> ( ( inf_in3249246906714053971_a_nat @ A @ B )
= bot_bo1033123847703346641_a_nat ) ) ).
% Int_emptyI
thf(fact_379_Int__emptyI,axiom,
! [A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ! [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A )
=> ~ ( member8690443509505302927_a_nat @ X4 @ B ) )
=> ( ( inf_in8399021836546144180_a_nat @ A @ B )
= bot_bo6441361344521902642_a_nat ) ) ).
% Int_emptyI
thf(fact_380_Int__emptyI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ~ ( member_nat @ X4 @ B ) )
=> ( ( inf_inf_set_nat @ A @ B )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_381_Int__emptyI,axiom,
! [A: set_o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ~ ( member_o @ X4 @ B ) )
=> ( ( inf_inf_set_o @ A @ B )
= bot_bot_set_o ) ) ).
% Int_emptyI
thf(fact_382_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_383_image__iff,axiom,
! [Z2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Z2 @ ( image_7676043921908783909_a_nat @ F @ A ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_384_image__iff,axiom,
! [Z2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ Z2 @ ( image_5081948215111134021_a_nat @ F @ A ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_385_Int__assoc,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ).
% Int_assoc
thf(fact_386_equals0I,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ! [Y: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ Y @ A )
=> ( A = bot_bo1033123847703346641_a_nat ) ) ).
% equals0I
thf(fact_387_equals0I,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ! [Y: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ Y @ A )
=> ( A = bot_bo6441361344521902642_a_nat ) ) ).
% equals0I
thf(fact_388_equals0I,axiom,
! [A: set_nat] :
( ! [Y: nat] :
~ ( member_nat @ Y @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_389_equals0I,axiom,
! [A: set_o] :
( ! [Y: $o] :
~ ( member_o @ Y @ A )
=> ( A = bot_bot_set_o ) ) ).
% equals0I
thf(fact_390_equals0D,axiom,
! [A: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat] :
( ( A = bot_bo1033123847703346641_a_nat )
=> ~ ( member408289922725080238_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_391_equals0D,axiom,
! [A: set_na3699693778330250182_a_nat,A2: nat > sum_sum_a_nat] :
( ( A = bot_bo6441361344521902642_a_nat )
=> ~ ( member8690443509505302927_a_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_392_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_393_equals0D,axiom,
! [A: set_o,A2: $o] :
( ( A = bot_bot_set_o )
=> ~ ( member_o @ A2 @ A ) ) ).
% equals0D
thf(fact_394_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_395_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > $o] :
( ( member_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_396_imageI,axiom,
! [X3: $o,A: set_o,F: $o > nat] :
( ( member_o @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_o_nat @ F @ A ) ) ) ).
% imageI
thf(fact_397_imageI,axiom,
! [X3: $o,A: set_o,F: $o > $o] :
( ( member_o @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_o_o @ F @ A ) ) ) ).
% imageI
thf(fact_398_imageI,axiom,
! [X3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_2535339886381165584at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_399_imageI,axiom,
! [X3: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o] :
( ( member408289922725080238_a_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_3984946558445957976_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_400_imageI,axiom,
! [X3: nat,A: set_nat,F: nat > list_Sum_sum_a_nat] :
( ( member_nat @ X3 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ ( image_6262589752765146990_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_401_imageI,axiom,
! [X3: $o,A: set_o,F: $o > list_Sum_sum_a_nat] :
( ( member_o @ X3 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X3 ) @ ( image_733135354807394034_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_402_imageI,axiom,
! [X3: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > nat] :
( ( member8690443509505302927_a_nat @ X3 @ A )
=> ( member_nat @ ( F @ X3 ) @ ( image_5786201776793816049at_nat @ F @ A ) ) ) ).
% imageI
thf(fact_403_imageI,axiom,
! [X3: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > $o] :
( ( member8690443509505302927_a_nat @ X3 @ A )
=> ( member_o @ ( F @ X3 ) @ ( image_2376713081370839351_nat_o @ F @ A ) ) ) ).
% imageI
thf(fact_404_emptyE,axiom,
! [A2: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ A2 @ bot_bo1033123847703346641_a_nat ) ).
% emptyE
thf(fact_405_emptyE,axiom,
! [A2: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ A2 @ bot_bo6441361344521902642_a_nat ) ).
% emptyE
thf(fact_406_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_407_emptyE,axiom,
! [A2: $o] :
~ ( member_o @ A2 @ bot_bot_set_o ) ).
% emptyE
thf(fact_408_IntD2,axiom,
! [C2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
=> ( member408289922725080238_a_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_409_IntD2,axiom,
! [C2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( inf_in8399021836546144180_a_nat @ A @ B ) )
=> ( member8690443509505302927_a_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_410_IntD2,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C2 @ B ) ) ).
% IntD2
thf(fact_411_IntD2,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ B ) ) ).
% IntD2
thf(fact_412_IntD1,axiom,
! [C2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
=> ( member408289922725080238_a_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_413_IntD1,axiom,
! [C2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( inf_in8399021836546144180_a_nat @ A @ B ) )
=> ( member8690443509505302927_a_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_414_IntD1,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
=> ( member_o @ C2 @ A ) ) ).
% IntD1
thf(fact_415_IntD1,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C2 @ A ) ) ).
% IntD1
thf(fact_416_IntE,axiom,
! [C2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ C2 @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
=> ~ ( ( member408289922725080238_a_nat @ C2 @ A )
=> ~ ( member408289922725080238_a_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_417_IntE,axiom,
! [C2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ C2 @ ( inf_in8399021836546144180_a_nat @ A @ B ) )
=> ~ ( ( member8690443509505302927_a_nat @ C2 @ A )
=> ~ ( member8690443509505302927_a_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_418_IntE,axiom,
! [C2: $o,A: set_o,B: set_o] :
( ( member_o @ C2 @ ( inf_inf_set_o @ A @ B ) )
=> ~ ( ( member_o @ C2 @ A )
=> ~ ( member_o @ C2 @ B ) ) ) ).
% IntE
thf(fact_419_IntE,axiom,
! [C2: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C2 @ A )
=> ~ ( member_nat @ C2 @ B ) ) ) ).
% IntE
thf(fact_420_inf__left__commute,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_421_inf_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C2 ) )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% inf.left_commute
thf(fact_422_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X ) ) ) ).
% inf_commute
thf(fact_423_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A4: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A4 ) ) ) ).
% inf.commute
thf(fact_424_inf__assoc,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_425_inf_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% inf.assoc
thf(fact_426_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X: set_nat,Y2: set_nat] : ( inf_inf_set_nat @ Y2 @ X ) ) ) ).
% inf_sup_aci(1)
thf(fact_427_inf__sup__aci_I2_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Z2 )
= ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_428_inf__sup__aci_I3_J,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ Y3 @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_429_inf__sup__aci_I4_J,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ X3 @ Y3 ) )
= ( inf_inf_set_nat @ X3 @ Y3 ) ) ).
% inf_sup_aci(4)
thf(fact_430_ad__agr__sets__comm,axiom,
! [FV: set_nat,S: set_nat,X5: set_a,Sigma2: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ad_agr_sets_a_nat @ FV @ S @ X5 @ Sigma2 @ Tau2 )
=> ( ad_agr_sets_a_nat @ FV @ S @ X5 @ Tau2 @ Sigma2 ) ) ).
% ad_agr_sets_comm
thf(fact_431_image__Int__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ A @ B ) ) @ ( inf_in3249246906714053971_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_432_image__Int__subset,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] : ( ord_le8138476598237931237_a_nat @ ( image_7676043921908783909_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ A @ B ) ) @ ( inf_in5367731912061063475_a_nat @ ( image_7676043921908783909_a_nat @ F @ A ) @ ( image_7676043921908783909_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_433_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_434_bot_Oextremum,axiom,
! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% bot.extremum
thf(fact_435_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_436_bot_Oextremum,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% bot.extremum
thf(fact_437_bot_Oextremum__unique,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
= ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_unique
thf(fact_438_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_439_bot_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
= ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_unique
thf(fact_440_bot_Oextremum__uniqueI,axiom,
! [A2: set_o] :
( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
=> ( A2 = bot_bot_set_o ) ) ).
% bot.extremum_uniqueI
thf(fact_441_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_442_bot_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
=> ( A2 = bot_bot_set_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_443_insort__insert__key__triv,axiom,
! [F: nat > nat,X3: nat,Xs: list_nat] :
( ( member_nat @ ( F @ X3 ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
=> ( ( linord1921536354676448932at_nat @ F @ X3 @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_444_insort__insert__key__triv,axiom,
! [F: nat > $o,X3: nat,Xs: list_nat] :
( ( member_o @ ( F @ X3 ) @ ( image_nat_o @ F @ ( set_nat2 @ Xs ) ) )
=> ( ( linord2970189811765474244_nat_o @ F @ X3 @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_445_insort__insert__key__triv,axiom,
! [F: sum_sum_a_nat > nat,X3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_nat @ ( F @ X3 ) @ ( image_2473878607534554506at_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) ) )
=> ( ( linord458367490959630669at_nat @ F @ X3 @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_446_insort__insert__key__triv,axiom,
! [F: sum_sum_a_nat > $o,X3: sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( member_o @ ( F @ X3 ) @ ( image_6095136190293192542_nat_o @ F @ ( set_Sum_sum_a_nat2 @ Xs ) ) )
=> ( ( linord8391790818113774811_nat_o @ F @ X3 @ Xs )
= Xs ) ) ).
% insort_insert_key_triv
thf(fact_447_inf__sup__ord_I2_J,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_448_inf__sup__ord_I2_J,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_sup_ord(2)
thf(fact_449_inf__sup__ord_I1_J,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_450_inf__sup__ord_I1_J,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_sup_ord(1)
thf(fact_451_inf__le1,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_452_inf__le1,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ X3 ) ).
% inf_le1
thf(fact_453_inf__le2,axiom,
! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_454_inf__le2,axiom,
! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ Y3 ) ).
% inf_le2
thf(fact_455_le__infE,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_456_le__infE,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ~ ( ord_less_eq_set_nat @ X3 @ B2 ) ) ) ).
% le_infE
thf(fact_457_le__infI,axiom,
! [X3: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( ( ord_less_eq_nat @ X3 @ B2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_458_le__infI,axiom,
! [X3: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ A2 )
=> ( ( ord_less_eq_set_nat @ X3 @ B2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_459_inf__mono,axiom,
! [A2: nat,C2: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_460_inf__mono,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).
% inf_mono
thf(fact_461_le__infI1,axiom,
! [A2: nat,X3: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_462_le__infI1,axiom,
! [A2: set_nat,X3: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI1
thf(fact_463_le__infI2,axiom,
! [B2: nat,X3: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X3 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_464_le__infI2,axiom,
! [B2: set_nat,X3: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X3 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X3 ) ) ).
% le_infI2
thf(fact_465_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_466_inf_OorderE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_467_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_468_inf_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_469_inf__unique,axiom,
! [F: nat > nat > nat,X3: nat,Y3: nat] :
( ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y ) @ X4 )
=> ( ! [X4: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y ) @ Y )
=> ( ! [X4: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ( ord_less_eq_nat @ X4 @ Z )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y @ Z ) ) ) )
=> ( ( inf_inf_nat @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_470_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X3: set_nat,Y3: set_nat] :
( ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X4 @ Y ) @ X4 )
=> ( ! [X4: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( F @ X4 @ Y ) @ Y )
=> ( ! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ( ord_less_eq_set_nat @ X4 @ Z )
=> ( ord_less_eq_set_nat @ X4 @ ( F @ Y @ Z ) ) ) )
=> ( ( inf_inf_set_nat @ X3 @ Y3 )
= ( F @ X3 @ Y3 ) ) ) ) ) ).
% inf_unique
thf(fact_471_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X: nat,Y2: nat] :
( ( inf_inf_nat @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_472_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X: set_nat,Y2: set_nat] :
( ( inf_inf_set_nat @ X @ Y2 )
= X ) ) ) ).
% le_iff_inf
thf(fact_473_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_474_inf_Oabsorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_475_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_476_inf_Oabsorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_477_inf__absorb1,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( inf_inf_nat @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_478_inf__absorb1,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( inf_inf_set_nat @ X3 @ Y3 )
= X3 ) ) ).
% inf_absorb1
thf(fact_479_inf__absorb2,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( inf_inf_nat @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_480_inf__absorb2,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( inf_inf_set_nat @ X3 @ Y3 )
= Y3 ) ) ).
% inf_absorb2
thf(fact_481_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_482_inf_OboundedE,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% inf.boundedE
thf(fact_483_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_484_inf_OboundedI,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% inf.boundedI
thf(fact_485_inf__greatest,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_486_inf__greatest,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_487_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( A4
= ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_488_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( A4
= ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% inf.order_iff
thf(fact_489_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_490_inf_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_491_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_492_inf_Ocobounded2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_493_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_494_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A4: set_nat,B4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= A4 ) ) ) ).
% inf.absorb_iff1
thf(fact_495_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( inf_inf_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_496_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B4: set_nat,A4: set_nat] :
( ( inf_inf_set_nat @ A4 @ B4 )
= B4 ) ) ) ).
% inf.absorb_iff2
thf(fact_497_inf_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_498_inf_OcoboundedI1,axiom,
! [A2: set_nat,C2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI1
thf(fact_499_inf_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_500_inf_OcoboundedI2,axiom,
! [B2: set_nat,C2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C2 ) ) ).
% inf.coboundedI2
thf(fact_501_image__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_502_image__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ord_le8138476598237931237_a_nat @ ( image_7676043921908783909_a_nat @ F @ A ) @ ( image_7676043921908783909_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_503_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_504_image__subsetI,axiom,
! [A: set_nat,F: nat > $o,B: set_o] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_nat_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_505_image__subsetI,axiom,
! [A: set_o,F: $o > $o,B: set_o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_o_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_506_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_507_image__subsetI,axiom,
! [A: set_o,F: $o > nat,B: set_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_o_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_508_image__subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o,B: set_o] :
( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_3984946558445957976_nat_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_509_image__subsetI,axiom,
! [A: set_nat,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_6262589752765146990_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_510_image__subsetI,axiom,
! [A: set_o,F: $o > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( member408289922725080238_a_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le1147066620699065093_a_nat @ ( image_733135354807394034_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_511_image__subsetI,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat,B: set_nat] :
( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_2535339886381165584at_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_512_image__subsetI,axiom,
! [A: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > $o,B: set_o] :
( ! [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A )
=> ( member_o @ ( F @ X4 ) @ B ) )
=> ( ord_less_eq_set_o @ ( image_2376713081370839351_nat_o @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_513_image__subsetI,axiom,
! [A: set_nat,F: nat > nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member8690443509505302927_a_nat @ ( F @ X4 ) @ B ) )
=> ( ord_le8108555184339247974_a_nat @ ( image_1051037728736664655_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_514_subset__imageE,axiom,
! [B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ~ ! [C3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C3 @ A )
=> ( B
!= ( image_5081948215111134021_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_515_subset__imageE,axiom,
! [B: set_se4330304633200676677_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le8138476598237931237_a_nat @ B @ ( image_7676043921908783909_a_nat @ F @ A ) )
=> ~ ! [C3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ C3 @ A )
=> ( B
!= ( image_7676043921908783909_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_516_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_517_image__subset__iff,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( image_7676043921908783909_a_nat @ F @ A ) @ B )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ( member5553968465346197646_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_518_image__subset__iff,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_519_subset__image__iff,axiom,
! [B: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A ) )
= ( ? [AA: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ AA @ A )
& ( B
= ( image_5081948215111134021_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_520_subset__image__iff,axiom,
! [B: set_se4330304633200676677_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ord_le8138476598237931237_a_nat @ B @ ( image_7676043921908783909_a_nat @ F @ A ) )
= ( ? [AA: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ AA @ A )
& ( B
= ( image_7676043921908783909_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_521_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_522_distrib__imp1,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X4 @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X4 @ Y ) @ ( inf_inf_set_nat @ X4 @ Z ) ) )
=> ( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp1
thf(fact_523_distrib__imp2,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ! [X4: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X4 @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X4 @ Y ) @ ( sup_sup_set_nat @ X4 @ Z ) ) )
=> ( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ) ).
% distrib_imp2
thf(fact_524_inf__sup__distrib1,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% inf_sup_distrib1
thf(fact_525_inf__sup__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% inf_sup_distrib2
thf(fact_526_sup__inf__distrib1,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% sup_inf_distrib1
thf(fact_527_sup__inf__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% sup_inf_distrib2
thf(fact_528_image__Un,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_529_image__Un,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
= ( sup_su499249268922660121_a_nat @ ( image_7676043921908783909_a_nat @ F @ A ) @ ( image_7676043921908783909_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_530_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_531_Int__mono,axiom,
! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D2 ) ) ) ) ).
% Int_mono
thf(fact_532_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_533_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_534_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_535_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_536_Int__greatest,axiom,
! [C: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C @ A )
=> ( ( ord_less_eq_set_nat @ C @ B )
=> ( ord_less_eq_set_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_537_Int__Collect__mono,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o,Q: list_Sum_sum_a_nat > $o] :
( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( inf_in3249246906714053971_a_nat @ A @ ( collec7555443234367654128_a_nat @ P ) ) @ ( inf_in3249246906714053971_a_nat @ B @ ( collec7555443234367654128_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_538_Int__Collect__mono,axiom,
! [A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,P: ( nat > sum_sum_a_nat ) > $o,Q: ( nat > sum_sum_a_nat ) > $o] :
( ( ord_le8108555184339247974_a_nat @ A @ B )
=> ( ! [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_le8108555184339247974_a_nat @ ( inf_in8399021836546144180_a_nat @ A @ ( collec5629555741568564177_a_nat @ P ) ) @ ( inf_in8399021836546144180_a_nat @ B @ ( collec5629555741568564177_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_539_Int__Collect__mono,axiom,
! [A: set_o,B: set_o,P: $o > $o,Q: $o > $o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_o @ ( inf_inf_set_o @ A @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B @ ( collect_o @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_540_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( P @ X4 )
=> ( Q @ X4 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_541_Un__Int__crazy,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ B @ C ) ) @ ( inf_inf_set_nat @ C @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ B @ C ) ) @ ( sup_sup_set_nat @ C @ A ) ) ) ).
% Un_Int_crazy
thf(fact_542_Int__Un__distrib,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ C ) ) ) ).
% Int_Un_distrib
thf(fact_543_Un__Int__distrib,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ A @ C ) ) ) ).
% Un_Int_distrib
thf(fact_544_Int__Un__distrib2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A ) @ ( inf_inf_set_nat @ C @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_545_Un__Int__distrib2,axiom,
! [B: set_nat,C: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ C ) @ A )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ A ) @ ( sup_sup_set_nat @ C @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_546_Un__empty__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B )
= B ) ).
% Un_empty_left
thf(fact_547_Un__empty__left,axiom,
! [B: set_o] :
( ( sup_sup_set_o @ bot_bot_set_o @ B )
= B ) ).
% Un_empty_left
thf(fact_548_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_549_Un__empty__right,axiom,
! [A: set_o] :
( ( sup_sup_set_o @ A @ bot_bot_set_o )
= A ) ).
% Un_empty_right
thf(fact_550_ad__agr__sets__mono,axiom,
! [X5: set_a,Y4: set_a,FV: set_nat,S: set_nat,Sigma2: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ X5 @ Y4 )
=> ( ( ad_agr_sets_a_nat @ FV @ S @ Y4 @ Sigma2 @ Tau2 )
=> ( ad_agr_sets_a_nat @ FV @ S @ X5 @ Sigma2 @ Tau2 ) ) ) ).
% ad_agr_sets_mono
thf(fact_551_distrib__inf__le,axiom,
! [X3: nat,Y3: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ ( inf_inf_nat @ X3 @ Z2 ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_552_distrib__inf__le,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) @ ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) ) ) ).
% distrib_inf_le
thf(fact_553_distrib__sup__le,axiom,
! [X3: nat,Y3: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y3 ) @ ( sup_sup_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_554_distrib__sup__le,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% distrib_sup_le
thf(fact_555_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_556_image__set,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,Xs: list_l4703314356710769291_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ ( set_li2392974972034027290_a_nat @ Xs ) )
= ( set_se268393680088689402_a_nat @ ( map_li4181295100893080420_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_557_image__set,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( image_5908522764169838868_a_nat @ F @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_558_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_559_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_560_image__set,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( image_2473878607534554506at_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( set_nat2 @ ( map_Su5227373005390213643at_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_561_image__set,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_562_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C ) ) )
= ( ord_less_eq_set_nat @ C @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_563_ad__agr__sets__mono_H,axiom,
! [S: set_nat,S2: set_nat,FV: set_nat,X5: set_a,Sigma2: nat > sum_sum_a_nat,Tau2: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ S @ S2 )
=> ( ( ad_agr_sets_a_nat @ FV @ S2 @ X5 @ Sigma2 @ Tau2 )
=> ( ad_agr_sets_a_nat @ FV @ S @ X5 @ Sigma2 @ Tau2 ) ) ) ).
% ad_agr_sets_mono'
thf(fact_564_map__inj__on,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ F @ Ys ) )
=> ( ( inj_on663950023194574504_a_nat @ F @ ( sup_su4565425741012426374_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs ) @ ( set_Pr2196415342531872975_a_nat @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_565_map__inj__on,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
=> ( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_566_map__inj__on,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ F @ Ys ) )
=> ( ( inj_on4348161877322679292_a_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_567_inj__on__map__eq__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( inj_on663950023194574504_a_nat @ F @ ( sup_su4565425741012426374_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs ) @ ( set_Pr2196415342531872975_a_nat @ Ys ) ) )
=> ( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_568_inj__on__map__eq__map,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_569_inj__on__map__eq__map,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on4348161877322679292_a_nat @ F @ ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) )
=> ( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_570_distinct__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( distin2701893636801681144_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) )
= ( ( distin5548549501492119275_a_nat @ Xs )
& ( inj_on663950023194574504_a_nat @ F @ ( set_Pr2196415342531872975_a_nat @ Xs ) ) ) ) ).
% distinct_map
thf(fact_571_distinct__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( distinct_nat @ ( map_nat_nat @ F @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on_nat_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_572_distinct__map,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( distin2701893636801681144_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( ( distinct_nat @ Xs )
& ( inj_on4348161877322679292_a_nat @ F @ ( set_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_573_distinct__map,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( distinct_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs ) )
= ( ( distin2701893636801681144_a_nat @ Xs )
& ( inj_on8752143810983750942at_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ).
% distinct_map
thf(fact_574_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_575_le__cases3,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X3 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
=> ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_576_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_577_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [X: set_nat,Y2: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y2 )
& ( ord_less_eq_set_nat @ Y2 @ X ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_578_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_579_ord__eq__le__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_580_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_581_ord__le__eq__trans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_582_order__antisym,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_583_order__antisym,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ).
% order_antisym
thf(fact_584_order_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ord_less_eq_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_585_order_Otrans,axiom,
! [A2: set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ord_less_eq_set_nat @ A2 @ C2 ) ) ) ).
% order.trans
thf(fact_586_order__trans,axiom,
! [X3: nat,Y3: nat,Z2: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_nat @ Y3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_587_order__trans,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X3 @ Y3 )
=> ( ( ord_less_eq_set_nat @ Y3 @ Z2 )
=> ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).
% order_trans
thf(fact_588_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_589_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_590_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ B4 @ A4 )
& ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_591_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_592_dual__order_Oantisym,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_593_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C2 @ B2 )
=> ( ord_less_eq_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_594_dual__order_Otrans,axiom,
! [B2: set_nat,A2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C2 @ B2 )
=> ( ord_less_eq_set_nat @ C2 @ A2 ) ) ) ).
% dual_order.trans
thf(fact_595_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_596_antisym,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_597_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_598_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: set_nat,Z3: set_nat] : ( Y5 = Z3 ) )
= ( ^ [A4: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A4 @ B4 )
& ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_599_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_600_order__subst1,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_601_order__subst1,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_602_order__subst1,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_603_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_604_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_605_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_606_order__subst2,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ ( F @ B2 ) @ C2 )
=> ( ! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_607_order__eq__refl,axiom,
! [X3: nat,Y3: nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_608_order__eq__refl,axiom,
! [X3: set_nat,Y3: set_nat] :
( ( X3 = Y3 )
=> ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% order_eq_refl
thf(fact_609_linorder__linear,axiom,
! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
| ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_linear
thf(fact_610_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_611_ord__eq__le__subst,axiom,
! [A2: set_nat,F: nat > set_nat,B2: nat,C2: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_612_ord__eq__le__subst,axiom,
! [A2: nat,F: set_nat > nat,B2: set_nat,C2: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_613_ord__eq__le__subst,axiom,
! [A2: set_nat,F: set_nat > set_nat,B2: set_nat,C2: set_nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_nat @ B2 @ C2 )
=> ( ! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ A2 @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_614_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_615_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > set_nat,C2: set_nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: nat,Y: nat] :
( ( ord_less_eq_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_616_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > nat,C2: nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_617_ord__le__eq__subst,axiom,
! [A2: set_nat,B2: set_nat,F: set_nat > set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X4 @ Y )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_set_nat @ ( F @ A2 ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_618_linorder__le__cases,axiom,
! [X3: nat,Y3: nat] :
( ~ ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% linorder_le_cases
thf(fact_619_order__antisym__conv,axiom,
! [Y3: nat,X3: nat] :
( ( ord_less_eq_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_620_order__antisym__conv,axiom,
! [Y3: set_nat,X3: set_nat] :
( ( ord_less_eq_set_nat @ Y3 @ X3 )
=> ( ( ord_less_eq_set_nat @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ).
% order_antisym_conv
thf(fact_621_fs__map__map__of,axiom,
( fs
= ( map_na823391071729141993_a_nat @ ( comp_o2721059924169077421at_nat @ the_Sum_sum_a_nat @ ( map_of1932138695521331722_a_nat @ ( zip_na2013496608136855606_a_nat @ fv_sub_comp @ fs ) ) ) @ fv_sub_comp ) ) ).
% fs_map_map_of
thf(fact_622_Fun_Oinj__on__empty,axiom,
! [F: nat > nat] : ( inj_on_nat_nat @ F @ bot_bot_set_nat ) ).
% Fun.inj_on_empty
thf(fact_623_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_left
thf(fact_624_boolean__algebra_Oconj__zero__left,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ bot_bot_set_o @ X3 )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_left
thf(fact_625_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_nat] :
( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% boolean_algebra.conj_zero_right
thf(fact_626_boolean__algebra_Oconj__zero__right,axiom,
! [X3: set_o] :
( ( inf_inf_set_o @ X3 @ bot_bot_set_o )
= bot_bot_set_o ) ).
% boolean_algebra.conj_zero_right
thf(fact_627_inj__on__image__Int,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,C: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F @ C )
=> ( ( ord_le1147066620699065093_a_nat @ A @ C )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C )
=> ( ( image_5081948215111134021_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
= ( inf_in3249246906714053971_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ ( image_5081948215111134021_a_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_628_inj__on__image__Int,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inj_on7795941589172611729_a_nat @ F @ C )
=> ( ( ord_le1147066620699065093_a_nat @ A @ C )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C )
=> ( ( image_7676043921908783909_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ A @ B ) )
= ( inf_in5367731912061063475_a_nat @ ( image_7676043921908783909_a_nat @ F @ A ) @ ( image_7676043921908783909_a_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_629_inj__on__image__Int,axiom,
! [F: nat > nat,C: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_630_inj__on__iff__surj,axiom,
! [A: set_se4330304633200676677_a_nat,A7: set_li6526943997496501093_a_nat] :
( ( A != bot_bo3237059034911209905_a_nat )
=> ( ( ? [F2: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( inj_on1831002258565622481_a_nat @ F2 @ A )
& ( ord_le1147066620699065093_a_nat @ ( image_1711104591301794661_a_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_631_inj__on__iff__surj,axiom,
! [A: set_li6526943997496501093_a_nat,A7: set_li6526943997496501093_a_nat] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ( ? [F2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F2 @ A )
& ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: list_Sum_sum_a_nat > list_Sum_sum_a_nat] :
( ( image_5081948215111134021_a_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_632_inj__on__iff__surj,axiom,
! [A: set_li6526943997496501093_a_nat,A7: set_se4330304633200676677_a_nat] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ( ? [F2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( inj_on7795941589172611729_a_nat @ F2 @ A )
& ( ord_le8138476598237931237_a_nat @ ( image_7676043921908783909_a_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( image_1711104591301794661_a_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_633_inj__on__iff__surj,axiom,
! [A: set_nat,A7: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F2: nat > nat] :
( ( inj_on_nat_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: nat > nat] :
( ( image_nat_nat @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_634_inj__on__iff__surj,axiom,
! [A: set_o,A7: set_nat] :
( ( A != bot_bot_set_o )
=> ( ( ? [F2: $o > nat] :
( ( inj_on_o_nat @ F2 @ A )
& ( ord_less_eq_set_nat @ ( image_o_nat @ F2 @ A ) @ A7 ) ) )
= ( ? [G2: nat > $o] :
( ( image_nat_o @ G2 @ A7 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_635_proj__tuple__merge,axiom,
! [Nxs: list_P5056861408695629236_a_nat,Mys: list_P5056861408695629236_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) )
& ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
=> ( ( ( inf_inf_set_nat @ ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) ) @ ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) )
= bot_bot_set_nat )
=> ( ( proj_tuple_a @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ ( merge_Sum_sum_a_nat @ Nxs @ Mys ) )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Nxs ) ) ) ) ) ).
% proj_tuple_merge
thf(fact_636_assms_I8_J,axiom,
member408289922725080238_a_nat @ xs @ ( image_5081948215111134021_a_nat @ ( fo_nmlz_a @ ad ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ ( ext_tuple_a @ ad @ fv_sub @ fv_sub_comp ) @ ass ) ) ) ).
% assms(8)
thf(fact_637_inj__on__Un__image__eq__iff,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
=> ( ( ( image_5081948215111134021_a_nat @ F @ A )
= ( image_5081948215111134021_a_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_638_inj__on__Un__image__eq__iff,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inj_on7795941589172611729_a_nat @ F @ ( sup_su4083067149120280889_a_nat @ A @ B ) )
=> ( ( ( image_7676043921908783909_a_nat @ F @ A )
= ( image_7676043921908783909_a_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_639_inj__on__Un__image__eq__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_640_comp__apply,axiom,
( comp_o2721059924169077421at_nat
= ( ^ [F2: option_Sum_sum_a_nat > sum_sum_a_nat,G2: nat > option_Sum_sum_a_nat,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_641_comp__apply,axiom,
( comp_nat_nat_nat
= ( ^ [F2: nat > nat,G2: nat > nat,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_apply
thf(fact_642_List_Omap_Ocomp,axiom,
! [F: nat > nat,G: nat > nat] :
( ( comp_l7223822213492037765st_nat @ ( map_nat_nat @ F ) @ ( map_nat_nat @ G ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_643_List_Omap_Ocomp,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat] :
( ( comp_l5575366868326257725st_nat @ ( map_op8672217432457595070_a_nat @ F ) @ ( map_na4778969939102885679_a_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_o2721059924169077421at_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_644_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_645_List_Omap_Ocomp,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( comp_l6851066656763219287_a_nat @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_Pr2515564165894599445_a_nat @ G ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_S8059809979792035783_a_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_646_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_647_List_Omap_Ocomp,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat] :
( ( comp_l7562833077401626166_a_nat @ ( map_na823391071729141993_a_nat @ F ) @ ( map_Pr2058079554187054650at_nat @ G ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_n6276394432912108976_a_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_648_List_Omap_Ocomp,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: nat > produc5454855657571582244_a_nat] :
( ( comp_l6946783421279048718st_nat @ ( map_Pr2515564165894599445_a_nat @ F ) @ ( map_na4983825868277236666_a_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_P2433158451856970632at_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_649_List_Omap_Ocomp,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > produc5454855657571582244_a_nat] :
( ( comp_l1389448023631161314_a_nat @ ( map_Pr2515564165894599445_a_nat @ F ) @ ( map_Pr4115244106412896910_a_nat @ G ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_P9187809017694532828_a_nat @ F @ G ) ) ) ).
% List.map.comp
thf(fact_650_map__comp__map,axiom,
! [F: nat > nat,G: nat > nat] :
( ( comp_l7223822213492037765st_nat @ ( map_nat_nat @ F ) @ ( map_nat_nat @ G ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_651_map__comp__map,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat] :
( ( comp_l5575366868326257725st_nat @ ( map_op8672217432457595070_a_nat @ F ) @ ( map_na4778969939102885679_a_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_o2721059924169077421at_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_652_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_653_map__comp__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( ( comp_l6851066656763219287_a_nat @ ( map_Su2790769393171190532_a_nat @ F ) @ ( map_Pr2515564165894599445_a_nat @ G ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_S8059809979792035783_a_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_654_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_655_map__comp__map,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat] :
( ( comp_l7562833077401626166_a_nat @ ( map_na823391071729141993_a_nat @ F ) @ ( map_Pr2058079554187054650at_nat @ G ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_n6276394432912108976_a_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_656_map__comp__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: nat > produc5454855657571582244_a_nat] :
( ( comp_l6946783421279048718st_nat @ ( map_Pr2515564165894599445_a_nat @ F ) @ ( map_na4983825868277236666_a_nat @ G ) )
= ( map_na823391071729141993_a_nat @ ( comp_P2433158451856970632at_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_657_map__comp__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > produc5454855657571582244_a_nat] :
( ( comp_l1389448023631161314_a_nat @ ( map_Pr2515564165894599445_a_nat @ F ) @ ( map_Pr4115244106412896910_a_nat @ G ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_P9187809017694532828_a_nat @ F @ G ) ) ) ).
% map_comp_map
thf(fact_658_list_Omap__comp,axiom,
! [G: nat > nat,F: nat > nat,V: list_nat] :
( ( map_nat_nat @ G @ ( map_nat_nat @ F @ V ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_659_list_Omap__comp,axiom,
! [G: option_Sum_sum_a_nat > sum_sum_a_nat,F: nat > option_Sum_sum_a_nat,V: list_nat] :
( ( map_op8672217432457595070_a_nat @ G @ ( map_na4778969939102885679_a_nat @ F @ V ) )
= ( map_na823391071729141993_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_660_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_661_list_Omap__comp,axiom,
! [G: sum_sum_a_nat > sum_sum_a_nat,F: produc5454855657571582244_a_nat > sum_sum_a_nat,V: list_P5056861408695629236_a_nat] :
( ( map_Su2790769393171190532_a_nat @ G @ ( map_Pr2515564165894599445_a_nat @ F @ V ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_S8059809979792035783_a_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_662_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_663_list_Omap__comp,axiom,
! [G: nat > sum_sum_a_nat,F: produc5454855657571582244_a_nat > nat,V: list_P5056861408695629236_a_nat] :
( ( map_na823391071729141993_a_nat @ G @ ( map_Pr2058079554187054650at_nat @ F @ V ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_n6276394432912108976_a_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_664_list_Omap__comp,axiom,
! [G: produc5454855657571582244_a_nat > sum_sum_a_nat,F: nat > produc5454855657571582244_a_nat,V: list_nat] :
( ( map_Pr2515564165894599445_a_nat @ G @ ( map_na4983825868277236666_a_nat @ F @ V ) )
= ( map_na823391071729141993_a_nat @ ( comp_P2433158451856970632at_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_665_list_Omap__comp,axiom,
! [G: produc5454855657571582244_a_nat > sum_sum_a_nat,F: produc5454855657571582244_a_nat > produc5454855657571582244_a_nat,V: list_P5056861408695629236_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ G @ ( map_Pr4115244106412896910_a_nat @ F @ V ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_P9187809017694532828_a_nat @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_666_List_Omap_Ocompositionality,axiom,
! [F: nat > nat,G: nat > nat,List: list_nat] :
( ( map_nat_nat @ F @ ( map_nat_nat @ G @ List ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_667_List_Omap_Ocompositionality,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,List: list_nat] :
( ( map_op8672217432457595070_a_nat @ F @ ( map_na4778969939102885679_a_nat @ G @ List ) )
= ( map_na823391071729141993_a_nat @ ( comp_o2721059924169077421at_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_668_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_669_List_Omap_Ocompositionality,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat,List: list_P5056861408695629236_a_nat] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_Pr2515564165894599445_a_nat @ G @ List ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_S8059809979792035783_a_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_670_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_671_List_Omap_Ocompositionality,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,List: list_P5056861408695629236_a_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( map_Pr2058079554187054650at_nat @ G @ List ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_n6276394432912108976_a_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_672_List_Omap_Ocompositionality,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: nat > produc5454855657571582244_a_nat,List: list_nat] :
( ( map_Pr2515564165894599445_a_nat @ F @ ( map_na4983825868277236666_a_nat @ G @ List ) )
= ( map_na823391071729141993_a_nat @ ( comp_P2433158451856970632at_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_673_List_Omap_Ocompositionality,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > produc5454855657571582244_a_nat,List: list_P5056861408695629236_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ F @ ( map_Pr4115244106412896910_a_nat @ G @ List ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_P9187809017694532828_a_nat @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_674_map__map,axiom,
! [F: nat > nat,G: nat > nat,Xs: list_nat] :
( ( map_nat_nat @ F @ ( map_nat_nat @ G @ Xs ) )
= ( map_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_675_map__map,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,Xs: list_nat] :
( ( map_op8672217432457595070_a_nat @ F @ ( map_na4778969939102885679_a_nat @ G @ Xs ) )
= ( map_na823391071729141993_a_nat @ ( comp_o2721059924169077421at_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_676_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_677_map__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( map_Su2790769393171190532_a_nat @ F @ ( map_Pr2515564165894599445_a_nat @ G @ Xs ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_S8059809979792035783_a_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_678_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_679_map__map,axiom,
! [F: nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > nat,Xs: list_P5056861408695629236_a_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( map_Pr2058079554187054650at_nat @ G @ Xs ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_n6276394432912108976_a_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_680_map__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: nat > produc5454855657571582244_a_nat,Xs: list_nat] :
( ( map_Pr2515564165894599445_a_nat @ F @ ( map_na4983825868277236666_a_nat @ G @ Xs ) )
= ( map_na823391071729141993_a_nat @ ( comp_P2433158451856970632at_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_681_map__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,G: produc5454855657571582244_a_nat > produc5454855657571582244_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ F @ ( map_Pr4115244106412896910_a_nat @ G @ Xs ) )
= ( map_Pr2515564165894599445_a_nat @ ( comp_P9187809017694532828_a_nat @ F @ G ) @ Xs ) ) ).
% map_map
thf(fact_682_image__comp,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,R: set_nat] :
( ( image_6550528888111247359_a_nat @ F @ ( image_2070734891404281902_a_nat @ G @ R ) )
= ( image_7293268710728258664_a_nat @ ( comp_o2721059924169077421at_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_683_image__comp,axiom,
! [F: nat > nat,G: nat > nat,R: set_nat] :
( ( image_nat_nat @ F @ ( image_nat_nat @ G @ R ) )
= ( image_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_684_image__comp,axiom,
! [F: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,R: set_li6526943997496501093_a_nat] :
( ( image_1711104591301794661_a_nat @ F @ ( image_7676043921908783909_a_nat @ G @ R ) )
= ( image_5081948215111134021_a_nat @ ( comp_s2318311659520683906_a_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_685_image__comp,axiom,
! [F: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,R: set_li6526943997496501093_a_nat] :
( ( image_3472601871771700037_a_nat @ F @ ( image_7676043921908783909_a_nat @ G @ R ) )
= ( image_7676043921908783909_a_nat @ ( comp_s3377174507952090018_a_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_686_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,R: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ R ) )
= ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_687_image__comp,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,R: set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ ( image_5081948215111134021_a_nat @ G @ R ) )
= ( image_7676043921908783909_a_nat @ ( comp_l6800670225517141954_a_nat @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_688_inj__on__imageI2,axiom,
! [F3: option_Sum_sum_a_nat > sum_sum_a_nat,F: nat > option_Sum_sum_a_nat,A: set_nat] :
( ( inj_on4348161877322679292_a_nat @ ( comp_o2721059924169077421at_nat @ F3 @ F ) @ A )
=> ( inj_on8649581803800938946_a_nat @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_689_inj__on__imageI2,axiom,
! [F3: nat > nat,F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_imageI2
thf(fact_690_image__eq__imp__comp,axiom,
! [F: nat > option_Sum_sum_a_nat,A: set_nat,G: nat > option_Sum_sum_a_nat,B: set_nat,H: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( ( image_2070734891404281902_a_nat @ F @ A )
= ( image_2070734891404281902_a_nat @ G @ B ) )
=> ( ( image_7293268710728258664_a_nat @ ( comp_o2721059924169077421at_nat @ H @ F ) @ A )
= ( image_7293268710728258664_a_nat @ ( comp_o2721059924169077421at_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_691_image__eq__imp__comp,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat,B: set_nat,H: nat > nat] :
( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ G @ B ) )
=> ( ( image_nat_nat @ ( comp_nat_nat_nat @ H @ F ) @ A )
= ( image_nat_nat @ ( comp_nat_nat_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_692_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: 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 @ A )
= ( image_5081948215111134021_a_nat @ G @ B ) )
=> ( ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ H @ F ) @ A )
= ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_693_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: 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 > set_li6526943997496501093_a_nat] :
( ( ( image_5081948215111134021_a_nat @ F @ A )
= ( image_5081948215111134021_a_nat @ G @ B ) )
=> ( ( image_7676043921908783909_a_nat @ ( comp_l6800670225517141954_a_nat @ H @ F ) @ A )
= ( image_7676043921908783909_a_nat @ ( comp_l6800670225517141954_a_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_694_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,H: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( ( image_7676043921908783909_a_nat @ F @ A )
= ( image_7676043921908783909_a_nat @ G @ B ) )
=> ( ( image_5081948215111134021_a_nat @ ( comp_s2318311659520683906_a_nat @ H @ F ) @ A )
= ( image_5081948215111134021_a_nat @ ( comp_s2318311659520683906_a_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_695_image__eq__imp__comp,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,H: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat] :
( ( ( image_7676043921908783909_a_nat @ F @ A )
= ( image_7676043921908783909_a_nat @ G @ B ) )
=> ( ( image_7676043921908783909_a_nat @ ( comp_s3377174507952090018_a_nat @ H @ F ) @ A )
= ( image_7676043921908783909_a_nat @ ( comp_s3377174507952090018_a_nat @ H @ G ) @ B ) ) ) ).
% image_eq_imp_comp
thf(fact_696_comp__eq__dest__lhs,axiom,
! [A2: option_Sum_sum_a_nat > sum_sum_a_nat,B2: nat > option_Sum_sum_a_nat,C2: nat > sum_sum_a_nat,V: nat] :
( ( ( comp_o2721059924169077421at_nat @ A2 @ B2 )
= C2 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_697_comp__eq__dest__lhs,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A2 @ B2 )
= C2 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_698_comp__eq__elim,axiom,
! [A2: option_Sum_sum_a_nat > sum_sum_a_nat,B2: nat > option_Sum_sum_a_nat,C2: option_Sum_sum_a_nat > sum_sum_a_nat,D: nat > option_Sum_sum_a_nat] :
( ( ( comp_o2721059924169077421at_nat @ A2 @ B2 )
= ( comp_o2721059924169077421at_nat @ C2 @ D ) )
=> ! [V2: nat] :
( ( A2 @ ( B2 @ V2 ) )
= ( C2 @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_699_comp__eq__elim,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat,D: nat > nat] :
( ( ( comp_nat_nat_nat @ A2 @ B2 )
= ( comp_nat_nat_nat @ C2 @ D ) )
=> ! [V2: nat] :
( ( A2 @ ( B2 @ V2 ) )
= ( C2 @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_700_comp__eq__dest,axiom,
! [A2: option_Sum_sum_a_nat > sum_sum_a_nat,B2: nat > option_Sum_sum_a_nat,C2: option_Sum_sum_a_nat > sum_sum_a_nat,D: nat > option_Sum_sum_a_nat,V: nat] :
( ( ( comp_o2721059924169077421at_nat @ A2 @ B2 )
= ( comp_o2721059924169077421at_nat @ C2 @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_701_comp__eq__dest,axiom,
! [A2: nat > nat,B2: nat > nat,C2: nat > nat,D: nat > nat,V: nat] :
( ( ( comp_nat_nat_nat @ A2 @ B2 )
= ( comp_nat_nat_nat @ C2 @ D ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_702_comp__assoc,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,H: nat > nat] :
( ( comp_n1522318729830440540at_nat @ ( comp_o2721059924169077421at_nat @ F @ G ) @ H )
= ( comp_o2721059924169077421at_nat @ F @ ( comp_n848383228264675746at_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_703_comp__assoc,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat,H: nat > option_Sum_sum_a_nat] :
( ( comp_o2721059924169077421at_nat @ ( comp_S2748962947160188642_a_nat @ F @ G ) @ H )
= ( comp_S2395038772458240499at_nat @ F @ ( comp_o2721059924169077421at_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_704_comp__assoc,axiom,
! [F: option_Sum_sum_a_nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > option_Sum_sum_a_nat,H: nat > option_Sum_sum_a_nat] :
( ( comp_o2721059924169077421at_nat @ ( comp_o8686311273808465576_a_nat @ F @ G ) @ H )
= ( comp_o2721059924169077421at_nat @ F @ ( comp_o5487264423576322163at_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_705_comp__assoc,axiom,
! [F: nat > nat,G: nat > nat,H: nat > nat] :
( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ G ) @ H )
= ( comp_nat_nat_nat @ F @ ( comp_nat_nat_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_706_comp__def,axiom,
( comp_o2721059924169077421at_nat
= ( ^ [F2: option_Sum_sum_a_nat > sum_sum_a_nat,G2: nat > option_Sum_sum_a_nat,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_707_comp__def,axiom,
( comp_nat_nat_nat
= ( ^ [F2: nat > nat,G2: nat > nat,X: nat] : ( F2 @ ( G2 @ X ) ) ) ) ).
% comp_def
thf(fact_708_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_709_bot__set__def,axiom,
( bot_bot_set_o
= ( collect_o @ bot_bot_o_o ) ) ).
% bot_set_def
thf(fact_710_comp__inj__on__iff,axiom,
! [F: nat > option_Sum_sum_a_nat,A: set_nat,F3: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( inj_on8649581803800938946_a_nat @ F @ A )
=> ( ( inj_on5841394699700873579_a_nat @ F3 @ ( image_2070734891404281902_a_nat @ F @ A ) )
= ( inj_on4348161877322679292_a_nat @ ( comp_o2721059924169077421at_nat @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_711_comp__inj__on__iff,axiom,
! [F: nat > nat,A: set_nat,F3: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ F3 @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ ( comp_nat_nat_nat @ F3 @ F ) @ A ) ) ) ).
% comp_inj_on_iff
thf(fact_712_inj__on__imageI,axiom,
! [G: option_Sum_sum_a_nat > sum_sum_a_nat,F: nat > option_Sum_sum_a_nat,A: set_nat] :
( ( inj_on4348161877322679292_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ A )
=> ( inj_on5841394699700873579_a_nat @ G @ ( image_2070734891404281902_a_nat @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_713_inj__on__imageI,axiom,
! [G: nat > nat,F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A )
=> ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) ) ).
% inj_on_imageI
thf(fact_714_comp__inj__on,axiom,
! [F: nat > option_Sum_sum_a_nat,A: set_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( inj_on8649581803800938946_a_nat @ F @ A )
=> ( ( inj_on5841394699700873579_a_nat @ G @ ( image_2070734891404281902_a_nat @ F @ A ) )
=> ( inj_on4348161877322679292_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_715_comp__inj__on,axiom,
! [F: nat > nat,A: set_nat,G: nat > nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
=> ( inj_on_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% comp_inj_on
thf(fact_716_inf__Sup,axiom,
! [A2: set_nat,B: set_set_nat] :
( ( inf_inf_set_nat @ A2 @ ( comple7399068483239264473et_nat @ B ) )
= ( comple7399068483239264473et_nat @ ( image_7916887816326733075et_nat @ ( inf_inf_set_nat @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_717_inf__Sup,axiom,
! [A2: set_li6526943997496501093_a_nat,B: set_se4330304633200676677_a_nat] :
( ( inf_in3249246906714053971_a_nat @ A2 @ ( comple1686717674086456018_a_nat @ B ) )
= ( comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ ( inf_in3249246906714053971_a_nat @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_718_inf__Sup,axiom,
! [A2: $o,B: set_o] :
( ( inf_inf_o @ A2 @ ( complete_Sup_Sup_o @ B ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ ( inf_inf_o @ A2 ) @ B ) ) ) ).
% inf_Sup
thf(fact_719_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_nat,A2: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ B ) @ A2 )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ B )
=> ( ( inf_inf_set_nat @ X @ A2 )
= bot_bot_set_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_720_Sup__inf__eq__bot__iff,axiom,
! [B: set_set_o,A2: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ B ) @ A2 )
= bot_bot_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ B )
=> ( ( inf_inf_set_o @ X @ A2 )
= bot_bot_set_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_721_Sup__inf__eq__bot__iff,axiom,
! [B: set_se4330304633200676677_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ B ) @ A2 )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ B )
=> ( ( inf_in3249246906714053971_a_nat @ X @ A2 )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_722_Sup__inf__eq__bot__iff,axiom,
! [B: set_o,A2: $o] :
( ( ( inf_inf_o @ ( complete_Sup_Sup_o @ B ) @ A2 )
= bot_bot_o )
= ( ! [X: $o] :
( ( member_o @ X @ B )
=> ( ( inf_inf_o @ X @ A2 )
= bot_bot_o ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_723_inj__on__mapI,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,A: set_li5408732054060274282_a_nat] :
( ( inj_on663950023194574504_a_nat @ F @ ( comple3314747135086582061_a_nat @ ( image_3068022823300244885_a_nat @ set_Pr2196415342531872975_a_nat @ A ) ) )
=> ( inj_on543452247319097886_a_nat @ ( map_Pr2515564165894599445_a_nat @ F ) @ A ) ) ).
% inj_on_mapI
thf(fact_724_inj__on__mapI,axiom,
! [F: nat > nat,A: set_list_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ ( image_1775855109352712557et_nat @ set_nat2 @ A ) ) )
=> ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ A ) ) ).
% inj_on_mapI
thf(fact_725_inj__on__mapI,axiom,
! [F: nat > sum_sum_a_nat,A: set_list_nat] :
( ( inj_on4348161877322679292_a_nat @ F @ ( comple7399068483239264473et_nat @ ( image_1775855109352712557et_nat @ set_nat2 @ A ) ) )
=> ( inj_on1175180479830800114_a_nat @ ( map_na823391071729141993_a_nat @ F ) @ A ) ) ).
% inj_on_mapI
thf(fact_726_pair__list__eqI,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs )
= ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Ys ) )
=> ( ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Xs )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_727_zip__map__fst__snd,axiom,
! [Zs: list_P5056861408695629236_a_nat] :
( ( zip_na2013496608136855606_a_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Zs ) @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Zs ) )
= Zs ) ).
% zip_map_fst_snd
thf(fact_728_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_729_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_730_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B
= ( inf_inf_set_nat @ K @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_731_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( inf_inf_set_nat @ K @ A2 ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_732_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( G @ ( F @ X4 ) )
= X4 ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_on_inverseI
thf(fact_733_inj__on__contraD,axiom,
! [F: nat > nat,A: set_nat,X3: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( X3 != Y3 )
=> ( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( F @ X3 )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_734_inj__on__eq__iff,axiom,
! [F: nat > nat,A: set_nat,X3: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_735_inj__on__cong,axiom,
! [A: set_nat,F: nat > nat,G: nat > nat] :
( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ( ( F @ A6 )
= ( G @ A6 ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_736_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F2: nat > nat,A3: set_nat] :
! [X: nat] :
( ( member_nat @ X @ A3 )
=> ! [Y2: nat] :
( ( member_nat @ Y2 @ A3 )
=> ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
=> ( X = Y2 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_737_inj__onI,axiom,
! [A: set_nat,F: nat > nat] :
( ! [X4: nat,Y: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F @ X4 )
= ( F @ Y ) )
=> ( X4 = Y ) ) ) )
=> ( inj_on_nat_nat @ F @ A ) ) ).
% inj_onI
thf(fact_738_inj__onD,axiom,
! [F: nat > nat,A: set_nat,X3: nat,Y3: nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( ( member_nat @ X3 @ A )
=> ( ( member_nat @ Y3 @ A )
=> ( X3 = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_739_proj__tuple__idle,axiom,
! [Nxs: list_P5056861408695629236_a_nat] :
( ( proj_tuple_a @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ Nxs )
= ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Nxs ) ) ).
% proj_tuple_idle
thf(fact_740_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_nat] :
( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_741_boolean__algebra_Odisj__zero__right,axiom,
! [X3: set_o] :
( ( sup_sup_set_o @ X3 @ bot_bot_set_o )
= X3 ) ).
% boolean_algebra.disj_zero_right
thf(fact_742_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ Z2 ) @ X3 )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ X3 ) @ ( sup_sup_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_743_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y3: set_nat,Z2: set_nat,X3: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y3 @ Z2 ) @ X3 )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y3 @ X3 ) @ ( inf_inf_set_nat @ Z2 @ X3 ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_744_boolean__algebra_Odisj__conj__distrib,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y3 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y3 ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_745_boolean__algebra_Oconj__disj__distrib,axiom,
! [X3: set_nat,Y3: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y3 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y3 ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_746_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ( G @ ( F @ X4 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X4 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_747_subset__inj__on,axiom,
! [F: nat > nat,B: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_748_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( inj_on_nat_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_749_inj__on__Int,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ( inj_on_nat_nat @ F @ A )
| ( inj_on_nat_nat @ F @ B ) )
=> ( inj_on_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) ) ).
% inj_on_Int
thf(fact_750_subset__image__inj,axiom,
! [S: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ S @ ( image_5081948215111134021_a_nat @ F @ T3 ) )
= ( ? [U: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ U @ T3 )
& ( inj_on6609798167860701873_a_nat @ F @ U )
& ( S
= ( image_5081948215111134021_a_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_751_subset__image__inj,axiom,
! [S: set_se4330304633200676677_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,T3: set_li6526943997496501093_a_nat] :
( ( ord_le8138476598237931237_a_nat @ S @ ( image_7676043921908783909_a_nat @ F @ T3 ) )
= ( ? [U: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ U @ T3 )
& ( inj_on7795941589172611729_a_nat @ F @ U )
& ( S
= ( image_7676043921908783909_a_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_752_subset__image__inj,axiom,
! [S: set_nat,F: nat > nat,T3: set_nat] :
( ( ord_less_eq_set_nat @ S @ ( image_nat_nat @ F @ T3 ) )
= ( ? [U: set_nat] :
( ( ord_less_eq_set_nat @ U @ T3 )
& ( inj_on_nat_nat @ F @ U )
& ( S
= ( image_nat_nat @ F @ U ) ) ) ) ) ).
% subset_image_inj
thf(fact_753_inj__on__image__mem__iff,axiom,
! [F: $o > nat,B: set_o,A2: $o,A: set_o] :
( ( inj_on_o_nat @ F @ B )
=> ( ( member_o @ A2 @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_o_nat @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_754_inj__on__image__mem__iff,axiom,
! [F: $o > $o,B: set_o,A2: $o,A: set_o] :
( ( inj_on_o_o @ F @ B )
=> ( ( member_o @ A2 @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ( member_o @ ( F @ A2 ) @ ( image_o_o @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_755_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_756_inj__on__image__mem__iff,axiom,
! [F: nat > $o,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_o @ F @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_o @ ( F @ A2 ) @ ( image_nat_o @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_757_inj__on__image__mem__iff,axiom,
! [F: list_Sum_sum_a_nat > nat,B: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inj_on6397736275557941668at_nat @ F @ B )
=> ( ( member408289922725080238_a_nat @ A2 @ B )
=> ( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_2535339886381165584at_nat @ F @ A ) )
= ( member408289922725080238_a_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_758_inj__on__image__mem__iff,axiom,
! [F: list_Sum_sum_a_nat > $o,B: set_li6526943997496501093_a_nat,A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inj_on2262799350278082756_nat_o @ F @ B )
=> ( ( member408289922725080238_a_nat @ A2 @ B )
=> ( ( ord_le1147066620699065093_a_nat @ A @ B )
=> ( ( member_o @ ( F @ A2 ) @ ( image_3984946558445957976_nat_o @ F @ A ) )
= ( member408289922725080238_a_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_759_inj__on__image__mem__iff,axiom,
! [F: $o > list_Sum_sum_a_nat,B: set_o,A2: $o,A: set_o] :
( ( inj_on8234360183494294622_a_nat @ F @ B )
=> ( ( member_o @ A2 @ B )
=> ( ( ord_less_eq_set_o @ A @ B )
=> ( ( member408289922725080238_a_nat @ ( F @ A2 ) @ ( image_733135354807394034_a_nat @ F @ A ) )
= ( member_o @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_760_inj__on__image__mem__iff,axiom,
! [F: nat > list_Sum_sum_a_nat,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on901614105087147266_a_nat @ F @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member408289922725080238_a_nat @ ( F @ A2 ) @ ( image_6262589752765146990_a_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_761_inj__on__image__mem__iff,axiom,
! [F: ( nat > sum_sum_a_nat ) > nat,B: set_na3699693778330250182_a_nat,A2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat] :
( ( inj_on3141676652335697285at_nat @ F @ B )
=> ( ( member8690443509505302927_a_nat @ A2 @ B )
=> ( ( ord_le8108555184339247974_a_nat @ A @ B )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_5786201776793816049at_nat @ F @ A ) )
= ( member8690443509505302927_a_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_762_inj__on__image__mem__iff,axiom,
! [F: ( nat > sum_sum_a_nat ) > $o,B: set_na3699693778330250182_a_nat,A2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat] :
( ( inj_on1633075686209370787_nat_o @ F @ B )
=> ( ( member8690443509505302927_a_nat @ A2 @ B )
=> ( ( ord_le8108555184339247974_a_nat @ A @ B )
=> ( ( member_o @ ( F @ A2 ) @ ( image_2376713081370839351_nat_o @ F @ A ) )
= ( member8690443509505302927_a_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_763_inj__on__image__eq__iff,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,C: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F @ C )
=> ( ( ord_le1147066620699065093_a_nat @ A @ C )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C )
=> ( ( ( image_5081948215111134021_a_nat @ F @ A )
= ( image_5081948215111134021_a_nat @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_764_inj__on__image__eq__iff,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,C: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( inj_on7795941589172611729_a_nat @ F @ C )
=> ( ( ord_le1147066620699065093_a_nat @ A @ C )
=> ( ( ord_le1147066620699065093_a_nat @ B @ C )
=> ( ( ( image_7676043921908783909_a_nat @ F @ A )
= ( image_7676043921908783909_a_nat @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_765_inj__on__image__eq__iff,axiom,
! [F: nat > nat,C: set_nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ C )
=> ( ( ord_less_eq_set_nat @ A @ C )
=> ( ( ord_less_eq_set_nat @ B @ C )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B ) )
= ( A = B ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_766_Sup__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Sup_empty
thf(fact_767_Sup__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Sup_empty
thf(fact_768_Sup__empty,axiom,
( ( comple1686717674086456018_a_nat @ bot_bo3237059034911209905_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% Sup_empty
thf(fact_769_Sup__empty,axiom,
( ( complete_Sup_Sup_o @ bot_bot_set_o )
= bot_bot_o ) ).
% Sup_empty
thf(fact_770_UN__ball__bex__simps_I4_J,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
& ( P @ X ) ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ? [Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y2 @ ( B @ X ) )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_771_UN__ball__bex__simps_I2_J,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
=> ( P @ X ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ! [Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y2 @ ( B @ X ) )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_772_bex__UN,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
& ( P @ X ) ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ? [Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y2 @ ( B @ X ) )
& ( P @ Y2 ) ) ) ) ) ).
% bex_UN
thf(fact_773_ball__UN,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
=> ( P @ X ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ! [Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y2 @ ( B @ X ) )
=> ( P @ Y2 ) ) ) ) ) ).
% ball_UN
thf(fact_774_Union__Un__distrib,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_775_Union__Un__distrib,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( comple1686717674086456018_a_nat @ ( sup_su499249268922660121_a_nat @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ ( comple1686717674086456018_a_nat @ B ) ) ) ).
% Union_Un_distrib
thf(fact_776_UN__ball__bex__simps_I3_J,axiom,
! [A: set_se4330304633200676677_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ A ) )
& ( P @ X ) ) )
= ( ? [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A )
& ? [Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y2 @ X )
& ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_777_UN__ball__bex__simps_I1_J,axiom,
! [A: set_se4330304633200676677_a_nat,P: list_Sum_sum_a_nat > $o] :
( ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ A ) )
=> ( P @ X ) ) )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A )
=> ! [Y2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Y2 @ X )
=> ( P @ Y2 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_778_UnionI,axiom,
! [X5: set_na3699693778330250182_a_nat,C: set_se5822283258546872870_a_nat,A: nat > sum_sum_a_nat] :
( ( member3060896489619847151_a_nat @ X5 @ C )
=> ( ( member8690443509505302927_a_nat @ A @ X5 )
=> ( member8690443509505302927_a_nat @ A @ ( comple8643769897048643123_a_nat @ C ) ) ) ) ).
% UnionI
thf(fact_779_UnionI,axiom,
! [X5: set_nat,C: set_set_nat,A: nat] :
( ( member_set_nat @ X5 @ C )
=> ( ( member_nat @ A @ X5 )
=> ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) ) ) ) ).
% UnionI
thf(fact_780_UnionI,axiom,
! [X5: set_o,C: set_set_o,A: $o] :
( ( member_set_o @ X5 @ C )
=> ( ( member_o @ A @ X5 )
=> ( member_o @ A @ ( comple90263536869209701_set_o @ C ) ) ) ) ).
% UnionI
thf(fact_781_UnionI,axiom,
! [X5: set_li6526943997496501093_a_nat,C: set_se4330304633200676677_a_nat,A: list_Sum_sum_a_nat] :
( ( member5553968465346197646_a_nat @ X5 @ C )
=> ( ( member408289922725080238_a_nat @ A @ X5 )
=> ( member408289922725080238_a_nat @ A @ ( comple1686717674086456018_a_nat @ C ) ) ) ) ).
% UnionI
thf(fact_782_Union__iff,axiom,
! [A: nat > sum_sum_a_nat,C: set_se5822283258546872870_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( comple8643769897048643123_a_nat @ C ) )
= ( ? [X: set_na3699693778330250182_a_nat] :
( ( member3060896489619847151_a_nat @ X @ C )
& ( member8690443509505302927_a_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_783_Union__iff,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
= ( ? [X: set_nat] :
( ( member_set_nat @ X @ C )
& ( member_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_784_Union__iff,axiom,
! [A: $o,C: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C ) )
= ( ? [X: set_o] :
( ( member_set_o @ X @ C )
& ( member_o @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_785_Union__iff,axiom,
! [A: list_Sum_sum_a_nat,C: set_se4330304633200676677_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( comple1686717674086456018_a_nat @ C ) )
= ( ? [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ C )
& ( member408289922725080238_a_nat @ A @ X ) ) ) ) ).
% Union_iff
thf(fact_786_Sup__bot__conv_I2_J,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_787_Sup__bot__conv_I2_J,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_788_Sup__bot__conv_I2_J,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( comple1686717674086456018_a_nat @ A ) )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A )
=> ( X = bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_789_Sup__bot__conv_I2_J,axiom,
! [A: set_o] :
( ( bot_bot_o
= ( complete_Sup_Sup_o @ A ) )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( X = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_790_Sup__bot__conv_I1_J,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_791_Sup__bot__conv_I1_J,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = bot_bot_set_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_792_Sup__bot__conv_I1_J,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( ( comple1686717674086456018_a_nat @ A )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A )
=> ( X = bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_793_Sup__bot__conv_I1_J,axiom,
! [A: set_o] :
( ( ( complete_Sup_Sup_o @ A )
= bot_bot_o )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( X = bot_bot_o ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_794_Inf_OINF__cong,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat > list_Sum_sum_a_nat,D2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A = B )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_5081948215111134021_a_nat @ C @ A ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_795_Inf_OINF__cong,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,D2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,Inf: set_se4330304633200676677_a_nat > set_li6526943997496501093_a_nat] :
( ( A = B )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Inf @ ( image_7676043921908783909_a_nat @ C @ A ) )
= ( Inf @ ( image_7676043921908783909_a_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_796_Sup_OSUP__cong,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat > list_Sum_sum_a_nat,D2: list_Sum_sum_a_nat > list_Sum_sum_a_nat,Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat] :
( ( A = B )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_5081948215111134021_a_nat @ C @ A ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_797_Sup_OSUP__cong,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,D2: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,Sup: set_se4330304633200676677_a_nat > set_li6526943997496501093_a_nat] :
( ( A = B )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( Sup @ ( image_7676043921908783909_a_nat @ C @ A ) )
= ( Sup @ ( image_7676043921908783909_a_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_798_UnionE,axiom,
! [A: nat > sum_sum_a_nat,C: set_se5822283258546872870_a_nat] :
( ( member8690443509505302927_a_nat @ A @ ( comple8643769897048643123_a_nat @ C ) )
=> ~ ! [X6: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ A @ X6 )
=> ~ ( member3060896489619847151_a_nat @ X6 @ C ) ) ) ).
% UnionE
thf(fact_799_UnionE,axiom,
! [A: nat,C: set_set_nat] :
( ( member_nat @ A @ ( comple7399068483239264473et_nat @ C ) )
=> ~ ! [X6: set_nat] :
( ( member_nat @ A @ X6 )
=> ~ ( member_set_nat @ X6 @ C ) ) ) ).
% UnionE
thf(fact_800_UnionE,axiom,
! [A: $o,C: set_set_o] :
( ( member_o @ A @ ( comple90263536869209701_set_o @ C ) )
=> ~ ! [X6: set_o] :
( ( member_o @ A @ X6 )
=> ~ ( member_set_o @ X6 @ C ) ) ) ).
% UnionE
thf(fact_801_UnionE,axiom,
! [A: list_Sum_sum_a_nat,C: set_se4330304633200676677_a_nat] :
( ( member408289922725080238_a_nat @ A @ ( comple1686717674086456018_a_nat @ C ) )
=> ~ ! [X6: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ A @ X6 )
=> ~ ( member5553968465346197646_a_nat @ X6 @ C ) ) ) ).
% UnionE
thf(fact_802_Sup__upper2,axiom,
! [U2: set_nat,A: set_set_nat,V: set_nat] :
( ( member_set_nat @ U2 @ A )
=> ( ( ord_less_eq_set_nat @ V @ U2 )
=> ( ord_less_eq_set_nat @ V @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_803_Sup__upper2,axiom,
! [U2: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat,V: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ U2 @ A )
=> ( ( ord_le1147066620699065093_a_nat @ V @ U2 )
=> ( ord_le1147066620699065093_a_nat @ V @ ( comple1686717674086456018_a_nat @ A ) ) ) ) ).
% Sup_upper2
thf(fact_804_Sup__upper2,axiom,
! [U2: $o,A: set_o,V: $o] :
( ( member_o @ U2 @ A )
=> ( ( ord_less_eq_o @ V @ U2 )
=> ( ord_less_eq_o @ V @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% Sup_upper2
thf(fact_805_Sup__le__iff,axiom,
! [A: set_set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ B2 )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( ord_less_eq_set_nat @ X @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_806_Sup__le__iff,axiom,
! [A: set_se4330304633200676677_a_nat,B2: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ B2 )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A )
=> ( ord_le1147066620699065093_a_nat @ X @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_807_Sup__le__iff,axiom,
! [A: set_o,B2: $o] :
( ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ B2 )
= ( ! [X: $o] :
( ( member_o @ X @ A )
=> ( ord_less_eq_o @ X @ B2 ) ) ) ) ).
% Sup_le_iff
thf(fact_808_Sup__upper,axiom,
! [X3: set_nat,A: set_set_nat] :
( ( member_set_nat @ X3 @ A )
=> ( ord_less_eq_set_nat @ X3 @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Sup_upper
thf(fact_809_Sup__upper,axiom,
! [X3: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ X3 @ A )
=> ( ord_le1147066620699065093_a_nat @ X3 @ ( comple1686717674086456018_a_nat @ A ) ) ) ).
% Sup_upper
thf(fact_810_Sup__upper,axiom,
! [X3: $o,A: set_o] :
( ( member_o @ X3 @ A )
=> ( ord_less_eq_o @ X3 @ ( complete_Sup_Sup_o @ A ) ) ) ).
% Sup_upper
thf(fact_811_Sup__least,axiom,
! [A: set_set_nat,Z2: set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_812_Sup__least,axiom,
! [A: set_se4330304633200676677_a_nat,Z2: set_li6526943997496501093_a_nat] :
( ! [X4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ A )
=> ( ord_le1147066620699065093_a_nat @ X4 @ Z2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_813_Sup__least,axiom,
! [A: set_o,Z2: $o] :
( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ Z2 ) ) ).
% Sup_least
thf(fact_814_Sup__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [A6: set_nat] :
( ( member_set_nat @ A6 @ A )
=> ? [X2: set_nat] :
( ( member_set_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ A6 @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_mono
thf(fact_815_Sup__mono,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ! [A6: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ A6 @ A )
=> ? [X2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X2 @ B )
& ( ord_le1147066620699065093_a_nat @ A6 @ X2 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ ( comple1686717674086456018_a_nat @ B ) ) ) ).
% Sup_mono
thf(fact_816_Sup__mono,axiom,
! [A: set_o,B: set_o] :
( ! [A6: $o] :
( ( member_o @ A6 @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_o @ A6 @ X2 ) ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_mono
thf(fact_817_Sup__eqI,axiom,
! [A: set_set_nat,X3: set_nat] :
( ! [Y: set_nat] :
( ( member_set_nat @ Y @ A )
=> ( ord_less_eq_set_nat @ Y @ X3 ) )
=> ( ! [Y: set_nat] :
( ! [Z4: set_nat] :
( ( member_set_nat @ Z4 @ A )
=> ( ord_less_eq_set_nat @ Z4 @ Y ) )
=> ( ord_less_eq_set_nat @ X3 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_818_Sup__eqI,axiom,
! [A: set_se4330304633200676677_a_nat,X3: set_li6526943997496501093_a_nat] :
( ! [Y: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Y @ A )
=> ( ord_le1147066620699065093_a_nat @ Y @ X3 ) )
=> ( ! [Y: set_li6526943997496501093_a_nat] :
( ! [Z4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Z4 @ A )
=> ( ord_le1147066620699065093_a_nat @ Z4 @ Y ) )
=> ( ord_le1147066620699065093_a_nat @ X3 @ Y ) )
=> ( ( comple1686717674086456018_a_nat @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_819_Sup__eqI,axiom,
! [A: set_o,X3: $o] :
( ! [Y: $o] :
( ( member_o @ Y @ A )
=> ( ord_less_eq_o @ Y @ X3 ) )
=> ( ! [Y: $o] :
( ! [Z4: $o] :
( ( member_o @ Z4 @ A )
=> ( ord_less_eq_o @ Z4 @ Y ) )
=> ( ord_less_eq_o @ X3 @ Y ) )
=> ( ( complete_Sup_Sup_o @ A )
= X3 ) ) ) ).
% Sup_eqI
thf(fact_820_Sup_OSUP__image,axiom,
! [Sup: set_Sum_sum_a_nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat,F: nat > option_Sum_sum_a_nat,A: set_nat] :
( ( Sup @ ( image_6550528888111247359_a_nat @ G @ ( image_2070734891404281902_a_nat @ F @ A ) ) )
= ( Sup @ ( image_7293268710728258664_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_821_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: nat > nat,F: nat > nat,A: set_nat] :
( ( Sup @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_822_Sup_OSUP__image,axiom,
! [Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Sup @ ( image_1711104591301794661_a_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A ) ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ ( comp_s2318311659520683906_a_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_823_Sup_OSUP__image,axiom,
! [Sup: set_se4330304633200676677_a_nat > set_li6526943997496501093_a_nat,G: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Sup @ ( image_3472601871771700037_a_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A ) ) )
= ( Sup @ ( image_7676043921908783909_a_nat @ ( comp_s3377174507952090018_a_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_824_Sup_OSUP__image,axiom,
! [Sup: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Sup @ ( image_5081948215111134021_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( Sup @ ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_825_Sup_OSUP__image,axiom,
! [Sup: set_se4330304633200676677_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Sup @ ( image_7676043921908783909_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( Sup @ ( image_7676043921908783909_a_nat @ ( comp_l6800670225517141954_a_nat @ G @ F ) @ A ) ) ) ).
% Sup.SUP_image
thf(fact_826_Inf_OINF__image,axiom,
! [Inf: set_Sum_sum_a_nat > sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat,F: nat > option_Sum_sum_a_nat,A: set_nat] :
( ( Inf @ ( image_6550528888111247359_a_nat @ G @ ( image_2070734891404281902_a_nat @ F @ A ) ) )
= ( Inf @ ( image_7293268710728258664_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_827_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: nat > nat,F: nat > nat,A: set_nat] :
( ( Inf @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_828_Inf_OINF__image,axiom,
! [Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Inf @ ( image_1711104591301794661_a_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A ) ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ ( comp_s2318311659520683906_a_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_829_Inf_OINF__image,axiom,
! [Inf: set_se4330304633200676677_a_nat > set_li6526943997496501093_a_nat,G: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Inf @ ( image_3472601871771700037_a_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A ) ) )
= ( Inf @ ( image_7676043921908783909_a_nat @ ( comp_s3377174507952090018_a_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_830_Inf_OINF__image,axiom,
! [Inf: set_li6526943997496501093_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Inf @ ( image_5081948215111134021_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( Inf @ ( image_5081948215111134021_a_nat @ ( comp_l3689981812903912354_a_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_831_Inf_OINF__image,axiom,
! [Inf: set_se4330304633200676677_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( Inf @ ( image_7676043921908783909_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( Inf @ ( image_7676043921908783909_a_nat @ ( comp_l6800670225517141954_a_nat @ G @ F ) @ A ) ) ) ).
% Inf.INF_image
thf(fact_832_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C: nat > $o,D2: nat > $o] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_833_SUP__cong,axiom,
! [A: set_o,B: set_o,C: $o > $o,D2: $o > $o] :
( ( A = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_834_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C: nat > nat,D2: nat > nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_nat_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_835_SUP__cong,axiom,
! [A: set_o,B: set_o,C: $o > nat,D2: $o > nat] :
( ( A = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_o_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_o_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_836_SUP__cong,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat > $o,D2: list_Sum_sum_a_nat > $o] :
( ( A = B )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_837_SUP__cong,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat,C: list_Sum_sum_a_nat > nat,D2: list_Sum_sum_a_nat > nat] :
( ( A = B )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_838_SUP__cong,axiom,
! [A: set_nat,B: set_nat,C: nat > set_li6526943997496501093_a_nat,D2: nat > set_li6526943997496501093_a_nat] :
( ( A = B )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ C @ A ) )
= ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_839_SUP__cong,axiom,
! [A: set_o,B: set_o,C: $o > set_li6526943997496501093_a_nat,D2: $o > set_li6526943997496501093_a_nat] :
( ( A = B )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_2951602039182858962_a_nat @ C @ A ) )
= ( comple1686717674086456018_a_nat @ ( image_2951602039182858962_a_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_840_SUP__cong,axiom,
! [A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,C: ( nat > sum_sum_a_nat ) > $o,D2: ( nat > sum_sum_a_nat ) > $o] :
( ( A = B )
=> ( ! [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_2376713081370839351_nat_o @ C @ A ) )
= ( complete_Sup_Sup_o @ ( image_2376713081370839351_nat_o @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_841_SUP__cong,axiom,
! [A: set_na3699693778330250182_a_nat,B: set_na3699693778330250182_a_nat,C: ( nat > sum_sum_a_nat ) > nat,D2: ( nat > sum_sum_a_nat ) > nat] :
( ( A = B )
=> ( ! [X4: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X4 @ B )
=> ( ( C @ X4 )
= ( D2 @ X4 ) ) )
=> ( ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ C @ A ) )
= ( complete_Sup_Sup_nat @ ( image_5786201776793816049at_nat @ D2 @ B ) ) ) ) ) ).
% SUP_cong
thf(fact_842_empty__Union__conv,axiom,
! [A: set_set_nat] :
( ( bot_bot_set_nat
= ( comple7399068483239264473et_nat @ A ) )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_843_empty__Union__conv,axiom,
! [A: set_set_o] :
( ( bot_bot_set_o
= ( comple90263536869209701_set_o @ A ) )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = bot_bot_set_o ) ) ) ) ).
% empty_Union_conv
thf(fact_844_empty__Union__conv,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( comple1686717674086456018_a_nat @ A ) )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A )
=> ( X = bot_bo1033123847703346641_a_nat ) ) ) ) ).
% empty_Union_conv
thf(fact_845_Union__empty__conv,axiom,
! [A: set_set_nat] :
( ( ( comple7399068483239264473et_nat @ A )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ A )
=> ( X = bot_bot_set_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_846_Union__empty__conv,axiom,
! [A: set_set_o] :
( ( ( comple90263536869209701_set_o @ A )
= bot_bot_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ A )
=> ( X = bot_bot_set_o ) ) ) ) ).
% Union_empty_conv
thf(fact_847_Union__empty__conv,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( ( comple1686717674086456018_a_nat @ A )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ A )
=> ( X = bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Union_empty_conv
thf(fact_848_Union__empty,axiom,
( ( comple7399068483239264473et_nat @ bot_bot_set_set_nat )
= bot_bot_set_nat ) ).
% Union_empty
thf(fact_849_Union__empty,axiom,
( ( comple90263536869209701_set_o @ bot_bot_set_set_o )
= bot_bot_set_o ) ).
% Union_empty
thf(fact_850_Union__empty,axiom,
( ( comple1686717674086456018_a_nat @ bot_bo3237059034911209905_a_nat )
= bot_bo1033123847703346641_a_nat ) ).
% Union_empty
thf(fact_851_Union__subsetI,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ A )
=> ? [Y6: set_nat] :
( ( member_set_nat @ Y6 @ B )
& ( ord_less_eq_set_nat @ X4 @ Y6 ) ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_852_Union__subsetI,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ! [X4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ A )
=> ? [Y6: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ Y6 @ B )
& ( ord_le1147066620699065093_a_nat @ X4 @ Y6 ) ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ ( comple1686717674086456018_a_nat @ B ) ) ) ).
% Union_subsetI
thf(fact_853_Union__upper,axiom,
! [B: set_nat,A: set_set_nat] :
( ( member_set_nat @ B @ A )
=> ( ord_less_eq_set_nat @ B @ ( comple7399068483239264473et_nat @ A ) ) ) ).
% Union_upper
thf(fact_854_Union__upper,axiom,
! [B: set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ B @ A )
=> ( ord_le1147066620699065093_a_nat @ B @ ( comple1686717674086456018_a_nat @ A ) ) ) ).
% Union_upper
thf(fact_855_Union__least,axiom,
! [A: set_set_nat,C: set_nat] :
( ! [X6: set_nat] :
( ( member_set_nat @ X6 @ A )
=> ( ord_less_eq_set_nat @ X6 @ C ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ C ) ) ).
% Union_least
thf(fact_856_Union__least,axiom,
! [A: set_se4330304633200676677_a_nat,C: set_li6526943997496501093_a_nat] :
( ! [X6: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X6 @ A )
=> ( ord_le1147066620699065093_a_nat @ X6 @ C ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ C ) ) ).
% Union_least
thf(fact_857_Union__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_mono
thf(fact_858_Union__mono,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ A @ B )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ ( comple1686717674086456018_a_nat @ B ) ) ) ).
% Union_mono
thf(fact_859_SUP__eq,axiom,
! [A: set_nat,B: set_nat,F: nat > $o,G: nat > $o] :
( ! [I: nat] :
( ( member_nat @ I @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_860_SUP__eq,axiom,
! [A: set_nat,B: set_o,F: nat > $o,G: $o > $o] :
( ! [I: nat] :
( ( member_nat @ I @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: $o] :
( ( member_o @ J @ B )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_861_SUP__eq,axiom,
! [A: set_o,B: set_nat,F: $o > $o,G: nat > $o] :
( ! [I: $o] :
( ( member_o @ I @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_862_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > $o,G: $o > $o] :
( ! [I: $o] :
( ( member_o @ I @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: $o] :
( ( member_o @ J @ B )
=> ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_863_SUP__eq,axiom,
! [A: set_nat,B: set_nat,F: nat > set_nat,G: nat > set_nat] :
( ! [I: nat] :
( ( member_nat @ I @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_864_SUP__eq,axiom,
! [A: set_nat,B: set_o,F: nat > set_nat,G: $o > set_nat] :
( ! [I: nat] :
( ( member_nat @ I @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: $o] :
( ( member_o @ J @ B )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_865_SUP__eq,axiom,
! [A: set_o,B: set_nat,F: $o > set_nat,G: nat > set_nat] :
( ! [I: $o] :
( ( member_o @ I @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_866_SUP__eq,axiom,
! [A: set_o,B: set_o,F: $o > set_nat,G: $o > set_nat] :
( ! [I: $o] :
( ( member_o @ I @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_set_nat @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: $o] :
( ( member_o @ J @ B )
=> ? [X2: $o] :
( ( member_o @ X2 @ A )
& ( ord_less_eq_set_nat @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) )
= ( comple7399068483239264473et_nat @ ( image_o_set_nat @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_867_SUP__eq,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_nat,F: list_Sum_sum_a_nat > $o,G: nat > $o] :
( ! [I: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: nat] :
( ( member_nat @ J @ B )
=> ? [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_nat_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_868_SUP__eq,axiom,
! [A: set_li6526943997496501093_a_nat,B: set_o,F: list_Sum_sum_a_nat > $o,G: $o > $o] :
( ! [I: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I @ A )
=> ? [X2: $o] :
( ( member_o @ X2 @ B )
& ( ord_less_eq_o @ ( F @ I ) @ ( G @ X2 ) ) ) )
=> ( ! [J: $o] :
( ( member_o @ J @ B )
=> ? [X2: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X2 @ A )
& ( ord_less_eq_o @ ( G @ J ) @ ( F @ X2 ) ) ) )
=> ( ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ F @ A ) )
= ( complete_Sup_Sup_o @ ( image_o_o @ G @ B ) ) ) ) ) ).
% SUP_eq
thf(fact_869_less__eq__Sup,axiom,
! [A: set_set_nat,U2: set_nat] :
( ! [V3: set_nat] :
( ( member_set_nat @ V3 @ A )
=> ( ord_less_eq_set_nat @ U2 @ V3 ) )
=> ( ( A != bot_bot_set_set_nat )
=> ( ord_less_eq_set_nat @ U2 @ ( comple7399068483239264473et_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_870_less__eq__Sup,axiom,
! [A: set_se4330304633200676677_a_nat,U2: set_li6526943997496501093_a_nat] :
( ! [V3: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ V3 @ A )
=> ( ord_le1147066620699065093_a_nat @ U2 @ V3 ) )
=> ( ( A != bot_bo3237059034911209905_a_nat )
=> ( ord_le1147066620699065093_a_nat @ U2 @ ( comple1686717674086456018_a_nat @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_871_less__eq__Sup,axiom,
! [A: set_o,U2: $o] :
( ! [V3: $o] :
( ( member_o @ V3 @ A )
=> ( ord_less_eq_o @ U2 @ V3 ) )
=> ( ( A != bot_bot_set_o )
=> ( ord_less_eq_o @ U2 @ ( complete_Sup_Sup_o @ A ) ) ) ) ).
% less_eq_Sup
thf(fact_872_Sup__subset__mono,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( ord_le6893508408891458716et_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_873_Sup__subset__mono,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ A @ B )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ ( comple1686717674086456018_a_nat @ B ) ) ) ).
% Sup_subset_mono
thf(fact_874_Sup__subset__mono,axiom,
! [A: set_o,B: set_o] :
( ( ord_less_eq_set_o @ A @ B )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_subset_mono
thf(fact_875_SUP__eq__const,axiom,
! [I2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( I2 != bot_bo1033123847703346641_a_nat )
=> ( ! [I: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I @ I2 )
=> ( ( F @ I )
= X3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ I2 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_876_SUP__eq__const,axiom,
! [I2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( I2 != bot_bo6441361344521902642_a_nat )
=> ( ! [I: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I @ I2 )
=> ( ( F @ I )
= X3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_494503752901444484_a_nat @ F @ I2 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_877_SUP__eq__const,axiom,
! [I2: set_nat,F: nat > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ( F @ I )
= X3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ I2 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_878_SUP__eq__const,axiom,
! [I2: set_o,F: $o > set_li6526943997496501093_a_nat,X3: set_li6526943997496501093_a_nat] :
( ( I2 != bot_bot_set_o )
=> ( ! [I: $o] :
( ( member_o @ I @ I2 )
=> ( ( F @ I )
= X3 ) )
=> ( ( comple1686717674086456018_a_nat @ ( image_2951602039182858962_a_nat @ F @ I2 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_879_SUP__eq__const,axiom,
! [I2: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o,X3: $o] :
( ( I2 != bot_bo1033123847703346641_a_nat )
=> ( ! [I: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I @ I2 )
=> ( ( F @ I )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ F @ I2 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_880_SUP__eq__const,axiom,
! [I2: set_na3699693778330250182_a_nat,F: ( nat > sum_sum_a_nat ) > $o,X3: $o] :
( ( I2 != bot_bo6441361344521902642_a_nat )
=> ( ! [I: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I @ I2 )
=> ( ( F @ I )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_2376713081370839351_nat_o @ F @ I2 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_881_SUP__eq__const,axiom,
! [I2: set_nat,F: nat > $o,X3: $o] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ( F @ I )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I2 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_882_SUP__eq__const,axiom,
! [I2: set_o,F: $o > $o,X3: $o] :
( ( I2 != bot_bot_set_o )
=> ( ! [I: $o] :
( ( member_o @ I @ I2 )
=> ( ( F @ I )
= X3 ) )
=> ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I2 ) )
= X3 ) ) ) ).
% SUP_eq_const
thf(fact_883_SUP__image,axiom,
! [G: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A ) ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ ( comp_s3377174507952090018_a_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_884_SUP__image,axiom,
! [G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ ( comp_l6800670225517141954_a_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_885_SUP__image,axiom,
! [G: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ ( comp_l7931591902798318543_a_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_886_SUP__image,axiom,
! [G: set_li6526943997496501093_a_nat > $o,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( complete_Sup_Sup_o @ ( image_6425633437027283832_nat_o @ G @ ( image_7676043921908783909_a_nat @ F @ A ) ) )
= ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ ( comp_s7989947936688375727_a_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_887_SUP__image,axiom,
! [G: nat > nat,F: nat > nat,A: set_nat] :
( ( complete_Sup_Sup_nat @ ( image_nat_nat @ G @ ( image_nat_nat @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_888_SUP__image,axiom,
! [G: list_Sum_sum_a_nat > nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ G @ ( image_5081948215111134021_a_nat @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ ( comp_l6309227251058912209_a_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_889_SUP__image,axiom,
! [G: set_li6526943997496501093_a_nat > nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( complete_Sup_Sup_nat @ ( image_4755997627985507312at_nat @ G @ ( image_7676043921908783909_a_nat @ F @ A ) ) )
= ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ ( comp_s8210616430925637617_a_nat @ G @ F ) @ A ) ) ) ).
% SUP_image
thf(fact_890_Sup__union__distrib,axiom,
! [A: set_set_nat,B: set_set_nat] :
( ( comple7399068483239264473et_nat @ ( sup_sup_set_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_891_Sup__union__distrib,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] :
( ( comple1686717674086456018_a_nat @ ( sup_su499249268922660121_a_nat @ A @ B ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ ( comple1686717674086456018_a_nat @ B ) ) ) ).
% Sup_union_distrib
thf(fact_892_Sup__union__distrib,axiom,
! [A: set_o,B: set_o] :
( ( complete_Sup_Sup_o @ ( sup_sup_set_o @ A @ B ) )
= ( sup_sup_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_union_distrib
thf(fact_893_Union__disjoint,axiom,
! [C: set_set_nat,A: set_nat] :
( ( ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ C ) @ A )
= bot_bot_set_nat )
= ( ! [X: set_nat] :
( ( member_set_nat @ X @ C )
=> ( ( inf_inf_set_nat @ X @ A )
= bot_bot_set_nat ) ) ) ) ).
% Union_disjoint
thf(fact_894_Union__disjoint,axiom,
! [C: set_set_o,A: set_o] :
( ( ( inf_inf_set_o @ ( comple90263536869209701_set_o @ C ) @ A )
= bot_bot_set_o )
= ( ! [X: set_o] :
( ( member_set_o @ X @ C )
=> ( ( inf_inf_set_o @ X @ A )
= bot_bot_set_o ) ) ) ) ).
% Union_disjoint
thf(fact_895_Union__disjoint,axiom,
! [C: set_se4330304633200676677_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ C ) @ A )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X @ C )
=> ( ( inf_in3249246906714053971_a_nat @ X @ A )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% Union_disjoint
thf(fact_896_Union__Int__subset,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_897_Union__Int__subset,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] : ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( inf_in5367731912061063475_a_nat @ A @ B ) ) @ ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ ( comple1686717674086456018_a_nat @ B ) ) ) ).
% Union_Int_subset
thf(fact_898_inj__on__image,axiom,
! [F: nat > nat,A: set_set_nat] :
( ( inj_on_nat_nat @ F @ ( comple7399068483239264473et_nat @ A ) )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_899_inj__on__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_se4330304633200676677_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F @ ( comple1686717674086456018_a_nat @ A ) )
=> ( inj_on561899399213738673_a_nat @ ( image_5081948215111134021_a_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_900_inj__on__image,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat] :
( ( inj_on7795941589172611729_a_nat @ F @ ( comple1686717674086456018_a_nat @ A ) )
=> ( inj_on4382358243947751057_a_nat @ ( image_7676043921908783909_a_nat @ F ) @ A ) ) ).
% inj_on_image
thf(fact_901_SUP__eq__iff,axiom,
! [I2: set_nat,C2: $o,F: nat > $o] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ord_less_eq_o @ C2 @ ( F @ I ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_nat_o @ F @ I2 ) )
= C2 )
= ( ! [X: nat] :
( ( member_nat @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_902_SUP__eq__iff,axiom,
! [I2: set_o,C2: $o,F: $o > $o] :
( ( I2 != bot_bot_set_o )
=> ( ! [I: $o] :
( ( member_o @ I @ I2 )
=> ( ord_less_eq_o @ C2 @ ( F @ I ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_o_o @ F @ I2 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_903_SUP__eq__iff,axiom,
! [I2: set_nat,C2: set_nat,F: nat > set_nat] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ord_less_eq_set_nat @ C2 @ ( F @ I ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ I2 ) )
= C2 )
= ( ! [X: nat] :
( ( member_nat @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_904_SUP__eq__iff,axiom,
! [I2: set_o,C2: set_nat,F: $o > set_nat] :
( ( I2 != bot_bot_set_o )
=> ( ! [I: $o] :
( ( member_o @ I @ I2 )
=> ( ord_less_eq_set_nat @ C2 @ ( F @ I ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ I2 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_905_SUP__eq__iff,axiom,
! [I2: set_li6526943997496501093_a_nat,C2: $o,F: list_Sum_sum_a_nat > $o] :
( ( I2 != bot_bo1033123847703346641_a_nat )
=> ( ! [I: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I @ I2 )
=> ( ord_less_eq_o @ C2 @ ( F @ I ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ F @ I2 ) )
= C2 )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_906_SUP__eq__iff,axiom,
! [I2: set_li6526943997496501093_a_nat,C2: set_nat,F: list_Sum_sum_a_nat > set_nat] :
( ( I2 != bot_bo1033123847703346641_a_nat )
=> ( ! [I: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ I @ I2 )
=> ( ord_less_eq_set_nat @ C2 @ ( F @ I ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ F @ I2 ) )
= C2 )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_907_SUP__eq__iff,axiom,
! [I2: set_nat,C2: set_li6526943997496501093_a_nat,F: nat > set_li6526943997496501093_a_nat] :
( ( I2 != bot_bot_set_nat )
=> ( ! [I: nat] :
( ( member_nat @ I @ I2 )
=> ( ord_le1147066620699065093_a_nat @ C2 @ ( F @ I ) ) )
=> ( ( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ I2 ) )
= C2 )
= ( ! [X: nat] :
( ( member_nat @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_908_SUP__eq__iff,axiom,
! [I2: set_o,C2: set_li6526943997496501093_a_nat,F: $o > set_li6526943997496501093_a_nat] :
( ( I2 != bot_bot_set_o )
=> ( ! [I: $o] :
( ( member_o @ I @ I2 )
=> ( ord_le1147066620699065093_a_nat @ C2 @ ( F @ I ) ) )
=> ( ( ( comple1686717674086456018_a_nat @ ( image_2951602039182858962_a_nat @ F @ I2 ) )
= C2 )
= ( ! [X: $o] :
( ( member_o @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_909_SUP__eq__iff,axiom,
! [I2: set_na3699693778330250182_a_nat,C2: $o,F: ( nat > sum_sum_a_nat ) > $o] :
( ( I2 != bot_bo6441361344521902642_a_nat )
=> ( ! [I: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I @ I2 )
=> ( ord_less_eq_o @ C2 @ ( F @ I ) ) )
=> ( ( ( complete_Sup_Sup_o @ ( image_2376713081370839351_nat_o @ F @ I2 ) )
= C2 )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_910_SUP__eq__iff,axiom,
! [I2: set_na3699693778330250182_a_nat,C2: set_nat,F: ( nat > sum_sum_a_nat ) > set_nat] :
( ( I2 != bot_bo6441361344521902642_a_nat )
=> ( ! [I: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ I @ I2 )
=> ( ord_less_eq_set_nat @ C2 @ ( F @ I ) ) )
=> ( ( ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ F @ I2 ) )
= C2 )
= ( ! [X: nat > sum_sum_a_nat] :
( ( member8690443509505302927_a_nat @ X @ I2 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_911_Sup__inter__less__eq,axiom,
! [A: set_set_nat,B: set_set_nat] : ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( inf_inf_set_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( comple7399068483239264473et_nat @ A ) @ ( comple7399068483239264473et_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_912_Sup__inter__less__eq,axiom,
! [A: set_se4330304633200676677_a_nat,B: set_se4330304633200676677_a_nat] : ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( inf_in5367731912061063475_a_nat @ A @ B ) ) @ ( inf_in3249246906714053971_a_nat @ ( comple1686717674086456018_a_nat @ A ) @ ( comple1686717674086456018_a_nat @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_913_Sup__inter__less__eq,axiom,
! [A: set_o,B: set_o] : ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( inf_inf_set_o @ A @ B ) ) @ ( inf_inf_o @ ( complete_Sup_Sup_o @ A ) @ ( complete_Sup_Sup_o @ B ) ) ) ).
% Sup_inter_less_eq
thf(fact_914_map__of__inject__set,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs ) )
=> ( ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Ys ) )
=> ( ( ( map_of1932138695521331722_a_nat @ Xs )
= ( map_of1932138695521331722_a_nat @ Ys ) )
= ( ( set_Pr2196415342531872975_a_nat @ Xs )
= ( set_Pr2196415342531872975_a_nat @ Ys ) ) ) ) ) ).
% map_of_inject_set
thf(fact_915_map__of__eqI,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs ) )
= ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Ys ) ) )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Xs ) ) )
=> ( ( map_of1932138695521331722_a_nat @ Xs @ X4 )
= ( map_of1932138695521331722_a_nat @ Ys @ X4 ) ) )
=> ( ( map_of1932138695521331722_a_nat @ Xs )
= ( map_of1932138695521331722_a_nat @ Ys ) ) ) ) ).
% map_of_eqI
thf(fact_916_Union__image__empty,axiom,
! [A: set_nat,F: nat > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_917_Union__image__empty,axiom,
! [A: set_nat,F: $o > set_nat] :
( ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ bot_bot_set_o ) ) )
= A ) ).
% Union_image_empty
thf(fact_918_Union__image__empty,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ F @ bot_bo1033123847703346641_a_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_919_Union__image__empty,axiom,
! [A: set_li6526943997496501093_a_nat,F: nat > set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ bot_bot_set_nat ) ) )
= A ) ).
% Union_image_empty
thf(fact_920_Union__image__empty,axiom,
! [A: set_li6526943997496501093_a_nat,F: $o > set_li6526943997496501093_a_nat] :
( ( sup_su4083067149120280889_a_nat @ A @ ( comple1686717674086456018_a_nat @ ( image_2951602039182858962_a_nat @ F @ bot_bot_set_o ) ) )
= A ) ).
% Union_image_empty
thf(fact_921_cSUP__least,axiom,
! [A: set_nat,F: nat > $o,M: $o] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_nat_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_922_cSUP__least,axiom,
! [A: set_o,F: $o > $o,M: $o] :
( ( A != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_o_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_923_cSUP__least,axiom,
! [A: set_nat,F: nat > nat,M: nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_nat_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_924_cSUP__least,axiom,
! [A: set_o,F: $o > nat,M: nat] :
( ( A != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_o_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_925_cSUP__least,axiom,
! [A: set_nat,F: nat > set_nat,M: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_926_cSUP__least,axiom,
! [A: set_o,F: $o > set_nat,M: set_nat] :
( ( A != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_927_cSUP__least,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > $o,M: $o] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ( ord_less_eq_o @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ ( image_3984946558445957976_nat_o @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_928_cSUP__least,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > nat,M: nat] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ ( image_2535339886381165584at_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_929_cSUP__least,axiom,
! [A: set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_nat,M: set_nat] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ! [X4: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ ( F @ X4 ) @ M ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_930_cSUP__least,axiom,
! [A: set_nat,F: nat > set_li6526943997496501093_a_nat,M: set_li6526943997496501093_a_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ord_le1147066620699065093_a_nat @ ( F @ X4 ) @ M ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ F @ A ) ) @ M ) ) ) ).
% cSUP_least
thf(fact_931_type__copy__map__cong0,axiom,
! [M: nat > option_Sum_sum_a_nat,G: nat > nat,X3: nat,N2: option_Sum_sum_a_nat > option_Sum_sum_a_nat,H: nat > option_Sum_sum_a_nat,F: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_n1522318729830440540at_nat @ ( comp_o2721059924169077421at_nat @ F @ M ) @ G @ X3 )
= ( comp_o2721059924169077421at_nat @ ( comp_o8686311273808465576_a_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_932_type__copy__map__cong0,axiom,
! [M: option_Sum_sum_a_nat > option_Sum_sum_a_nat,G: nat > option_Sum_sum_a_nat,X3: nat,N2: nat > option_Sum_sum_a_nat,H: nat > nat,F: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_o2721059924169077421at_nat @ ( comp_o8686311273808465576_a_nat @ F @ M ) @ G @ X3 )
= ( comp_n1522318729830440540at_nat @ ( comp_o2721059924169077421at_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_933_type__copy__map__cong0,axiom,
! [M: nat > nat,G: nat > nat,X3: nat,N2: nat > nat,H: nat > nat,F: nat > nat] :
( ( ( M @ ( G @ X3 ) )
= ( N2 @ ( H @ X3 ) ) )
=> ( ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ M ) @ G @ X3 )
= ( comp_nat_nat_nat @ ( comp_nat_nat_nat @ F @ N2 ) @ H @ X3 ) ) ) ).
% type_copy_map_cong0
thf(fact_934_cSup__eq__maximum,axiom,
! [Z2: set_nat,X5: set_set_nat] :
( ( member_set_nat @ Z2 @ X5 )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_935_cSup__eq__maximum,axiom,
! [Z2: set_li6526943997496501093_a_nat,X5: set_se4330304633200676677_a_nat] :
( ( member5553968465346197646_a_nat @ Z2 @ X5 )
=> ( ! [X4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ X5 )
=> ( ord_le1147066620699065093_a_nat @ X4 @ Z2 ) )
=> ( ( comple1686717674086456018_a_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_936_cSup__eq__maximum,axiom,
! [Z2: $o,X5: set_o] :
( ( member_o @ Z2 @ X5 )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_937_cSup__eq__maximum,axiom,
! [Z2: nat,X5: set_nat] :
( ( member_nat @ Z2 @ X5 )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= Z2 ) ) ) ).
% cSup_eq_maximum
thf(fact_938_map__of__eq__dom,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_of1932138695521331722_a_nat @ Xs )
= ( map_of1932138695521331722_a_nat @ Ys ) )
=> ( ( image_1677410150705809403at_nat @ produc1510075366785312874_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xs ) )
= ( image_1677410150705809403at_nat @ produc1510075366785312874_a_nat @ ( set_Pr2196415342531872975_a_nat @ Ys ) ) ) ) ).
% map_of_eq_dom
thf(fact_939_cSup__eq__non__empty,axiom,
! [X5: set_set_nat,A2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ A2 ) )
=> ( ! [Y: set_nat] :
( ! [X2: set_nat] :
( ( member_set_nat @ X2 @ X5 )
=> ( ord_less_eq_set_nat @ X2 @ Y ) )
=> ( ord_less_eq_set_nat @ A2 @ Y ) )
=> ( ( comple7399068483239264473et_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_940_cSup__eq__non__empty,axiom,
! [X5: set_se4330304633200676677_a_nat,A2: set_li6526943997496501093_a_nat] :
( ( X5 != bot_bo3237059034911209905_a_nat )
=> ( ! [X4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ X5 )
=> ( ord_le1147066620699065093_a_nat @ X4 @ A2 ) )
=> ( ! [Y: set_li6526943997496501093_a_nat] :
( ! [X2: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X2 @ X5 )
=> ( ord_le1147066620699065093_a_nat @ X2 @ Y ) )
=> ( ord_le1147066620699065093_a_nat @ A2 @ Y ) )
=> ( ( comple1686717674086456018_a_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_941_cSup__eq__non__empty,axiom,
! [X5: set_o,A2: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ A2 ) )
=> ( ! [Y: $o] :
( ! [X2: $o] :
( ( member_o @ X2 @ X5 )
=> ( ord_less_eq_o @ X2 @ Y ) )
=> ( ord_less_eq_o @ A2 @ Y ) )
=> ( ( complete_Sup_Sup_o @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_942_cSup__eq__non__empty,axiom,
! [X5: set_nat,A2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ A2 ) )
=> ( ! [Y: nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( ord_less_eq_nat @ X2 @ Y ) )
=> ( ord_less_eq_nat @ A2 @ Y ) )
=> ( ( complete_Sup_Sup_nat @ X5 )
= A2 ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_943_cSup__least,axiom,
! [X5: set_set_nat,Z2: set_nat] :
( ( X5 != bot_bot_set_set_nat )
=> ( ! [X4: set_nat] :
( ( member_set_nat @ X4 @ X5 )
=> ( ord_less_eq_set_nat @ X4 @ Z2 ) )
=> ( ord_less_eq_set_nat @ ( comple7399068483239264473et_nat @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_944_cSup__least,axiom,
! [X5: set_se4330304633200676677_a_nat,Z2: set_li6526943997496501093_a_nat] :
( ( X5 != bot_bo3237059034911209905_a_nat )
=> ( ! [X4: set_li6526943997496501093_a_nat] :
( ( member5553968465346197646_a_nat @ X4 @ X5 )
=> ( ord_le1147066620699065093_a_nat @ X4 @ Z2 ) )
=> ( ord_le1147066620699065093_a_nat @ ( comple1686717674086456018_a_nat @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_945_cSup__least,axiom,
! [X5: set_o,Z2: $o] :
( ( X5 != bot_bot_set_o )
=> ( ! [X4: $o] :
( ( member_o @ X4 @ X5 )
=> ( ord_less_eq_o @ X4 @ Z2 ) )
=> ( ord_less_eq_o @ ( complete_Sup_Sup_o @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_946_cSup__least,axiom,
! [X5: set_nat,Z2: nat] :
( ( X5 != bot_bot_set_nat )
=> ( ! [X4: nat] :
( ( member_nat @ X4 @ X5 )
=> ( ord_less_eq_nat @ X4 @ Z2 ) )
=> ( ord_less_eq_nat @ ( complete_Sup_Sup_nat @ X5 ) @ Z2 ) ) ) ).
% cSup_least
thf(fact_947_sp__equiv__list__link,axiom,
! [Sigma2: nat > sum_sum_a_nat,Ns: list_nat,Tau2: nat > sum_sum_a_nat] :
( ( sp_equiv_list_a_nat @ ( map_na823391071729141993_a_nat @ Sigma2 @ Ns ) @ ( map_na823391071729141993_a_nat @ Tau2 @ Ns ) )
= ( sp_equiv_a_nat @ Sigma2 @ Tau2 @ ( set_nat2 @ Ns ) ) ) ).
% sp_equiv_list_link
thf(fact_948_bot__empty__eq,axiom,
( bot_bo9042073657639083596_nat_o
= ( ^ [X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ bot_bo1033123847703346641_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_949_bot__empty__eq,axiom,
( bot_bo3382309974966529835_nat_o
= ( ^ [X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ bot_bo6441361344521902642_a_nat ) ) ) ).
% bot_empty_eq
thf(fact_950_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_951_bot__empty__eq,axiom,
( bot_bot_o_o
= ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).
% bot_empty_eq
thf(fact_952_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_953_Collect__empty__eq__bot,axiom,
! [P: $o > $o] :
( ( ( collect_o @ P )
= bot_bot_set_o )
= ( P = bot_bot_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_954_ran__distinct,axiom,
! [Al: list_P5056861408695629236_a_nat] :
( ( distinct_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Al ) )
=> ( ( ran_na775312650358800638_a_nat @ ( map_of1932138695521331722_a_nat @ Al ) )
= ( image_5908522764169838868_a_nat @ produc1280857981117566888_a_nat @ ( set_Pr2196415342531872975_a_nat @ Al ) ) ) ) ).
% ran_distinct
thf(fact_955_prod_Oexpand,axiom,
! [Prod: produc5454855657571582244_a_nat,Prod2: produc5454855657571582244_a_nat] :
( ( ( ( produc1510075366785312874_a_nat @ Prod )
= ( produc1510075366785312874_a_nat @ Prod2 ) )
& ( ( produc1280857981117566888_a_nat @ Prod )
= ( produc1280857981117566888_a_nat @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_956_Sup__SUP__eq,axiom,
( comple8250499579560596074_nat_o
= ( ^ [S3: set_na8448764090737828173_nat_o,X: nat > sum_sum_a_nat] : ( member8690443509505302927_a_nat @ X @ ( comple8643769897048643123_a_nat @ ( image_1911308356169199528_a_nat @ collec5629555741568564177_a_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_957_Sup__SUP__eq,axiom,
( complete_Sup_Sup_o_o
= ( ^ [S3: set_o_o,X: $o] : ( member_o @ X @ ( comple90263536869209701_set_o @ ( image_o_o_set_o @ collect_o @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_958_Sup__SUP__eq,axiom,
( comple8317665133742190828_nat_o
= ( ^ [S3: set_nat_o,X: nat] : ( member_nat @ X @ ( comple7399068483239264473et_nat @ ( image_nat_o_set_nat @ collect_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_959_Sup__SUP__eq,axiom,
( comple1334631452232999051_nat_o
= ( ^ [S3: set_li4526012430949197550_nat_o,X: list_Sum_sum_a_nat] : ( member408289922725080238_a_nat @ X @ ( comple1686717674086456018_a_nat @ ( image_2943553480580064358_a_nat @ collec7555443234367654128_a_nat @ S3 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_960_prod__eq__iff,axiom,
( ( ^ [Y5: produc5454855657571582244_a_nat,Z3: produc5454855657571582244_a_nat] : ( Y5 = Z3 ) )
= ( ^ [S4: produc5454855657571582244_a_nat,T2: produc5454855657571582244_a_nat] :
( ( ( produc1510075366785312874_a_nat @ S4 )
= ( produc1510075366785312874_a_nat @ T2 ) )
& ( ( produc1280857981117566888_a_nat @ S4 )
= ( produc1280857981117566888_a_nat @ T2 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_961_prod__eqI,axiom,
! [P2: produc5454855657571582244_a_nat,Q2: produc5454855657571582244_a_nat] :
( ( ( produc1510075366785312874_a_nat @ P2 )
= ( produc1510075366785312874_a_nat @ Q2 ) )
=> ( ( ( produc1280857981117566888_a_nat @ P2 )
= ( produc1280857981117566888_a_nat @ Q2 ) )
=> ( P2 = Q2 ) ) ) ).
% prod_eqI
thf(fact_962_exE__realizer_H,axiom,
! [P: sum_sum_a_nat > nat > $o,P2: produc5454855657571582244_a_nat] :
( ( P @ ( produc1280857981117566888_a_nat @ P2 ) @ ( produc1510075366785312874_a_nat @ P2 ) )
=> ~ ! [X4: nat,Y: sum_sum_a_nat] :
~ ( P @ Y @ X4 ) ) ).
% exE_realizer'
thf(fact_963_subset__emptyI,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ! [X4: list_Sum_sum_a_nat] :
~ ( member408289922725080238_a_nat @ X4 @ A )
=> ( ord_le1147066620699065093_a_nat @ A @ bot_bo1033123847703346641_a_nat ) ) ).
% subset_emptyI
thf(fact_964_subset__emptyI,axiom,
! [A: set_na3699693778330250182_a_nat] :
( ! [X4: nat > sum_sum_a_nat] :
~ ( member8690443509505302927_a_nat @ X4 @ A )
=> ( ord_le8108555184339247974_a_nat @ A @ bot_bo6441361344521902642_a_nat ) ) ).
% subset_emptyI
thf(fact_965_subset__emptyI,axiom,
! [A: set_o] :
( ! [X4: $o] :
~ ( member_o @ X4 @ A )
=> ( ord_less_eq_set_o @ A @ bot_bot_set_o ) ) ).
% subset_emptyI
thf(fact_966_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X4: nat] :
~ ( member_nat @ X4 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_967_all__subset__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,P: set_li6526943997496501093_a_nat > $o] :
( ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ ( image_5081948215111134021_a_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ A )
=> ( P @ ( image_5081948215111134021_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_968_all__subset__image,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,P: set_se4330304633200676677_a_nat > $o] :
( ( ! [B3: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ B3 @ ( image_7676043921908783909_a_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ B3 @ A )
=> ( P @ ( image_7676043921908783909_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_969_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_970_map__snd__enumerate,axiom,
! [N3: nat,Xs: list_Sum_sum_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( enumer3164015132978342500_a_nat @ N3 @ Xs ) )
= Xs ) ).
% map_snd_enumerate
thf(fact_971_inj__on__image__Fpow,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F @ A )
=> ( inj_on561899399213738673_a_nat @ ( image_5081948215111134021_a_nat @ F ) @ ( finite3225531020566593003_a_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_972_inj__on__image__Fpow,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inj_on7795941589172611729_a_nat @ F @ A )
=> ( inj_on4382358243947751057_a_nat @ ( image_7676043921908783909_a_nat @ F ) @ ( finite3225531020566593003_a_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_973_inj__on__image__Fpow,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( inj_on4604407203859583615et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A ) ) ) ).
% inj_on_image_Fpow
thf(fact_974_map__map__of,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin5548549501492119275_a_nat @ Xs )
=> ( Ys
= ( map_Pr2515564165894599445_a_nat @ ( comp_o6804431283847804801_a_nat @ the_Sum_sum_a_nat @ ( map_of3400036372718715318_a_nat @ ( zip_Pr8089753336904788450_a_nat @ Xs @ Ys ) ) ) @ Xs ) ) ) ) ).
% map_map_of
thf(fact_975_map__map__of,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( Ys
= ( map_Su2790769393171190532_a_nat @ ( comp_o6940958636155918818_a_nat @ the_Sum_sum_a_nat @ ( map_of7357225391242338723_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) ) @ Xs ) ) ) ) ).
% map_map_of
thf(fact_976_map__map__of,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( Ys
= ( map_Su5227373005390213643at_nat @ ( comp_o571110771943018602_a_nat @ the_nat @ ( map_of6336120629182403372at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) ) ) @ Xs ) ) ) ) ).
% map_map_of
thf(fact_977_map__map__of,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ( Ys
= ( map_na823391071729141993_a_nat @ ( comp_o2721059924169077421at_nat @ the_Sum_sum_a_nat @ ( map_of1932138695521331722_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) ) ) @ Xs ) ) ) ) ).
% map_map_of
thf(fact_978_map__map__of,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ( Ys
= ( map_nat_nat @ ( comp_o4663963155070737957at_nat @ the_nat @ ( map_of_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) @ Xs ) ) ) ) ).
% map_map_of
thf(fact_979_image__Fpow__mono,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B )
=> ( ord_le8138476598237931237_a_nat @ ( image_3472601871771700037_a_nat @ ( image_5081948215111134021_a_nat @ F ) @ ( finite3225531020566593003_a_nat @ A ) ) @ ( finite3225531020566593003_a_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_980_image__Fpow__mono,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( image_7676043921908783909_a_nat @ F @ A ) @ B )
=> ( ord_le8587131541179289285_a_nat @ ( image_7557436503106877221_a_nat @ ( image_7676043921908783909_a_nat @ F ) @ ( finite3225531020566593003_a_nat @ A ) ) @ ( finite9077140459278341579_a_nat @ B ) ) ) ).
% image_Fpow_mono
thf(fact_981_length__as,axiom,
( ( size_s5283204784079214577_a_nat @ as )
= ( size_size_list_nat @ fv_sub ) ) ).
% length_as
thf(fact_982_length__fs,axiom,
( ( size_s5283204784079214577_a_nat @ fs )
= ( size_size_list_nat @ fv_sub_comp ) ) ).
% length_fs
thf(fact_983_len__in__ass,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Xs @ ass )
=> ( ( Xs
= ( fo_nmlz_a @ ad @ Xs ) )
& ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ fv_sub ) ) ) ) ).
% len_in_ass
thf(fact_984_length__map,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) )
= ( size_s4076174644546656840_a_nat @ Xs ) ) ).
% length_map
thf(fact_985_length__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_986_length__map,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_987_length__map,axiom,
! [F: sum_sum_a_nat > nat,Xs: list_Sum_sum_a_nat] :
( ( size_size_list_nat @ ( map_Su5227373005390213643at_nat @ F @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_988_length__map,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_map
thf(fact_989_length__enumerate,axiom,
! [N3: nat,Xs: list_Sum_sum_a_nat] :
( ( size_s4076174644546656840_a_nat @ ( enumer3164015132978342500_a_nat @ N3 @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_enumerate
thf(fact_990_length__enumerate,axiom,
! [N3: nat,Xs: list_nat] :
( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N3 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ).
% length_enumerate
thf(fact_991_map__fst__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_992_map__fst__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr6349035483858946235_a_nat @ produc5914057300446384524at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_993_map__fst__zip,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_994_map__fst__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_995_map__snd__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_996_map__snd__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr8434518024483680788at_nat @ produc5684839914778638538at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_997_map__snd__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_998_map__snd__zip,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_999_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_s4076174644546656840_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1000_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1001_map__eq__imp__length__eq,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Pr2515564165894599445_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1002_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s4076174644546656840_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1003_map__eq__imp__length__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1004_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: produc5454855657571582244_a_nat > sum_sum_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_Pr2515564165894599445_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s4076174644546656840_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1005_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: sum_sum_a_nat > sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_Su2790769393171190532_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1006_map__eq__imp__length__eq,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,G: nat > sum_sum_a_nat,Ys: list_nat] :
( ( ( map_na823391071729141993_a_nat @ F @ Xs )
= ( map_na823391071729141993_a_nat @ G @ Ys ) )
=> ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_1007_neq__if__length__neq,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
!= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1008_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_1009_Ex__list__of__length,axiom,
! [N3: nat] :
? [Xs3: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ Xs3 )
= N3 ) ).
% Ex_list_of_length
thf(fact_1010_Ex__list__of__length,axiom,
! [N3: nat] :
? [Xs3: list_nat] :
( ( size_size_list_nat @ Xs3 )
= N3 ) ).
% Ex_list_of_length
thf(fact_1011_fo__nmlz__length,axiom,
! [AD: set_a,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( fo_nmlz_a @ AD @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% fo_nmlz_length
thf(fact_1012_ad__agr__list__length,axiom,
! [X5: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X5 @ Xs @ Ys )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% ad_agr_list_length
thf(fact_1013_empty__in__Fpow,axiom,
! [A: set_nat] : ( member_set_nat @ bot_bot_set_nat @ ( finite_Fpow_nat @ A ) ) ).
% empty_in_Fpow
thf(fact_1014_empty__in__Fpow,axiom,
! [A: set_o] : ( member_set_o @ bot_bot_set_o @ ( finite_Fpow_o @ A ) ) ).
% empty_in_Fpow
thf(fact_1015_exists__map,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin5548549501492119275_a_nat @ Xs )
=> ? [F4: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( Ys
= ( map_Pr2515564165894599445_a_nat @ F4 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1016_exists__map,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ? [F4: sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( map_Su2790769393171190532_a_nat @ F4 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1017_exists__map,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ? [F4: sum_sum_a_nat > nat] :
( Ys
= ( map_Su5227373005390213643at_nat @ F4 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1018_exists__map,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ? [F4: nat > sum_sum_a_nat] :
( Ys
= ( map_na823391071729141993_a_nat @ F4 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1019_exists__map,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ? [F4: nat > nat] :
( Ys
= ( map_nat_nat @ F4 @ Xs ) ) ) ) ).
% exists_map
thf(fact_1020_exists__fo__nmlzd,axiom,
! [Xs: list_P5056861408695629236_a_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_s4076174644546656840_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin5548549501492119275_a_nat @ Xs )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F4: produc5454855657571582244_a_nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_Pr2515564165894599445_a_nat @ F4 @ Xs ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_1021_exists__fo__nmlzd,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F4: sum_sum_a_nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_Su2790769393171190532_a_nat @ F4 @ Xs ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_1022_exists__fo__nmlzd,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,AD: set_a] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ( ( fo_nmlzd_a @ AD @ Ys )
=> ? [F4: nat > sum_sum_a_nat] :
( Ys
= ( fo_nmlz_a @ AD @ ( map_na823391071729141993_a_nat @ F4 @ Xs ) ) ) ) ) ) ).
% exists_fo_nmlzd
thf(fact_1023_map__of__zip__inject,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat,Zs: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Zs )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( ( map_of7357225391242338723_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) )
= ( map_of7357225391242338723_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Zs ) ) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_1024_map__of__zip__inject,axiom,
! [Ys: list_Sum_sum_a_nat,Xs: list_nat,Zs: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( ( size_s5283204784079214577_a_nat @ Zs )
= ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ Xs )
=> ( ( ( map_of1932138695521331722_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) )
= ( map_of1932138695521331722_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Zs ) ) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_1025_map__of__zip__inject,axiom,
! [Ys: list_nat,Xs: list_Sum_sum_a_nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Ys )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( ( map_of6336120629182403372at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) )
= ( map_of6336120629182403372at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Zs ) ) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_1026_map__of__zip__inject,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( size_size_list_nat @ Ys )
= ( size_size_list_nat @ Xs ) )
=> ( ( ( size_size_list_nat @ Zs )
= ( size_size_list_nat @ Xs ) )
=> ( ( distinct_nat @ Xs )
=> ( ( ( map_of_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) )
= ( map_of_nat_nat @ ( zip_nat_nat @ Xs @ Zs ) ) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_1027_Fpow__mono,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le6893508408891458716et_nat @ ( finite_Fpow_nat @ A ) @ ( finite_Fpow_nat @ B ) ) ) ).
% Fpow_mono
thf(fact_1028_proj__tuple__length,axiom,
! [Ns: list_nat,Ms: list_nat,Xs: list_Sum_sum_a_nat] :
( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ns )
& ( distinct_nat @ Ns ) )
=> ( ( ( sorted_wrt_nat @ ord_less_eq_nat @ Ms )
& ( distinct_nat @ Ms ) )
=> ( ( ord_less_eq_set_nat @ ( set_nat2 @ Ns ) @ ( set_nat2 @ Ms ) )
=> ( ( ( size_size_list_nat @ Ms )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( ( size_s5283204784079214577_a_nat @ ( proj_tuple_a @ Ns @ ( zip_na2013496608136855606_a_nat @ Ms @ Xs ) ) )
= ( size_size_list_nat @ Ns ) ) ) ) ) ) ).
% proj_tuple_length
thf(fact_1029_zip__eq__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Zs: list_P1195027771636113901_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( zip_Su7355543910597222519_a_nat @ Xs @ Ys )
= Zs )
= ( ( ( map_Pr3527087583225239196_a_nat @ produc3940427395981429571_a_nat @ Zs )
= Xs )
& ( ( map_Pr3527087583225239196_a_nat @ produc8028804885397038213_a_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_1030_zip__eq__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,Zs: list_P6164600145584960654at_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( zip_Su6417478541797927256at_nat @ Xs @ Ys )
= Zs )
= ( ( ( map_Pr6349035483858946235_a_nat @ produc5914057300446384524at_nat @ Zs )
= Xs )
& ( ( map_Pr8434518024483680788at_nat @ produc5684839914778638538at_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_1031_zip__eq__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_P6011104703257516679at_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( zip_nat_nat @ Xs @ Ys )
= Zs )
= ( ( ( map_Pr3938374229010428429at_nat @ product_fst_nat_nat @ Zs )
= Xs )
& ( ( map_Pr3938374229010428429at_nat @ product_snd_nat_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_1032_zip__eq__conv,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,Zs: list_P5056861408695629236_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( zip_na2013496608136855606_a_nat @ Xs @ Ys )
= Zs )
= ( ( ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Zs )
= Xs )
& ( ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_1033_ran__map__of__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( ran_Su6606007654835531183_a_nat @ ( map_of7357225391242338723_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) ) )
= ( set_Sum_sum_a_nat2 @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_1034_ran__map__of__zip,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( ran_Su5179294584019872288at_nat @ ( map_of6336120629182403372at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) ) )
= ( set_nat2 @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_1035_ran__map__of__zip,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ( ( ran_na775312650358800638_a_nat @ ( map_of1932138695521331722_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) ) )
= ( set_Sum_sum_a_nat2 @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_1036_ran__map__of__zip,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( distinct_nat @ Xs )
=> ( ( ran_nat_nat @ ( map_of_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
= ( set_nat2 @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_1037__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062as_Afs_O_A_092_060lbrakk_062as_A_092_060in_062_Aass_059_Afs_A_092_060in_062_Anall__tuples__rec_AAD_A_Icard_A_IInr_A_N_096_Aset_Aas_J_J_A_Ilength_Afv__sub__comp_J_059_Axs_A_061_Afo__nmlz_AAD_A_Imap_Asnd_A_Imerge_A_Izip_Afv__sub_Aas_J_A_Izip_Afv__sub__comp_Afs_J_J_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [As: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ As @ ass )
=> ! [Fs: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ Fs @ ( nall_tuples_rec_a @ ad @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ As ) ) ) @ ( size_size_list_nat @ fv_sub_comp ) ) )
=> ( xs
!= ( fo_nmlz_a @ ad @ ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ fv_sub @ As ) @ ( zip_na2013496608136855606_a_nat @ fv_sub_comp @ Fs ) ) ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>as fs. \<lbrakk>as \<in> ass; fs \<in> nall_tuples_rec AD (card (Inr -` set as)) (length fv_sub_comp); xs = fo_nmlz AD (map snd (merge (zip fv_sub as) (zip fv_sub_comp fs)))\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1038_map__of__zip__is__None,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_Sum_sum_a_nat,X3: list_Sum_sum_a_nat] :
( ( ( size_s5212483967078203639_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( map_of928655414695205533_a_nat @ ( zip_li3824709458655879025_a_nat @ Xs @ Ys ) @ X3 )
= none_Sum_sum_a_nat )
= ( ~ ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1039_map__of__zip__is__None,axiom,
! [Xs: list_n989787106983797996_a_nat,Ys: list_Sum_sum_a_nat,X3: nat > sum_sum_a_nat] :
( ( ( size_s2119148862884149080_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( map_of4712045957748317308_a_nat @ ( zip_na5841697299243076944_a_nat @ Xs @ Ys ) @ X3 )
= none_Sum_sum_a_nat )
= ( ~ ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1040_map__of__zip__is__None,axiom,
! [Xs: list_o,Ys: list_Sum_sum_a_nat,X3: $o] :
( ( ( size_size_list_o @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( map_of7579214193106729930_a_nat @ ( zip_o_Sum_sum_a_nat @ Xs @ Ys ) @ X3 )
= none_Sum_sum_a_nat )
= ( ~ ( member_o @ X3 @ ( set_o2 @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1041_map__of__zip__is__None,axiom,
! [Xs: list_l4703314356710769291_a_nat,Ys: list_nat,X3: list_Sum_sum_a_nat] :
( ( ( size_s5212483967078203639_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( map_of3602546406765676722at_nat @ ( zip_li3415139562815969502at_nat @ Xs @ Ys ) @ X3 )
= none_nat )
= ( ~ ( member408289922725080238_a_nat @ X3 @ ( set_li2392974972034027290_a_nat @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1042_map__of__zip__is__None,axiom,
! [Xs: list_n989787106983797996_a_nat,Ys: list_nat,X3: nat > sum_sum_a_nat] :
( ( ( size_s2119148862884149080_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( map_of9065542035934512787at_nat @ ( zip_na6430312473843624639at_nat @ Xs @ Ys ) @ X3 )
= none_nat )
= ( ~ ( member8690443509505302927_a_nat @ X3 @ ( set_na645604395003041787_a_nat @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1043_map__of__zip__is__None,axiom,
! [Xs: list_o,Ys: list_nat,X3: $o] :
( ( ( size_size_list_o @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( map_of_o_nat @ ( zip_o_nat @ Xs @ Ys ) @ X3 )
= none_nat )
= ( ~ ( member_o @ X3 @ ( set_o2 @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1044_map__of__zip__is__None,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,X3: sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( map_of7357225391242338723_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Ys ) @ X3 )
= none_Sum_sum_a_nat )
= ( ~ ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1045_map__of__zip__is__None,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_nat,X3: sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( map_of6336120629182403372at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Ys ) @ X3 )
= none_nat )
= ( ~ ( member_Sum_sum_a_nat @ X3 @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1046_map__of__zip__is__None,axiom,
! [Xs: list_nat,Ys: list_Sum_sum_a_nat,X3: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( ( ( map_of1932138695521331722_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Ys ) @ X3 )
= none_Sum_sum_a_nat )
= ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1047_map__of__zip__is__None,axiom,
! [Xs: list_nat,Ys: list_nat,X3: nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( map_of_nat_nat @ ( zip_nat_nat @ Xs @ Ys ) @ X3 )
= none_nat )
= ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_1048_vimageI,axiom,
! [F: nat > nat,A2: nat,B2: nat,B: set_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1049_vimageI,axiom,
! [F: $o > nat,A2: $o,B2: nat,B: set_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_o @ A2 @ ( vimage_o_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1050_vimageI,axiom,
! [F: nat > $o,A2: nat,B2: $o,B: set_o] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_o @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_o @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1051_vimageI,axiom,
! [F: $o > $o,A2: $o,B2: $o,B: set_o] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_o @ B2 @ B )
=> ( member_o @ A2 @ ( vimage_o_o @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1052_vimageI,axiom,
! [F: nat > sum_sum_a_nat,A2: nat,B2: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_Sum_sum_a_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1053_vimageI,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: nat,B2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member408289922725080238_a_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1054_vimageI,axiom,
! [F: $o > list_Sum_sum_a_nat,A2: $o,B2: list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member408289922725080238_a_nat @ B2 @ B )
=> ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1055_vimageI,axiom,
! [F: list_Sum_sum_a_nat > nat,A2: list_Sum_sum_a_nat,B2: nat,B: set_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1056_vimageI,axiom,
! [F: list_Sum_sum_a_nat > $o,A2: list_Sum_sum_a_nat,B2: $o,B: set_o] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_o @ B2 @ B )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1057_vimageI,axiom,
! [F: nat > nat > sum_sum_a_nat,A2: nat,B2: nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member8690443509505302927_a_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage5995291031355576089_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_1058_vimage__eq,axiom,
! [A2: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1059_vimage__eq,axiom,
! [A2: nat,F: nat > $o,B: set_o] :
( ( member_nat @ A2 @ ( vimage_nat_o @ F @ B ) )
= ( member_o @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1060_vimage__eq,axiom,
! [A2: $o,F: $o > nat,B: set_nat] :
( ( member_o @ A2 @ ( vimage_o_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1061_vimage__eq,axiom,
! [A2: $o,F: $o > $o,B: set_o] :
( ( member_o @ A2 @ ( vimage_o_o @ F @ B ) )
= ( member_o @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1062_vimage__eq,axiom,
! [A2: nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B ) )
= ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1063_vimage__eq,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,B: set_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1064_vimage__eq,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,B: set_o] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ B ) )
= ( member_o @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1065_vimage__eq,axiom,
! [A2: nat,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ B ) )
= ( member408289922725080238_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1066_vimage__eq,axiom,
! [A2: $o,F: $o > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ B ) )
= ( member408289922725080238_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1067_vimage__eq,axiom,
! [A2: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat,B: set_nat] :
( ( member8690443509505302927_a_nat @ A2 @ ( vimage1507083042557951675at_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_1068_vimage__empty,axiom,
! [F: nat > sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ bot_bo3438331934148233675_a_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_1069_vimage__empty,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ bot_bot_set_nat )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_1070_vimage__empty,axiom,
! [F: $o > nat] :
( ( vimage_o_nat @ F @ bot_bot_set_nat )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_1071_vimage__empty,axiom,
! [F: nat > $o] :
( ( vimage_nat_o @ F @ bot_bot_set_o )
= bot_bot_set_nat ) ).
% vimage_empty
thf(fact_1072_vimage__empty,axiom,
! [F: $o > $o] :
( ( vimage_o_o @ F @ bot_bot_set_o )
= bot_bot_set_o ) ).
% vimage_empty
thf(fact_1073_vimage__Int,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_1074_vimage__Int,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( vimage_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_1075_vimage__Un,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_1076_vimage__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( vimage_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_1077_as__fs__def_I2_J,axiom,
member408289922725080238_a_nat @ fs @ ( nall_tuples_rec_a @ ad @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ as ) ) ) @ ( size_size_list_nat @ fv_sub_comp ) ) ).
% as_fs_def(2)
thf(fact_1078_card__Inr__vimage__le__length,axiom,
! [Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% card_Inr_vimage_le_length
thf(fact_1079_nall__tuples__rec__length,axiom,
! [Xs: list_Sum_sum_a_nat,AD: set_a,I3: nat,N3: nat] :
( ( member408289922725080238_a_nat @ Xs @ ( nall_tuples_rec_a @ AD @ I3 @ N3 ) )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= N3 ) ) ).
% nall_tuples_rec_length
thf(fact_1080_set_Ocomp,axiom,
! [F: nat > nat,G: nat > nat] :
( ( comp_s8964582002068861047et_nat @ ( vimage_nat_nat @ F ) @ ( vimage_nat_nat @ G ) )
= ( vimage_nat_nat @ ( comp_nat_nat_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_1081_set_Ocomp,axiom,
! [F: nat > option_Sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat] :
( ( comp_s3847575663057944277_a_nat @ ( vimage7014988194023193336_a_nat @ F ) @ ( vimage1635291442211470005_a_nat @ G ) )
= ( vimage3040984495076556338_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_1082_set_Ocomp,axiom,
! [F: nat > nat,G: nat > sum_sum_a_nat] :
( ( comp_s5818590256416363566_a_nat @ ( vimage_nat_nat @ F ) @ ( vimage3040984495076556338_a_nat @ G ) )
= ( vimage3040984495076556338_a_nat @ ( comp_n1522318729830440540at_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_1083_set_Ocomp,axiom,
! [F: nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat] :
( ( comp_s1691789666922741659_a_nat @ ( vimage3040984495076556338_a_nat @ F ) @ ( vimage6545432446589551483_a_nat @ G ) )
= ( vimage3040984495076556338_a_nat @ ( comp_S2395038772458240499at_nat @ G @ F ) ) ) ).
% set.comp
thf(fact_1084_set_Ocompositionality,axiom,
! [F: nat > nat,G: nat > nat,Set: set_nat] :
( ( vimage_nat_nat @ F @ ( vimage_nat_nat @ G @ Set ) )
= ( vimage_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_1085_set_Ocompositionality,axiom,
! [F: nat > option_Sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat,Set: set_Sum_sum_a_nat] :
( ( vimage7014988194023193336_a_nat @ F @ ( vimage1635291442211470005_a_nat @ G @ Set ) )
= ( vimage3040984495076556338_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_1086_set_Ocompositionality,axiom,
! [F: nat > nat,G: nat > sum_sum_a_nat,Set: set_Sum_sum_a_nat] :
( ( vimage_nat_nat @ F @ ( vimage3040984495076556338_a_nat @ G @ Set ) )
= ( vimage3040984495076556338_a_nat @ ( comp_n1522318729830440540at_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_1087_set_Ocompositionality,axiom,
! [F: nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,Set: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( vimage6545432446589551483_a_nat @ G @ Set ) )
= ( vimage3040984495076556338_a_nat @ ( comp_S2395038772458240499at_nat @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_1088_vimage__comp,axiom,
! [F: nat > nat,G: nat > nat,X3: set_nat] :
( ( vimage_nat_nat @ F @ ( vimage_nat_nat @ G @ X3 ) )
= ( vimage_nat_nat @ ( comp_nat_nat_nat @ G @ F ) @ X3 ) ) ).
% vimage_comp
thf(fact_1089_vimage__comp,axiom,
! [F: nat > option_Sum_sum_a_nat,G: option_Sum_sum_a_nat > sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( vimage7014988194023193336_a_nat @ F @ ( vimage1635291442211470005_a_nat @ G @ X3 ) )
= ( vimage3040984495076556338_a_nat @ ( comp_o2721059924169077421at_nat @ G @ F ) @ X3 ) ) ).
% vimage_comp
thf(fact_1090_vimage__comp,axiom,
! [F: nat > nat,G: nat > sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( vimage_nat_nat @ F @ ( vimage3040984495076556338_a_nat @ G @ X3 ) )
= ( vimage3040984495076556338_a_nat @ ( comp_n1522318729830440540at_nat @ G @ F ) @ X3 ) ) ).
% vimage_comp
thf(fact_1091_vimage__comp,axiom,
! [F: nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,X3: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( vimage6545432446589551483_a_nat @ G @ X3 ) )
= ( vimage3040984495076556338_a_nat @ ( comp_S2395038772458240499at_nat @ G @ F ) @ X3 ) ) ).
% vimage_comp
thf(fact_1092_vimageD,axiom,
! [A2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1093_vimageD,axiom,
! [A2: nat,F: nat > $o,A: set_o] :
( ( member_nat @ A2 @ ( vimage_nat_o @ F @ A ) )
=> ( member_o @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1094_vimageD,axiom,
! [A2: $o,F: $o > nat,A: set_nat] :
( ( member_o @ A2 @ ( vimage_o_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1095_vimageD,axiom,
! [A2: $o,F: $o > $o,A: set_o] :
( ( member_o @ A2 @ ( vimage_o_o @ F @ A ) )
=> ( member_o @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1096_vimageD,axiom,
! [A2: nat,F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ A ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1097_vimageD,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,A: set_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1098_vimageD,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,A: set_o] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ A ) )
=> ( member_o @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1099_vimageD,axiom,
! [A2: nat,F: nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ A ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1100_vimageD,axiom,
! [A2: $o,F: $o > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ A ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1101_vimageD,axiom,
! [A2: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat,A: set_nat] :
( ( member8690443509505302927_a_nat @ A2 @ ( vimage1507083042557951675at_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1102_vimageE,axiom,
! [A2: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1103_vimageE,axiom,
! [A2: nat,F: nat > $o,B: set_o] :
( ( member_nat @ A2 @ ( vimage_nat_o @ F @ B ) )
=> ( member_o @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1104_vimageE,axiom,
! [A2: $o,F: $o > nat,B: set_nat] :
( ( member_o @ A2 @ ( vimage_o_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1105_vimageE,axiom,
! [A2: $o,F: $o > $o,B: set_o] :
( ( member_o @ A2 @ ( vimage_o_o @ F @ B ) )
=> ( member_o @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1106_vimageE,axiom,
! [A2: nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1107_vimageE,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > nat,B: set_nat] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1108_vimageE,axiom,
! [A2: list_Sum_sum_a_nat,F: list_Sum_sum_a_nat > $o,B: set_o] :
( ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ B ) )
=> ( member_o @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1109_vimageE,axiom,
! [A2: nat,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ B ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1110_vimageE,axiom,
! [A2: $o,F: $o > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ B ) )
=> ( member408289922725080238_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1111_vimageE,axiom,
! [A2: nat > sum_sum_a_nat,F: ( nat > sum_sum_a_nat ) > nat,B: set_nat] :
( ( member8690443509505302927_a_nat @ A2 @ ( vimage1507083042557951675at_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_1112_vimageI2,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1113_vimageI2,axiom,
! [F: $o > nat,A2: $o,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage_o_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1114_vimageI2,axiom,
! [F: nat > $o,A2: nat,A: set_o] :
( ( member_o @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_o @ F @ A ) ) ) ).
% vimageI2
thf(fact_1115_vimageI2,axiom,
! [F: $o > $o,A2: $o,A: set_o] :
( ( member_o @ ( F @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage_o_o @ F @ A ) ) ) ).
% vimageI2
thf(fact_1116_vimageI2,axiom,
! [F: nat > sum_sum_a_nat,A2: nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1117_vimageI2,axiom,
! [F: nat > list_Sum_sum_a_nat,A2: nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage8387349081243083832_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1118_vimageI2,axiom,
! [F: $o > list_Sum_sum_a_nat,A2: $o,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ ( F @ A2 ) @ A )
=> ( member_o @ A2 @ ( vimage2884537194618305448_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1119_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > nat,A2: list_Sum_sum_a_nat,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage4660099214859102426at_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1120_vimageI2,axiom,
! [F: list_Sum_sum_a_nat > $o,A2: list_Sum_sum_a_nat,A: set_o] :
( ( member_o @ ( F @ A2 ) @ A )
=> ( member408289922725080238_a_nat @ A2 @ ( vimage6136348398256869390_nat_o @ F @ A ) ) ) ).
% vimageI2
thf(fact_1121_vimageI2,axiom,
! [F: nat > nat > sum_sum_a_nat,A2: nat,A: set_na3699693778330250182_a_nat] :
( ( member8690443509505302927_a_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage5995291031355576089_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_1122_vimage__Collect,axiom,
! [P: sum_sum_a_nat > $o,F: nat > sum_sum_a_nat,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ ( F @ X4 ) )
= ( Q @ X4 ) )
=> ( ( vimage3040984495076556338_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_1123_vimage__Collect,axiom,
! [P: nat > $o,F: nat > nat,Q: nat > $o] :
( ! [X4: nat] :
( ( P @ ( F @ X4 ) )
= ( Q @ X4 ) )
=> ( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_1124_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > list_Sum_sum_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_less_eq_set_nat @ A @ ( vimage8387349081243083832_a_nat @ F @ B ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member408289922725080238_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_1125_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > nat > sum_sum_a_nat,B: set_na3699693778330250182_a_nat] :
( ( ord_less_eq_set_nat @ A @ ( vimage5995291031355576089_a_nat @ F @ B ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member8690443509505302927_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_1126_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_1127_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > $o,B: set_o] :
( ( ord_less_eq_set_nat @ A @ ( vimage_nat_o @ F @ B ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_o @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_1128_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B ) )
= ( ! [X: nat] :
( ( member_nat @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_1129_vimage__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% vimage_mono
thf(fact_1130_vimage__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_mono
thf(fact_1131_vimage__inter__cong,axiom,
! [S: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ! [W: nat] :
( ( member_nat @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ F @ Y3 ) @ S )
= ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ G @ Y3 ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_1132_image__subset__iff__subset__vimage,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) @ B )
= ( ord_le1147066620699065093_a_nat @ A @ ( vimage7538740559230589179_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1133_image__subset__iff__subset__vimage,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_se4330304633200676677_a_nat] :
( ( ord_le8138476598237931237_a_nat @ ( image_7676043921908783909_a_nat @ F @ A ) @ B )
= ( ord_le1147066620699065093_a_nat @ A @ ( vimage5553558261632676059_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1134_image__subset__iff__subset__vimage,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1135_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_1136_image__vimage__subset,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] : ( ord_le1147066620699065093_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( vimage7538740559230589179_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_1137_image__vimage__subset,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_se4330304633200676677_a_nat] : ( ord_le8138476598237931237_a_nat @ ( image_7676043921908783909_a_nat @ F @ ( vimage5553558261632676059_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_1138_image__vimage__subset,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( vimage3040984495076556338_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_1139_card__image,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inj_on6609798167860701873_a_nat @ F @ A )
=> ( ( finite9161971191270313901_a_nat @ ( image_5081948215111134021_a_nat @ F @ A ) )
= ( finite9161971191270313901_a_nat @ A ) ) ) ).
% card_image
thf(fact_1140_card__image,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( inj_on7795941589172611729_a_nat @ F @ A )
=> ( ( finite152669285360869261_a_nat @ ( image_7676043921908783909_a_nat @ F @ A ) )
= ( finite9161971191270313901_a_nat @ A ) ) ) ).
% card_image
thf(fact_1141_card__image,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_card_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_1142_card__length,axiom,
! [Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% card_length
thf(fact_1143_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_1144_distinct__card,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ Xs )
=> ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_1145_distinct__card,axiom,
! [Xs: list_nat] :
( ( distinct_nat @ Xs )
=> ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) ) ) ).
% distinct_card
thf(fact_1146_card__distinct,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) )
=> ( distin2701893636801681144_a_nat @ Xs ) ) ).
% card_distinct
thf(fact_1147_card__distinct,axiom,
! [Xs: list_nat] :
( ( ( finite_card_nat @ ( set_nat2 @ Xs ) )
= ( size_size_list_nat @ Xs ) )
=> ( distinct_nat @ Xs ) ) ).
% card_distinct
thf(fact_1148_map__of__eq__None__iff,axiom,
! [Xys: list_P5056861408695629236_a_nat,X3: nat] :
( ( ( map_of1932138695521331722_a_nat @ Xys @ X3 )
= none_Sum_sum_a_nat )
= ( ~ ( member_nat @ X3 @ ( image_1677410150705809403at_nat @ produc1510075366785312874_a_nat @ ( set_Pr2196415342531872975_a_nat @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_1149_ext__tuple__eq,axiom,
! [Fv_sub: list_nat,As2: list_Sum_sum_a_nat,AD: set_a,Fv_sub_comp: list_nat] :
( ( ( size_size_list_nat @ Fv_sub )
= ( size_s5283204784079214577_a_nat @ As2 ) )
=> ( ( ext_tuple_a @ AD @ Fv_sub @ Fv_sub_comp @ As2 )
= ( image_5081948215111134021_a_nat
@ ^ [Fs2: list_Sum_sum_a_nat] : ( map_Pr2515564165894599445_a_nat @ produc1280857981117566888_a_nat @ ( merge_Sum_sum_a_nat @ ( zip_na2013496608136855606_a_nat @ Fv_sub @ As2 ) @ ( zip_na2013496608136855606_a_nat @ Fv_sub_comp @ Fs2 ) ) )
@ ( nall_tuples_rec_a @ AD @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ As2 ) ) ) @ ( size_size_list_nat @ Fv_sub_comp ) ) ) ) ) ).
% ext_tuple_eq
thf(fact_1150_merge__length,axiom,
! [Nxs: list_P5056861408695629236_a_nat,Mys: list_P5056861408695629236_a_nat] :
( ( size_s4076174644546656840_a_nat @ ( merge_Sum_sum_a_nat @ Nxs @ Mys ) )
= ( size_size_list_nat @ ( append_nat @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Nxs ) @ ( map_Pr2058079554187054650at_nat @ produc1510075366785312874_a_nat @ Mys ) ) ) ) ).
% merge_length
thf(fact_1151_ad__agr__list__def,axiom,
( ad_agr_list_a_nat
= ( ^ [X7: set_a,Xs2: list_Sum_sum_a_nat,Ys2: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs2 )
= ( size_s5283204784079214577_a_nat @ Ys2 ) )
& ( ad_equiv_list_a_nat @ X7 @ Xs2 @ Ys2 )
& ( sp_equiv_list_a_nat @ Xs2 @ Ys2 ) ) ) ) ).
% ad_agr_list_def
thf(fact_1152_same__append__eq,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( ( append_nat @ Xs @ Ys )
= ( append_nat @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_1153_append__same__eq,axiom,
! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
( ( ( append_nat @ Ys @ Xs )
= ( append_nat @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_1154_append__assoc,axiom,
! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
= ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_1155_append_Oassoc,axiom,
! [A2: list_nat,B2: list_nat,C2: list_nat] :
( ( append_nat @ ( append_nat @ A2 @ B2 ) @ C2 )
= ( append_nat @ A2 @ ( append_nat @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_1156_image__ident,axiom,
! [Y4: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : X
@ Y4 )
= Y4 ) ).
% image_ident
thf(fact_1157_vimage__Collect__eq,axiom,
! [F: nat > sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( vimage3040984495076556338_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_nat
@ ^ [Y2: nat] : ( P @ ( F @ Y2 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1158_vimage__Collect__eq,axiom,
! [F: nat > nat,P: nat > $o] :
( ( vimage_nat_nat @ F @ ( collect_nat @ P ) )
= ( collect_nat
@ ^ [Y2: nat] : ( P @ ( F @ Y2 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_1159_append__eq__append__conv,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) )
| ( ( size_s5283204784079214577_a_nat @ Us )
= ( size_s5283204784079214577_a_nat @ Vs ) ) )
=> ( ( ( append_Sum_sum_a_nat @ Xs @ Us )
= ( append_Sum_sum_a_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1160_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_1161_map__append,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_1162_map__append,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat,Ys: list_nat] :
( ( map_na823391071729141993_a_nat @ F @ ( append_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( map_na823391071729141993_a_nat @ F @ Xs ) @ ( map_na823391071729141993_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_1163_map__append,axiom,
! [F: produc5454855657571582244_a_nat > sum_sum_a_nat,Xs: list_P5056861408695629236_a_nat,Ys: list_P5056861408695629236_a_nat] :
( ( map_Pr2515564165894599445_a_nat @ F @ ( append338925788367110473_a_nat @ Xs @ Ys ) )
= ( append_Sum_sum_a_nat @ ( map_Pr2515564165894599445_a_nat @ F @ Xs ) @ ( map_Pr2515564165894599445_a_nat @ F @ Ys ) ) ) ).
% map_append
thf(fact_1164_SUP__identity__eq,axiom,
! [A: set_se4330304633200676677_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_3472601871771700037_a_nat
@ ^ [X: set_li6526943997496501093_a_nat] : X
@ A ) )
= ( comple1686717674086456018_a_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_1165_SUP__identity__eq,axiom,
! [A: set_o] :
( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X: $o] : X
@ A ) )
= ( complete_Sup_Sup_o @ A ) ) ).
% SUP_identity_eq
thf(fact_1166_SUP__identity__eq,axiom,
! [A: set_nat] :
( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X: nat] : X
@ A ) )
= ( complete_Sup_Sup_nat @ A ) ) ).
% SUP_identity_eq
thf(fact_1167_UN__I,axiom,
! [A2: nat,A: set_nat,B2: nat,B: nat > set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1168_UN__I,axiom,
! [A2: nat,A: set_nat,B2: $o,B: nat > set_o] :
( ( member_nat @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1169_UN__I,axiom,
! [A2: $o,A: set_o,B2: nat,B: $o > set_nat] :
( ( member_o @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1170_UN__I,axiom,
! [A2: $o,A: set_o,B2: $o,B: $o > set_o] :
( ( member_o @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1171_UN__I,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: nat,B: list_Sum_sum_a_nat > set_nat] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_2109848253374568390et_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1172_UN__I,axiom,
! [A2: list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat,B2: $o,B: list_Sum_sum_a_nat > set_o] :
( ( member408289922725080238_a_nat @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_4041951406191573560_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1173_UN__I,axiom,
! [A2: nat,A: set_nat,B2: list_Sum_sum_a_nat,B: nat > set_li6526943997496501093_a_nat] :
( ( member_nat @ A2 @ A )
=> ( ( member408289922725080238_a_nat @ B2 @ ( B @ A2 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1174_UN__I,axiom,
! [A2: $o,A: set_o,B2: list_Sum_sum_a_nat,B: $o > set_li6526943997496501093_a_nat] :
( ( member_o @ A2 @ A )
=> ( ( member408289922725080238_a_nat @ B2 @ ( B @ A2 ) )
=> ( member408289922725080238_a_nat @ B2 @ ( comple1686717674086456018_a_nat @ ( image_2951602039182858962_a_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1175_UN__I,axiom,
! [A2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B2: nat,B: ( nat > sum_sum_a_nat ) > set_nat] :
( ( member8690443509505302927_a_nat @ A2 @ A )
=> ( ( member_nat @ B2 @ ( B @ A2 ) )
=> ( member_nat @ B2 @ ( comple7399068483239264473et_nat @ ( image_5790795286017029031et_nat @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1176_UN__I,axiom,
! [A2: nat > sum_sum_a_nat,A: set_na3699693778330250182_a_nat,B2: $o,B: ( nat > sum_sum_a_nat ) > set_o] :
( ( member8690443509505302927_a_nat @ A2 @ A )
=> ( ( member_o @ B2 @ ( B @ A2 ) )
=> ( member_o @ B2 @ ( comple90263536869209701_set_o @ ( image_9210942892871078935_set_o @ B @ A ) ) ) ) ) ).
% UN_I
thf(fact_1177_UN__iff,axiom,
! [B2: list_Sum_sum_a_nat,B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( member408289922725080238_a_nat @ B2 @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
= ( ? [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
& ( member408289922725080238_a_nat @ B2 @ ( B @ X ) ) ) ) ) ).
% UN_iff
thf(fact_1178_set__append,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_1179_set__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_append
thf(fact_1180_zip__append,axiom,
! [Xs: list_Sum_sum_a_nat,Us: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Vs: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Us ) )
=> ( ( zip_Su7355543910597222519_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us @ Vs ) )
= ( append1996214168388709506_a_nat @ ( zip_Su7355543910597222519_a_nat @ Xs @ Us ) @ ( zip_Su7355543910597222519_a_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1181_zip__append,axiom,
! [Xs: list_Sum_sum_a_nat,Us: list_nat,Ys: list_Sum_sum_a_nat,Vs: list_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_Su6417478541797927256at_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append2142653904031976739at_nat @ ( zip_Su6417478541797927256at_nat @ Xs @ Us ) @ ( zip_Su6417478541797927256at_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1182_zip__append,axiom,
! [Xs: list_nat,Us: list_Sum_sum_a_nat,Ys: list_nat,Vs: list_Sum_sum_a_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Us ) )
=> ( ( zip_na2013496608136855606_a_nat @ ( append_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_nat @ Us @ Vs ) )
= ( append338925788367110473_a_nat @ ( zip_na2013496608136855606_a_nat @ Xs @ Us ) @ ( zip_na2013496608136855606_a_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1183_zip__append,axiom,
! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Us ) )
=> ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
= ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_1184_SUP__bot,axiom,
! [A: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : bot_bo1033123847703346641_a_nat
@ A ) )
= bot_bo1033123847703346641_a_nat ) ).
% SUP_bot
thf(fact_1185_SUP__bot__conv_I1_J,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ( ( B @ X )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_1186_SUP__bot__conv_I2_J,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ( ( B @ X )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_1187_SUP__const,axiom,
! [A: set_li6526943997496501093_a_nat,F: set_li6526943997496501093_a_nat] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [I4: list_Sum_sum_a_nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1188_SUP__const,axiom,
! [A: set_nat,F: set_li6526943997496501093_a_nat] :
( ( A != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [I4: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1189_SUP__const,axiom,
! [A: set_o,F: set_li6526943997496501093_a_nat] :
( ( A != bot_bot_set_o )
=> ( ( comple1686717674086456018_a_nat
@ ( image_2951602039182858962_a_nat
@ ^ [I4: $o] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1190_SUP__const,axiom,
! [A: set_nat,F: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [I4: nat] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1191_SUP__const,axiom,
! [A: set_o,F: $o] :
( ( A != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [I4: $o] : F
@ A ) )
= F ) ) ).
% SUP_const
thf(fact_1192_cSUP__const,axiom,
! [A: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( A != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1193_cSUP__const,axiom,
! [A: set_nat,C2: set_li6526943997496501093_a_nat] :
( ( A != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1194_cSUP__const,axiom,
! [A: set_o,C2: set_li6526943997496501093_a_nat] :
( ( A != bot_bot_set_o )
=> ( ( comple1686717674086456018_a_nat
@ ( image_2951602039182858962_a_nat
@ ^ [X: $o] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1195_cSUP__const,axiom,
! [A: set_nat,C2: $o] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_o
@ ( image_nat_o
@ ^ [X: nat] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1196_cSUP__const,axiom,
! [A: set_o,C2: $o] :
( ( A != bot_bot_set_o )
=> ( ( complete_Sup_Sup_o
@ ( image_o_o
@ ^ [X: $o] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1197_cSUP__const,axiom,
! [A: set_nat,C2: nat] :
( ( A != bot_bot_set_nat )
=> ( ( complete_Sup_Sup_nat
@ ( image_nat_nat
@ ^ [X: nat] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1198_cSUP__const,axiom,
! [A: set_o,C2: nat] :
( ( A != bot_bot_set_o )
=> ( ( complete_Sup_Sup_nat
@ ( image_o_nat
@ ^ [X: $o] : C2
@ A ) )
= C2 ) ) ).
% cSUP_const
thf(fact_1199_if__image__distrib,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,G: list_Sum_sum_a_nat > list_Sum_sum_a_nat,S: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( if_lis4685338526944683083_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su4083067149120280889_a_nat @ ( image_5081948215111134021_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ S @ ( collec7555443234367654128_a_nat @ P ) ) )
@ ( image_5081948215111134021_a_nat @ G
@ ( inf_in3249246906714053971_a_nat @ S
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1200_if__image__distrib,axiom,
! [P: list_Sum_sum_a_nat > $o,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,G: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,S: set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( if_set7709265119413304363_a_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_su499249268922660121_a_nat @ ( image_7676043921908783909_a_nat @ F @ ( inf_in3249246906714053971_a_nat @ S @ ( collec7555443234367654128_a_nat @ P ) ) )
@ ( image_7676043921908783909_a_nat @ G
@ ( inf_in3249246906714053971_a_nat @ S
@ ( collec7555443234367654128_a_nat
@ ^ [X: list_Sum_sum_a_nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1201_if__image__distrib,axiom,
! [P: nat > $o,F: nat > nat,G: nat > nat,S: set_nat] :
( ( image_nat_nat
@ ^ [X: nat] : ( if_nat @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ S @ ( collect_nat @ P ) ) )
@ ( image_nat_nat @ G
@ ( inf_inf_set_nat @ S
@ ( collect_nat
@ ^ [X: nat] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_1202_UN__constant,axiom,
! [A: set_nat,C2: set_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1203_UN__constant,axiom,
! [A: set_nat,C2: set_o] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y2: nat] : C2
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1204_UN__constant,axiom,
! [A: set_o,C2: set_nat] :
( ( ( A = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y2: $o] : C2
@ A ) )
= bot_bot_set_nat ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [Y2: $o] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1205_UN__constant,axiom,
! [A: set_o,C2: set_o] :
( ( ( A = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C2
@ A ) )
= bot_bot_set_o ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [Y2: $o] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1206_UN__constant,axiom,
! [A: set_li6526943997496501093_a_nat,C2: set_li6526943997496501093_a_nat] :
( ( ( A = bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [Y2: list_Sum_sum_a_nat] : C2
@ A ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( A != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [Y2: list_Sum_sum_a_nat] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1207_UN__constant,axiom,
! [A: set_nat,C2: set_li6526943997496501093_a_nat] :
( ( ( A = bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [Y2: nat] : C2
@ A ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( A != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [Y2: nat] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1208_UN__constant,axiom,
! [A: set_o,C2: set_li6526943997496501093_a_nat] :
( ( ( A = bot_bot_set_o )
=> ( ( comple1686717674086456018_a_nat
@ ( image_2951602039182858962_a_nat
@ ^ [Y2: $o] : C2
@ A ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( A != bot_bot_set_o )
=> ( ( comple1686717674086456018_a_nat
@ ( image_2951602039182858962_a_nat
@ ^ [Y2: $o] : C2
@ A ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_1209_UN__Un,axiom,
! [M: nat > set_nat,A: set_nat,B: set_nat] :
( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ A ) ) @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_1210_UN__Un,axiom,
! [M: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ M @ ( sup_su4083067149120280889_a_nat @ A @ B ) ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ M @ A ) ) @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_1211_UN__Un,axiom,
! [M: nat > set_li6526943997496501093_a_nat,A: set_nat,B: set_nat] :
( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ M @ ( sup_sup_set_nat @ A @ B ) ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ M @ A ) ) @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ M @ B ) ) ) ) ).
% UN_Un
thf(fact_1212_UN__simps_I2_J,axiom,
! [C: set_nat,A: nat > set_nat,B: set_nat] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A @ X ) @ B )
@ C ) )
= bot_bot_set_nat ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ ( A @ X ) @ B )
@ C ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ A @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1213_UN__simps_I2_J,axiom,
! [C: set_nat,A: nat > set_o,B: set_o] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X: nat] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X: nat] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ A @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1214_UN__simps_I2_J,axiom,
! [C: set_o,A: $o > set_nat,B: set_nat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( sup_sup_set_nat @ ( A @ X ) @ B )
@ C ) )
= bot_bot_set_nat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( sup_sup_set_nat @ ( A @ X ) @ B )
@ C ) )
= ( sup_sup_set_nat @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ A @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1215_UN__simps_I2_J,axiom,
! [C: set_o,A: $o > set_o,B: set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( sup_sup_set_o @ ( A @ X ) @ B )
@ C ) )
= ( sup_sup_set_o @ ( comple90263536869209701_set_o @ ( image_o_set_o @ A @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1216_UN__simps_I2_J,axiom,
! [C: set_li6526943997496501093_a_nat,A: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( C = bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ ( A @ X ) @ B )
@ C ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ ( A @ X ) @ B )
@ C ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ A @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1217_UN__simps_I2_J,axiom,
! [C: set_nat,A: nat > set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ ( A @ X ) @ B )
@ C ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ ( A @ X ) @ B )
@ C ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ A @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1218_UN__simps_I2_J,axiom,
! [C: set_o,A: $o > set_li6526943997496501093_a_nat,B: set_li6526943997496501093_a_nat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple1686717674086456018_a_nat
@ ( image_2951602039182858962_a_nat
@ ^ [X: $o] : ( sup_su4083067149120280889_a_nat @ ( A @ X ) @ B )
@ C ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple1686717674086456018_a_nat
@ ( image_2951602039182858962_a_nat
@ ^ [X: $o] : ( sup_su4083067149120280889_a_nat @ ( A @ X ) @ B )
@ C ) )
= ( sup_su4083067149120280889_a_nat @ ( comple1686717674086456018_a_nat @ ( image_2951602039182858962_a_nat @ A @ C ) ) @ B ) ) ) ) ).
% UN_simps(2)
thf(fact_1219_UN__simps_I3_J,axiom,
! [C: set_nat,A: set_nat,B: nat > set_nat] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A @ ( B @ X ) )
@ C ) )
= bot_bot_set_nat ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple7399068483239264473et_nat
@ ( image_nat_set_nat
@ ^ [X: nat] : ( sup_sup_set_nat @ A @ ( B @ X ) )
@ C ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1220_UN__simps_I3_J,axiom,
! [C: set_nat,A: set_o,B: nat > set_o] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X: nat] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple90263536869209701_set_o
@ ( image_nat_set_o
@ ^ [X: nat] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C ) )
= ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ ( image_nat_set_o @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1221_UN__simps_I3_J,axiom,
! [C: set_o,A: set_nat,B: $o > set_nat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( sup_sup_set_nat @ A @ ( B @ X ) )
@ C ) )
= bot_bot_set_nat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple7399068483239264473et_nat
@ ( image_o_set_nat
@ ^ [X: $o] : ( sup_sup_set_nat @ A @ ( B @ X ) )
@ C ) )
= ( sup_sup_set_nat @ A @ ( comple7399068483239264473et_nat @ ( image_o_set_nat @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1222_UN__simps_I3_J,axiom,
! [C: set_o,A: set_o,B: $o > set_o] :
( ( ( C = bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C ) )
= bot_bot_set_o ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple90263536869209701_set_o
@ ( image_o_set_o
@ ^ [X: $o] : ( sup_sup_set_o @ A @ ( B @ X ) )
@ C ) )
= ( sup_sup_set_o @ A @ ( comple90263536869209701_set_o @ ( image_o_set_o @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1223_UN__simps_I3_J,axiom,
! [C: set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( ( C = bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ A @ ( B @ X ) )
@ C ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C != bot_bo1033123847703346641_a_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( sup_su4083067149120280889_a_nat @ A @ ( B @ X ) )
@ C ) )
= ( sup_su4083067149120280889_a_nat @ A @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1224_UN__simps_I3_J,axiom,
! [C: set_nat,A: set_li6526943997496501093_a_nat,B: nat > set_li6526943997496501093_a_nat] :
( ( ( C = bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ A @ ( B @ X ) )
@ C ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C != bot_bot_set_nat )
=> ( ( comple1686717674086456018_a_nat
@ ( image_6073155208182920014_a_nat
@ ^ [X: nat] : ( sup_su4083067149120280889_a_nat @ A @ ( B @ X ) )
@ C ) )
= ( sup_su4083067149120280889_a_nat @ A @ ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1225_UN__simps_I3_J,axiom,
! [C: set_o,A: set_li6526943997496501093_a_nat,B: $o > set_li6526943997496501093_a_nat] :
( ( ( C = bot_bot_set_o )
=> ( ( comple1686717674086456018_a_nat
@ ( image_2951602039182858962_a_nat
@ ^ [X: $o] : ( sup_su4083067149120280889_a_nat @ A @ ( B @ X ) )
@ C ) )
= bot_bo1033123847703346641_a_nat ) )
& ( ( C != bot_bot_set_o )
=> ( ( comple1686717674086456018_a_nat
@ ( image_2951602039182858962_a_nat
@ ^ [X: $o] : ( sup_su4083067149120280889_a_nat @ A @ ( B @ X ) )
@ C ) )
= ( sup_su4083067149120280889_a_nat @ A @ ( comple1686717674086456018_a_nat @ ( image_2951602039182858962_a_nat @ B @ C ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_1226_distinct__append,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( distin2701893636801681144_a_nat @ ( append_Sum_sum_a_nat @ Xs @ Ys ) )
= ( ( distin2701893636801681144_a_nat @ Xs )
& ( distin2701893636801681144_a_nat @ Ys )
& ( ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) )
= bot_bo3438331934148233675_a_nat ) ) ) ).
% distinct_append
thf(fact_1227_distinct__append,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( distinct_nat @ ( append_nat @ Xs @ Ys ) )
= ( ( distinct_nat @ Xs )
& ( distinct_nat @ Ys )
& ( ( inf_inf_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
= bot_bot_set_nat ) ) ) ).
% distinct_append
thf(fact_1228_distinct__append,axiom,
! [Xs: list_o,Ys: list_o] :
( ( distinct_o @ ( append_o @ Xs @ Ys ) )
= ( ( distinct_o @ Xs )
& ( distinct_o @ Ys )
& ( ( inf_inf_set_o @ ( set_o2 @ Xs ) @ ( set_o2 @ Ys ) )
= bot_bot_set_o ) ) ) ).
% distinct_append
thf(fact_1229_vimage__UN,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( vimage7538740559230589179_a_nat @ F @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( vimage7538740559230589179_a_nat @ F @ ( B @ X ) )
@ A ) ) ) ).
% vimage_UN
thf(fact_1230_vimage__def,axiom,
( vimage8387349081243083832_a_nat
= ( ^ [F2: nat > list_Sum_sum_a_nat,B3: set_li6526943997496501093_a_nat] :
( collect_nat
@ ^ [X: nat] : ( member408289922725080238_a_nat @ ( F2 @ X ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_1231_vimage__def,axiom,
( vimage5995291031355576089_a_nat
= ( ^ [F2: nat > nat > sum_sum_a_nat,B3: set_na3699693778330250182_a_nat] :
( collect_nat
@ ^ [X: nat] : ( member8690443509505302927_a_nat @ ( F2 @ X ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_1232_vimage__def,axiom,
( vimage_nat_nat
= ( ^ [F2: nat > nat,B3: set_nat] :
( collect_nat
@ ^ [X: nat] : ( member_nat @ ( F2 @ X ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_1233_vimage__def,axiom,
( vimage_nat_o
= ( ^ [F2: nat > $o,B3: set_o] :
( collect_nat
@ ^ [X: nat] : ( member_o @ ( F2 @ X ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_1234_vimage__def,axiom,
( vimage3040984495076556338_a_nat
= ( ^ [F2: nat > sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( collect_nat
@ ^ [X: nat] : ( member_Sum_sum_a_nat @ ( F2 @ X ) @ B3 ) ) ) ) ).
% vimage_def
thf(fact_1235_insort__insert__triv,axiom,
! [X3: $o,Xs: list_o] :
( ( member_o @ X3 @ ( set_o2 @ Xs ) )
=> ( ( linord6472470733373143810ey_o_o
@ ^ [X: $o] : X
@ X3
@ Xs )
= Xs ) ) ).
% insort_insert_triv
thf(fact_1236_insort__insert__triv,axiom,
! [X3: nat,Xs: list_nat] :
( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
=> ( ( linord1921536354676448932at_nat
@ ^ [X: nat] : X
@ X3
@ Xs )
= Xs ) ) ).
% insort_insert_triv
thf(fact_1237_UN__extend__simps_I10_J,axiom,
! [B: set_li6526943997496501093_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [A4: list_Sum_sum_a_nat] : ( B @ ( F @ A4 ) )
@ A ) )
= ( comple1686717674086456018_a_nat @ ( image_3472601871771700037_a_nat @ B @ ( image_7676043921908783909_a_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1238_UN__extend__simps_I10_J,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,A: set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [A4: list_Sum_sum_a_nat] : ( B @ ( F @ A4 ) )
@ A ) )
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ ( image_5081948215111134021_a_nat @ F @ A ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_1239_image__UN,axiom,
! [F: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( image_7676043921908783909_a_nat @ F @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
= ( comple1778571073001984690_a_nat
@ ( image_7324140262315149061_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( image_7676043921908783909_a_nat @ F @ ( B @ X ) )
@ A ) ) ) ).
% image_UN
thf(fact_1240_image__UN,axiom,
! [F: list_Sum_sum_a_nat > list_Sum_sum_a_nat,B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( image_5081948215111134021_a_nat @ F @ ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
= ( comple1686717674086456018_a_nat
@ ( image_7676043921908783909_a_nat
@ ^ [X: list_Sum_sum_a_nat] : ( image_5081948215111134021_a_nat @ F @ ( B @ X ) )
@ A ) ) ) ).
% image_UN
thf(fact_1241_UNION__empty__conv_I2_J,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) )
= bot_bo1033123847703346641_a_nat )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ( ( B @ X )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_1242_UNION__empty__conv_I1_J,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat,A: set_li6526943997496501093_a_nat] :
( ( bot_bo1033123847703346641_a_nat
= ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ A ) ) )
= ( ! [X: list_Sum_sum_a_nat] :
( ( member408289922725080238_a_nat @ X @ A )
=> ( ( B @ X )
= bot_bo1033123847703346641_a_nat ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_1243_UN__empty,axiom,
! [B: list_Sum_sum_a_nat > set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_7676043921908783909_a_nat @ B @ bot_bo1033123847703346641_a_nat ) )
= bot_bo1033123847703346641_a_nat ) ).
% UN_empty
thf(fact_1244_UN__empty,axiom,
! [B: nat > set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_6073155208182920014_a_nat @ B @ bot_bot_set_nat ) )
= bot_bo1033123847703346641_a_nat ) ).
% UN_empty
thf(fact_1245_UN__empty,axiom,
! [B: $o > set_li6526943997496501093_a_nat] :
( ( comple1686717674086456018_a_nat @ ( image_2951602039182858962_a_nat @ B @ bot_bot_set_o ) )
= bot_bo1033123847703346641_a_nat ) ).
% UN_empty
thf(fact_1246_fv__all__sort,axiom,
( fv_all
= ( linord738340561235409698at_nat
@ ^ [X: nat] : X
@ ( append_nat @ fv_sub @ fv_sub_comp ) ) ) ).
% fv_all_sort
thf(fact_1247_Sup__bool__def,axiom,
( complete_Sup_Sup_o
= ( member_o @ $true ) ) ).
% Sup_bool_def
thf(fact_1248_UNIV__bool,axiom,
( top_top_set_o
= ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% UNIV_bool
thf(fact_1249_Sup__nat__empty,axiom,
( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
= zero_zero_nat ) ).
% Sup_nat_empty
thf(fact_1250_le0,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% le0
thf(fact_1251_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1252_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1253_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y6: nat] :
( ( P @ Y6 )
=> ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1254_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_1255_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_1256_eq__imp__le,axiom,
! [M2: nat,N3: nat] :
( ( M2 = N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% eq_imp_le
thf(fact_1257_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_1258_le__refl,axiom,
! [N3: nat] : ( ord_less_eq_nat @ N3 @ N3 ) ).
% le_refl
thf(fact_1259_less__eq__nat_Osimps_I1_J,axiom,
! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N3 ) ).
% less_eq_nat.simps(1)
thf(fact_1260_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1261_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1262_le__0__eq,axiom,
! [N3: nat] :
( ( ord_less_eq_nat @ N3 @ zero_zero_nat )
= ( N3 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1263_Suc__le__mono,axiom,
! [N3: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N3 @ M2 ) ) ).
% Suc_le_mono
thf(fact_1264_card__Collect__le__nat,axiom,
! [N3: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N3 ) ) )
= ( suc @ N3 ) ) ).
% card_Collect_le_nat
thf(fact_1265_card_Ocomp__fun__commute__on,axiom,
( ( comp_nat_nat_nat @ suc @ suc )
= ( comp_nat_nat_nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_1266_Suc__leD,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( ord_less_eq_nat @ M2 @ N3 ) ) ).
% Suc_leD
thf(fact_1267_le__SucE,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N3 )
=> ( M2
= ( suc @ N3 ) ) ) ) ).
% le_SucE
thf(fact_1268_le__SucI,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) ) ) ).
% le_SucI
thf(fact_1269_Suc__le__D,axiom,
! [N3: nat,M3: nat] :
( ( ord_less_eq_nat @ ( suc @ N3 ) @ M3 )
=> ? [M4: nat] :
( M3
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1270_le__Suc__eq,axiom,
! [M2: nat,N3: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N3 ) )
= ( ( ord_less_eq_nat @ M2 @ N3 )
| ( M2
= ( suc @ N3 ) ) ) ) ).
% le_Suc_eq
thf(fact_1271_Suc__n__not__le__n,axiom,
! [N3: nat] :
~ ( ord_less_eq_nat @ ( suc @ N3 ) @ N3 ) ).
% Suc_n_not_le_n
thf(fact_1272_not__less__eq__eq,axiom,
! [M2: nat,N3: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N3 ) )
= ( ord_less_eq_nat @ ( suc @ N3 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_1273_full__nat__induct,axiom,
! [P: nat > $o,N3: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N3 ) ) ).
% full_nat_induct
thf(fact_1274_nat__induct__at__least,axiom,
! [M2: nat,N3: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ M2 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N3 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1275_transitive__stepwise__le,axiom,
! [M2: nat,N3: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ! [X4: nat] : ( R2 @ X4 @ X4 )
=> ( ! [X4: nat,Y: nat,Z: nat] :
( ( R2 @ X4 @ Y )
=> ( ( R2 @ Y @ Z )
=> ( R2 @ X4 @ Z ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M2 @ N3 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1276_inj__Suc,axiom,
! [N2: set_nat] : ( inj_on_nat_nat @ suc @ N2 ) ).
% inj_Suc
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y3: nat] :
( ( if_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X3: nat,Y3: nat] :
( ( if_nat @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X3: sum_sum_a_nat,Y3: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_T,axiom,
! [X3: sum_sum_a_nat,Y3: sum_sum_a_nat] :
( ( if_Sum_sum_a_nat @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X3: list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_T,axiom,
! [X3: list_Sum_sum_a_nat,Y3: list_Sum_sum_a_nat] :
( ( if_lis4685338526944683083_a_nat @ $true @ X3 @ Y3 )
= X3 ) ).
thf(help_If_3_1_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_T,axiom,
! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( if_set7709265119413304363_a_nat @ $false @ X3 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Set__Oset_It__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_T,axiom,
! [X3: set_li6526943997496501093_a_nat,Y3: set_li6526943997496501093_a_nat] :
( ( if_set7709265119413304363_a_nat @ $true @ X3 @ Y3 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( proj_tuple_a @ fv_sub @ ( zip_na2013496608136855606_a_nat @ fv_all @ xs ) )
= ( map_na823391071729141993_a_nat @ sigma @ fv_sub ) ) ).
%------------------------------------------------------------------------------